The second part of this section will be dedicated to a more algorithmic approach to these problems, and we will focus on the problem of computing the symmetricrank of a given homogeneous polynomial.
The last subsection of this section is dedicated to an overview of open problems.
2.1. On Dimensions of Secant Varieties of Veronese Varieties
This section is entirely devoted to computing the symmetricrank of a generic form, i.e., to the computation of the generic symmetricrank. As anticipated, we approach the problem by computing dimensions of secant varieties of Veronese varieties. Recall that, in algebraic geometry, we say that a property holds for a generic form of degree d if it holds on a Zariski open, hence dense, subset of $\mathbb{P}({S}^{d}{V}^{\ast})$.
2.1.1. Waring Problem for Forms
The problem that we are presenting here takes its name from an old question in number theory. In 1770, E. Waring in [
9] stated (without proofs) that:
“Every natural number can be written as sum of at most 9 positive cubes, Every natural number can be written as sum of at most 19 biquadratics.”
Moreover, he believed that:
“For all integers $d\ge 2$, there exists a number $g(d)$ such that each positive integer $n\in {\mathbb{Z}}^{+}$ can be written as sum of the dth powers of $g(d)$ many positive integers, i.e., $n={a}_{1}^{d}+\cdots +{a}_{g(d)}^{d}$ with ${a}_{i}\ge 0$.”
E. Waring’s belief was shown to be true by D. Hilbert in 1909, who proved that such a
$g(d)$ indeed exists for every
$d\ge 2$. In fact, we know from the famous foursquares Lagrange theorem (1770) that
$g(2)=4$, and more recently, it has been proven that
$g(3)=9$ and
$g(4)=19$. However, the exact number for higher powers is not yet known in general. In [
32], H. Davenport proved that any sufficiently large integer can be written as a sum of 16 fourth powers. As a consequence, for any integer
$d\ge 2$, a new number
$G(d)$ has been defined, as the least number of
dth powers of positive integers to write any sufficiently large positive integer as their sum. Previously, C. F. Gauss proved that any integer congruent to seven modulo eight can be written as a sum of four squares, establishing that
$G(2)=g(2)=4$. Again, the exact value
$G(d)$ for higher powers is not known in general.
This fascinating problem of number theory was then formulated for homogeneous polynomials as follows.
Let $\mathbb{k}$ be an algebraicallyclosed field of characteristic zero. We will work over the projective space ${\mathbb{P}}^{n}=\mathbb{P}V$ where V is an $(n+1)$dimensional vector space over $\mathbb{k}$. We consider the polynomial ring $S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ with the graded structure $S={\u2a01}_{d\ge 0}{S}_{d}$, where ${S}_{d}=\langle {x}_{0}^{d},{x}_{0}^{d1}{x}_{1},\dots ,{x}_{n}^{d}\rangle $ is the vector space of homogeneous polynomials, or forms, of degree d, which, as we said, can be also seen as the space ${S}^{d}V$ of symmetric tensors of order d over V. In geometric language, those vector spaces ${S}_{d}$ are called complete linear systems of hypersurfaces of degree d in ${\mathbb{P}}^{n}$. Sometimes, we will write $\mathbb{P}{S}_{d}$ in order to mean the projectivization of ${S}_{d}$, namely $\mathbb{P}{S}_{d}$ will be a ${\mathbb{P}}^{\left(\genfrac{}{}{0pt}{}{n+d}{d}\right)1}$ whose elements are classes of forms of degree d modulo scalar multiplication, i.e., $[F]\in \mathbb{P}{S}_{d}$ with $F\in {S}_{d}$.
In analogy to the Waring problem for integer numbers, the socalled little Waring problem for forms is the following.
Problem 1 (little Waring problem)
. Find the minimum $s\in \mathbb{Z}$ such that all forms $F\in {S}_{d}$ can be written as the sum of at most s dth powers of linear forms.
The answer to the latter question is analogous to the number $g(d)$ in the Waring problem for integers. At the same time, we can define an analogous number $G(d)$, which considers decomposition in sums of powers of all numbers, but finitely many. In particular, the big Waring problem for forms can be formulated as follows.
Problem 2 (big Waring problem)
. Find the minimum $s\in \mathbb{Z}$ such that the generic form $F\in {S}_{d}$ can be written as a sum of at most s dth powers of linear forms.
In order to know which elements of
${S}_{d}$ can be written as a sum of
s dth powers of linear forms, we study the image of the map:
In terms of maps
${\varphi}_{d,s}$, the little Waring problem (Problem 1) is to find the smallest
s, such that
$\mathrm{Im}({\varphi}_{d,s})={S}_{d}$. Analogously, to solve the big Waring problem (Problem 2), we require
$\overline{\mathrm{Im}({\varphi}_{d,s})}={S}_{d}$, which is equivalent to finding the minimal
s such that
$dim(\mathrm{Im}({\varphi}_{d,s}))=dim{S}_{d}$.
The map
${\varphi}_{d,s}$ can be viewed as a polynomial map between affine spaces:
In order to know the dimension of the image of such a map, we look at its differential at a general point
P of the domain:
Let
$P=({L}_{1},\dots ,{L}_{s})\in {\mathbb{A}}^{s(n+1)}$ and
$v=({M}_{1},\dots ,{M}_{s})\in {T}_{P}({\mathbb{A}}^{s(n+1)})\simeq {\mathbb{A}}^{s(n+1)}$, where
${L}_{i},{M}_{i}\in {S}_{1}$ for
$i=1,\dots ,s$. Let us consider the following parameterizations
$t\u27fc({L}_{1}+{M}_{1}t,\dots ,{L}_{s}+{M}_{s}t)$ of a line
$\mathcal{C}$ passing through
P whose tangent vector at
P is
M. The image of
$\mathcal{C}$ via
${\varphi}_{d,s}$ is
${\varphi}_{d,s}({L}_{1}+{M}_{1}t,\dots ,{L}_{s}+{M}_{s}t)={\sum}_{i=1}^{s}{({L}_{i}+{M}_{i}t)}^{d}$. The tangent vector to
${\varphi}_{d,s}(\mathcal{C})$ in
${\varphi}_{d,s}(P)$ is:
Now, as
$v=({M}_{1},\dots ,{M}_{s})$ varies in
${\mathbb{A}}^{s(n+1)}$, the tangent vectors that we get span
$\langle {L}_{1}^{d1}{S}_{1},\dots ,{L}_{s}^{d1}{S}_{1}\rangle $. Therefore, we just proved the following.
Proposition 1. Let ${L}_{1},\dots ,{L}_{s}$ be linear forms in $S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$, where ${L}_{i}={a}_{i,0}{x}_{0}+\cdots +{a}_{i,n}{x}_{n}$, and consider the map:then: It is very interesting to see how the problem of determining the latter dimension has been solved, because the solution involves many algebraic and geometric tools.
2.1.2. Veronese Varieties
The first geometric objects that are related to our problem are the Veronese varieties. We recall that a
Veronese variety can be viewed as (is projectively equivalent to) the image of the following
dpleembedding of
${\mathbb{P}}^{n}$, where all degree
d monomials in
$n+1$ variables appear in lexicographic order:
With a slight abuse of notation, we can describe the Veronese map as follows:
Let
${X}_{n,d}:={\nu}_{d}({\mathbb{P}}^{n})$ denote a Veronese variety.
Clearly, “
${\nu}_{d}$ as defined in (
4)” and “
${\nu}_{d}$ as defined in (
5)” are not the same map; indeed, from (
5),
However, the two images are projectively equivalent. In order to see that, it is enough to consider the monomial basis of
${S}_{d}$ given by:
Given a set of variables ${x}_{0},\dots ,{x}_{n}$, we let ${\mathbf{x}}^{\alpha}$ denote the monomial ${x}_{0}^{{\alpha}_{0}}\cdots {x}_{n}^{{\alpha}_{n}}$, for any $\alpha \in {\mathbb{N}}^{n+1}$. Moreover, we write $\alpha ={\alpha}_{0}+\dots +{\alpha}_{n}$ for its degree. Furthermore, if $\alpha =d$, we use the standard notation $\left(\genfrac{}{}{0pt}{}{d}{\alpha}\right)$ for the multinomial coefficient $\frac{d!}{{\alpha}_{0}!\cdots {\alpha}_{n}!}$.
Therefore, we can view the Veronese variety either as the variety that parametrizes dth powers of linear forms or as the one parameterizing completely decomposable symmetric tensors.
Example 1 (Twisted cubic)
. Let $V={\mathbb{k}}^{2}$ and $d=3$, then:If we take $\{{z}_{0},\dots ,{z}_{3}\}$ to be homogeneous coordinates in ${\mathbb{P}}^{3}$, then the Veronese curve in ${\mathbb{P}}^{3}$ (classically known as twisted cubic) is given by the solutions of the following system of equations:Observe that those equations can be obtained as the vanishing of all the maximal minors of the following matrix:Notice that the matrix (6) can be obtained also as the defining matrix of the linear map:where $F={\sum}_{i=0}^{3}{\left(\genfrac{}{}{0pt}{}{d}{i}\right)}^{1}{z}_{i}{x}_{0}^{3i}{x}_{1}^{i}$ and ${\partial}_{{x}_{i}}:=\frac{\partial}{\partial {x}_{i}}$. Another equivalent way to obtain (6) is to use the socalled flattenings. We give here an intuitive idea about flattenings, which works only for this specific example. Write the $2\times 2\times 2$ tensor by putting in position $ijk$ the variable ${z}_{i+j+k}$. This is an element of ${V}^{\ast}\otimes {V}^{\ast}\otimes {V}^{\ast}$. There is an obvious isomorphism among the space of $2\times 2\times 2$ tensors ${V}^{\ast}\otimes {V}^{\ast}\otimes {V}^{\ast}$ and the space of $4\times 2$ matrices $({V}^{\ast}\otimes {V}^{\ast})\otimes {V}^{\ast}$. Intuitively, this can be done by slicing the $2\times 2\times 2$ tensor, keeping fixed the third index. This is one of the three obvious possible flattenings of a $2\times 2\times 2$ tensor: the other two flattenings are obtained by considering as fixed the first or the second index. Now, after having written all the possible three flattenings of the tensor, one could remove the redundant repeated columns and compute all maximal minors of the three matrices obtained by this process, and they will give the same ideal.
The phenomenon described in Example 1 is a general fact. Indeed, Veronese varieties are always defined by
$2\times 2$ minors of matrices constructed as (
6), which are usually called catalecticant matrices.
Definition 1. Let $F\in {S}_{d}$ be a homogeneous polynomial of degree d in the polynomial ring $S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$. For any $i=0,\dots ,d$, the$(i,di)$th catalecticant matrix
associated to F is the matrix representing the following linear maps in the standard monomial basis, i.e.,where, for any $\alpha \in {\mathbb{N}}^{n+1}$ with $\alpha =di$, we denote ${\partial}_{{\mathbf{x}}^{\alpha}}^{di}:=\frac{{\partial}^{di}}{\partial {x}_{0}^{{\alpha}_{0}}\cdots \partial {x}_{n}^{{\alpha}_{n}}}$. Let $\{{z}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\alpha \in {\mathbb{N}}^{n+1},\phantom{\rule{3.33333pt}{0ex}}\alpha =d\}$ be the set of coordinates on $\mathbb{P}{S}^{d}V$, where V is $(n+1)$dimensional. The$(i,di)$th catalecticant matrix
of V is the $\left(\genfrac{}{}{0pt}{}{n+i}{n}\right)\times \left(\genfrac{}{}{0pt}{}{n+di}{n}\right)$ matrix whose rows are labeled by ${\mathcal{B}}_{i}=\{\beta \in {\mathbb{N}}^{n+1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =i\}$ and columns are labeled by ${\mathcal{B}}_{di}=\{\beta \in {\mathbb{N}}^{n+1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =di\}$, given by: Remark 1. Clearly, the catalecticant matrix representing $Ca{t}_{di,i}(F)$ is the transpose of $Ca{t}_{i,di}(F)$. Moreover, the most possible square catalecticant matrix is $Ca{t}_{\lfloor d/2\rfloor ,\lceil d/2\rceil}(F)$ (and its transpose).
Let us describe briefly how to compute the ideal of any Veronese variety.
Definition 2. A hypermatrix $A={({a}_{{i}_{1},\dots ,{i}_{d}})}_{0\le {i}_{j}\le n,\phantom{\rule{0.166667em}{0ex}}j=1,\dots ,d}$ is said to be symmetric, or completely symmetric, if ${a}_{{i}_{1},\dots ,{i}_{d}}={a}_{{i}_{\sigma (1)},\dots ,{i}_{\sigma (d)}}$ for all $\sigma \in {\mathfrak{S}}_{d}$, where ${\mathfrak{S}}_{d}$ is the permutation group of $\{1,\dots ,d\}$.
Definition 3. Let $H\subset {V}^{\otimes d}$ be the $\left(\genfrac{}{}{0pt}{}{n+d}{d}\right)$dimensional subspace of completely symmetric tensors of ${V}^{\otimes d}$, i.e., H is isomorphic to the symmetric algebra ${S}^{d}V$ or the space of homogeneous polynomials of degree d in $n+1$ variables. Let S be a ring of coordinates of ${\mathbb{P}}^{\left(\genfrac{}{}{0pt}{}{n+d}{d}\right)1}=\mathbb{P}H$ obtained as the quotient $S=\tilde{S}/I$ where $\tilde{S}=\mathbb{k}{[{x}_{{i}_{1},\dots ,{i}_{d}}]}_{0\le {i}_{j}\le n,\phantom{\rule{0.166667em}{0ex}}j=1,\dots ,d}$ and I is the ideal generated by all:The hypermatrix ${({\overline{x}}_{{i}_{1},\dots ,{i}_{d}})}_{0\le {i}_{j}\le n,\phantom{\rule{0.166667em}{0ex}}j=1,\dots ,d}$, whose entries are the generators of S, is said to be a generic symmetric hypermatrix.
Let
$A={({x}_{{i}_{1},\dots ,{i}_{d}})}_{0\le {i}_{j}\le n,\phantom{\rule{0.166667em}{0ex}}j=1,\dots ,d}$ be a generic symmetric hypermatrix, then it is a known result that the ideal of any Veronese variety is generated in degree two by the
$2\times 2$ minors of a generic symmetric hypermatrix, i.e.,
See [
33] for the set theoretical point of view. In [
34], the author proved that the ideal of the Veronese variety is generated by the twominors of a particular catalecticant matrix. In his PhD thesis [
35], A. Parolin showed that the ideal generated by the twominors of that catalecticant matrix is actually
${I}_{2}(A)$, where
A is a generic symmetric hypermatrix.
2.1.3. Secant Varieties
Now, we recall the basics on secant varieties.
Definition 4. Let $X\subset {\mathbb{P}}^{N}$ be a projective variety of dimension n. We define the sth secant variety
${\sigma}_{s}(X)$ of X as the closure of the union of all linear spaces spanned by s points lying on X, i.e.,For any $\mathcal{F}\subset {\mathbb{P}}^{n}$, $\langle \mathcal{F}\rangle $ denotes the linear span of $\mathcal{F}$, i.e., the smallest projective linear space containing $\mathcal{F}$. Remark 2. The closure in the definition of secant varieties is necessary. Indeed, let ${L}_{1},{L}_{2}\in {S}_{1}$ be two homogeneous linear forms. The polynomial ${L}_{1}^{d1}{L}_{2}$ is clearly in ${\sigma}_{2}({\nu}_{d}(\mathbb{P}(V)))$ since we can write:however, if $d>2$, there are no ${M}_{1},{M}_{2}\in {S}_{1}$ such that ${L}_{1}^{d1}{L}_{2}={M}_{1}^{d}+{M}_{2}^{d}$. This computation represents a very standard concept of basic calculus: tangent lines are the limit of secant lines. Indeed, by (3), the lefthand side of (8) is a point on the tangent line to the Veronese variety at $[{L}_{1}^{d}]$, while the elements inside the limit on the righthand side of (8) are lines secant to the Veronese variety at $[{L}_{1}^{d}]$ and another moving point; see Figure 1. From this definition, it is evident that the generic element of ${\sigma}_{s}(X)$ is an element of some $\langle {P}_{1},\dots ,{P}_{s}\rangle $, with ${P}_{i}\in X$; hence, it is a linear combination of s elements of X. This is why secant varieties are used to model problems concerning additive decompositions, which motivates the following general definition.
Definition 5. Let $X\subset {\mathbb{P}}^{N}$ be a projective variety. For any $P\in {\mathbb{P}}^{N}$, we define the Xrank
of P asand we define the border
Xrank
of P as If
X is a nondegenerate variety, i.e., it is not contained in a proper linear subspace of the ambient space, we obtain a chain of inclusions
Definition 6. The smallest $s\in \mathbb{Z}$ such that ${\sigma}_{s}(X)={\mathbb{P}}^{N}$ is called the generic Xrank. This is the Xrank of the generic point of the ambient space.
The generic Xrank of X is an invariant of the embedded variety X.
As we described in (
5), the image of the
duple Veronese embedding of
${\mathbb{P}}^{n}=\mathbb{P}{S}_{1}$ can be viewed as the subvariety of
$\mathbb{P}{S}_{d}$ made by all forms, which can be written as
dth powers of linear forms. From this point of view, the generic rank
s of the Veronese variety is the minimum integer such that the generic form of degree
d in
$n+1$ variables can be written as a sum of
s powers of linear forms. In other words,
the answer to the Big Waring problem (Problem 2) is the generic rank with respect to the duple Veronese embedding in $\mathbb{P}{S}_{d}$.
This is the reason why we want to study the problem of determining the dimension of sth secant varieties of an ndimensional projective variety $X\subset {\mathbb{P}}^{N}$.
Let
${X}^{s}:={\underbrace{X\times \cdots \times X}}_{s}$,
${X}_{0}\subset X$ be the open subset of regular points of
X and:
Therefore, for all
$({P}_{1},\dots ,{P}_{s})\in {U}_{s}(X)$, since the
${P}_{i}$’s are linearly independent, the linear span
$H=\langle {P}_{1},\dots ,{P}_{s}\rangle $ is a
${\mathbb{P}}^{s1}$. Consider the following incidence variety:
If
$s\le N+1$, the dimension of that incidence variety is:
With this definition, we can consider the projection on the first factor:
the
sth secant variety of
X is just the closure of the image of this map, i.e.,
Now, if
$dim(X)=n$, it is clear that, while
$dim({\mathfrak{I}}^{s}(X))=ns+s1$, the dimension of
${\sigma}_{s}(X)$ can be smaller: it suffices that the generic fiber of
${\pi}_{1}$ has positive dimension to impose
$dim({\sigma}_{s}(X))<n(s1)+n+s1$. Therefore, it is a general fact that, if
$X\subset {\mathbb{P}}^{N}$ and
$dim(X)=n$, then,
Definition 7. A projective variety $X\subset {\mathbb{P}}^{N}$ of dimension n is said to be sdefective
if $dim({\sigma}_{s}(X))<min\{N,sn+s1\}$. If so, we call sth defect
of X the difference:Moreover, if X is sdefective, then ${\sigma}_{s}(X)$ is said to be defective
. If ${\sigma}_{s}(X)$ is not defective, i.e., ${\delta}_{s}(X)=0$, then it is said to be regular
or of expected dimension.
Alexander–Hirschowitz Theorem ([
8]) tells us that the dimension of the
sth secant varieties to Veronese varieties is not always the expected one; moreover, they exhibit the list of all the defective cases.
Theorem 2 (Alexander–Hirschowitz Theorem)
. Let ${X}_{n,d}={\nu}_{d}({\mathbb{P}}^{n})$, for $d\ge 2$, be a Veronese variety. Then:except for the following cases:  (1)
$d=2$, $n\ge 2$, $s\le n$, where $dim({\sigma}_{s}(X))=min\left\{\left(\genfrac{}{}{0pt}{}{n+2}{2}\right)1,2n+1\left(\genfrac{}{}{0pt}{}{s}{2}\right)\right\}$;
 (2)
$d=3$, $n=4$, $s=7$, where ${\delta}_{s}=1$;
 (3)
$d=4$, $n=2$, $s=5$, where ${\delta}_{s}=1$;
 (4)
$d=4$, $n=3$, $s=9$, where ${\delta}_{s}=1$;
 (5)
$d=4$, $n=4$, $s=14$, where ${\delta}_{s}=1$.
Due to the importance of this theorem, we firstly give a historical review, then we will give the main steps of the idea of the proof. For this purpose, we will need to introduce many mathematical tools (apolarity in
Section 2.1.4 and fat points together with the Horace method in
Section 2.2) and some other excursuses on a very interesting and famous conjecture (the socalled SHGHconjecture; see Conjectures 1 and 2) related to the techniques used in the proof of this theorem.
The following historical review can also be found in [
36].
The quadric cases (
$d=2$) are classical. The first nontrivial exceptional case
$d=4$ and
$n=2$ was known already by Clebsch in 1860 [
37]. He thought of the quartic as a quadric of quadrics and found that
${\sigma}_{5}({\nu}_{4}({\mathbb{P}}^{2}))\u228a{\mathbb{P}}^{14}$, whose dimension was not the expected one. Moreover, he found the condition that the elements of
${\sigma}_{5}({\nu}_{4}({\mathbb{P}}^{2}))$ have to satisfy, i.e., he found the equation of the hypersurface
${\sigma}_{5}({\nu}_{4}({\mathbb{P}}^{2}))\u228a{\mathbb{P}}^{14}$: that condition was the vanishing of a
$6\times 6$ determinant of a certain catalecticant matrix.
To our knowledge, the first list of all exceptional cases was described by Richmond in [
38], who showed all the defectivities, case by case, without finding any general method to describe all of them. It is remarkable that he could describe also the most difficult case of general quartics of
${\mathbb{P}}^{4}$. The same problem, but from a more geometric point of view, was at the same time studied and solved by Palatini in 1902–1903; see [
39,
40]. In particular, Palatini studied the general problem, proved the defectivity of the space of cubics in
${\mathbb{P}}^{4}$ and studied the case of
$n=2$. He was also able to list all the defective cases.
The first work where the problem was treated in general is due to Campbell (in 1891; therefore, his work preceded those of Palatini, but in Palatini’s papers, there is no evidence of knowledge of Campbell’s work), who in [
41], found almost all the defective cases (except the last one) with very interesting, but not always correct arguments (the fact that the Campbell argument was wrong for
$n=3$ was claimed also in [
4] in 1915).
His approach is very close to the infinitesimal one of Terracini, who introduced in [
3] a very simple and elegant argument (today known as Terracini’s lemmas, the first of which will be displayed here as Lemma 1), which offered a completely new point of view in the field. Terracini showed again the case of
$n=2$ in [
3]. In [
42], he proved that the exceptional case of cubics in
${\mathbb{P}}^{4}$ can be solved by considering that the rational quartic through seven given points in
${\mathbb{P}}^{4}$ is the singular locus of its secant variety, which is a cubic hypersurface. In [
4], Terracini finally proved the case
$n=3$ (in 2001, Roé, Zappalà and Baggio revised Terracini’s argument, and they where able to present a rigorous proof for the case
$n=3$; see [
43]).
In 1931, Bronowski [
44] tried to tackle the problem checking if a linear system has a vanishing Jacobian by a numerical criterion, but his argument was incomplete.
In 1985, Hirschowitz ([
45]) proved again the cases
$n=2,3$, and he introduced for the first time in the study of this problem the use of zerodimensional schemes, which is the key point towards a complete solution of the problem (this will be the idea that we will follow in these notes). Alexander used this new and powerful idea of Hirschowitz, and in [
46], he proved the theorem for
$d\ge 5$.
Finally, in [
8,
47] (1992–1995), J. Alexander and A. Hirschowitz joined forces to complete the proof of Theorem 2. After this result, simplifications of the proof followed [
48,
49].
After this historical excursus, we can now review the main steps of the proof of the Alexander–Hirschowitz theorem. As already mentioned, one of the main ingredients to prove is Terracini’s lemma (see [
3] or [
50]), which gives an extremely powerful technique to compute the dimension of any secant variety.
Lemma 1 (Terracini’s lemma)
. Let X be an irreducible nondegenerate variety in ${\mathbb{P}}^{N}$, and let ${P}_{1},\dots ,{P}_{s}$ be s generic points on X. Then, the tangent space to ${\sigma}_{s}(X)$ at a generic point $Q\in \langle {P}_{1},\dots ,{P}_{s}\rangle $ is the linear span in ${\mathbb{P}}^{N}$ of the tangent spaces ${T}_{{P}_{i}}(X)$ to X at ${P}_{i}$, $i=1,\dots ,s$, i.e., This “lemma” (we believe it is very reductive to call it a “lemma”) can be proven in many ways (for example, without any assumption on the characteristic of
$\mathbb{k}$, or following Zak’s book [
7]). Here, we present a proof “made by hand”.
Proof. We give this proof in the case of $\mathbb{k}=\mathbb{C}$, even though it works in general for any algebraicallyclosed field of characteristic zero.
We have already used the notation
${X}^{s}$ for
$X\times \cdots \times X$ taken
s times. Suppose that
$dim(X)=n$. Let us consider the following incidence variety,
and the two following projections,
The dimension of
${X}^{s}$ is clearly
$sn$. If
$({P}_{1},\dots ,{P}_{s})\in {X}^{s}$, the fiber
${\pi}_{2}^{1}(({P}_{1},\dots ,{P}_{s}))$ is generically a
${\mathbb{P}}^{s1}$,
$s<N$. Then,
$dim(\mathfrak{I})=sn+s1$.
If the generic fiber of ${\pi}_{1}$ is finite, then ${\sigma}_{s}(X)$ is regular. i.e., it has the expected dimension; otherwise, it is defective with a value of the defect that is exactly the dimension of the generic fiber.
Let
$({P}_{1},\dots ,{P}_{s})\in {X}^{s}$ and suppose that each
${P}_{i}\in X\subset {\mathbb{P}}^{N}$ has coordinates
${P}_{i}=[{a}_{i,0}:\dots :{a}_{i,N}]$, for
$i=1,\dots ,s$. In an affine neighborhood
${U}_{i}$ of
${P}_{i}$, for any
i, the variety
X can be locally parametrized with some rational functions
${f}_{i,j}:{\mathbb{k}}^{n+1}\to \mathbb{k}$, with
$j=0,\dots ,N$, that are zero at the origin. Hence, we write:
Now, we need a parametrization
$\phi $ for
${\sigma}_{s}(X)$. Consider the subspace spanned by
s points of
X, i.e.,
where for simplicity of notation, we omit the dependence of the
${f}_{i,j}$ on the variables
${u}_{i,j}$; thus, an element of this subspace is of the form:
for some
${\lambda}_{1},\dots ,{\lambda}_{s}\in \mathbb{k}$. We can assume
${\lambda}_{1}=1$. Therefore, a parametrization of the
sth secant variety to
X in an affine neighborhood of the point
${P}_{1}+{\lambda}_{2}{P}_{2}+\dots +{\lambda}_{s}{P}_{s}$ is given by:
for some parameters
${t}_{2},\dots ,{t}_{s}$. Therefore, in coordinates, the parametrization of
${\sigma}_{s}(X)$ that we are looking for is the map
$\phi :{\mathbb{k}}^{s(n+1)+s1}\to {\mathbb{k}}^{N+1}$ given by:
where for simplicity, we have written only the
jth element of the image. Therefore, we are able to write the Jacobian of
$\phi $. We are writing it in three blocks: the first one is
$(N+1)\times (n+1)$; the second one is
$(N+1)\times (s1)(n+1)$; and the third one is
$(N+1)\times (s1)$:
with
$i=2,\dots ,s$;
$j=0,\dots ,N$ and
$k=0,\dots ,n$. Now, the first block is a basis of the (affine) tangent space to
X at
${P}_{1}$, and in the second block, we can find the bases for the tangent spaces to
X at
${P}_{2},\dots ,{P}_{s}$; the rows of:
give a basis for the (affine) tangent space of
X at
${P}_{i}$. □
The importance of Terracini’s lemma to compute the dimension of any secant variety is extremely evident. One of the main ideas of Alexander and Hirshowitz in order to tackle the specific case of Veronese variety was to take advantage of the fact that Veronese varieties are embedded in the projective space of homogeneous polynomials. They firstly moved the problem from computing the dimension of a vector space (the tangent space to a secant variety) to the computation of the dimension of its dual (see
Section 2.1.4 for the precise notion of duality used in this context). Secondly, their punchline was to identify such a dual space with a certain degree part of a zerodimensional scheme, whose Hilbert function can be computed by induction (almost always). We will be more clear on the whole technique in the sequel. Now we need to use the language of schemes.
Remark 3. Schemes are locallyringed spaces isomorphic to the spectrum of a commutative ring. Of course, this is not the right place to give a complete introduction to schemes. The reader interested in studying schemes can find the fundamental material in [51,52,53]. In any case, it is worth noting that we will always use only zerodimensional schemes, i.e., “points”; therefore, for our purpose, it is sufficient to think of zerodimensional schemes as points with a certain structure given by the vanishing of the polynomial equations appearing in the defining ideal. For example, a homogeneous ideal I contained in $\mathbb{k}[x,y,z]$, which is defined by the forms vanishing on a degree d plane curve $\mathcal{C}$ and on a tangent line to $\mathcal{C}$ at one of its smooth points P, represents a zerodimensional subscheme of the plane supported at P and of length two, since the degree of intersection among the curve and the tangent line is two at P (schemes of this kind are sometimes called jets). Definition 8. A fat point $Z\subset {\mathbb{P}}^{n}$ is a zerodimensional scheme, whose defining ideal is of the form ${\wp}^{m}$, where ℘ is the ideal of a simple point and m is a positive integer. In this case, we also say that Z is a mfat point, and we usually denote it as $mP$. We call the scheme of fat points a union of fat points ${m}_{1}{P}_{1}+\cdots +{m}_{s}{P}_{s}$, i.e., the zerodimensional scheme defined by the ideal ${\wp}_{1}^{{m}_{1}}\cap \cdots \cap {\wp}_{s}^{{m}_{s}}$, where ${\wp}_{i}$ is the prime ideal defining the point ${P}_{i}$, and the ${m}_{i}$’s are positive integers.
Remark 4. In the same notation as the latter definition, it is easy to show that $F\in {\wp}^{m}$ if and only if $\partial (F)(P)=0$, for any partial differential ∂ of order $\le m1$. In other words, the hypersurfaces “vanishing” at the mfat point $mP$ are the hypersurfaces that are passing through P with multiplicity m, i.e., are singular at P of order m.
Corollary 1. Let $(X,\mathcal{L})$ be an integral, polarized scheme. If $\mathcal{L}$ embeds X as a closed scheme in ${\mathbb{P}}^{N}$, then:where Z is the union of sgeneric twofat points in X. Proof. By Terracini’s lemma, we have that, for generic points ${P}_{1},\dots ,{P}_{s}\in X$, $dim({\sigma}_{s}(X))=dim(\langle {T}_{{P}_{1}}(X),\dots ,{T}_{{P}_{s}}(X)\rangle )$. Since X is embedded in ${\mathbb{P}}^{}({H}^{0}{(X,\mathcal{L})}^{\ast})$ of dimension N, we can view the elements of ${H}^{0}(X,\mathcal{L})$ as hyperplanes in ${\mathbb{P}}^{N}$. The hyperplanes that contain a space ${T}_{{P}_{i}}(X)$ correspond to elements in ${H}^{0}({\mathcal{I}}_{2{P}_{i},X}\otimes \mathcal{L})$, since they intersect X in a subscheme containing the first infinitesimal neighborhood of ${P}_{i}$. Hence, the hyperplanes of ${\mathbb{P}}^{N}$ containing $\langle {T}_{{P}_{1}}(X),\dots ,{T}_{{P}_{s}}(X)\rangle $ are the sections of ${H}^{0}({\mathcal{I}}_{Z,X}\otimes \mathcal{L})$, where Z is the scheme union of the first infinitesimal neighborhoods in X of the points ${P}_{i}$’s. □
Remark 5. A hyperplane H contains the tangent space to a nondegenerate projective variety X at a smooth point P if and only if the intersection $X\cap H$ has a singular point at P. In fact, the tangent space ${T}_{P}(X)$ to X at P has the same dimension of X and ${T}_{P}(X\cap H)=H\cap {T}_{P}(X)$. Moreover, P is singular in $H\cap X$ if and only if $dim({T}_{P}(X\cap H))\ge dim(X\cap H)=dim(X)1$, and this happens if and only if $H\supset {T}_{P}(X)$.
Example 2 (The Veronese surface of P5 is defective)
. Consider the Veronese surface ${X}_{2,2}={\nu}_{2}({\mathbb{P}}^{2})$ in ${\mathbb{P}}^{5}$. We want to show that it is twodefective, with ${\delta}_{2}=1$. In other words, since the expected dimension of ${\sigma}_{2}({X}_{2,2})$ is $2\xb72+1$, i.e., we expect that ${\sigma}_{2}({X}_{2,2})$ fills the ambient space, we want to prove that it is actually a hypersurface. This will imply that actually, it is not possible to write a generic ternary quadric as a sum of two squares, as expected by counting parameters, but at least three squares are necessary instead.
Let P be a general point on the linear span $\langle R,Q\rangle $ of two general points $R,Q\in X$; hence, $P\in {\sigma}_{2}({X}_{2,2})$. By Terracini’s lemma, ${T}_{P}({\sigma}_{2}({X}_{2,2}))=\langle {T}_{R}({X}_{2,2}),{T}_{Q}({X}_{2,2})\rangle $. The expected dimension for ${\sigma}_{2}({X}_{2,2})$ is five, so $dim({T}_{P}({\sigma}_{2}({X}_{2,2})))<5$ if and only if there exists a hyperplane H containing ${T}_{P}({\sigma}_{2}({X}_{2,2}))$. The previous remark tells us that this happens if and only if there exists a hyperplane H such that $H\cap {X}_{2,2}$ is singular at $R,Q$. Now, ${X}_{2,2}$ is the image of ${\mathbb{P}}^{2}$ via the map defined by the complete linear system of quadrics; hence, ${X}_{2,2}\cap H$ is the image of a plane conic. Let ${R}^{\prime},{Q}^{\prime}$ be the preimages via ${\nu}_{2}$ of $R,Q$ respectively. Then, the double line defined by ${R}^{\prime},{Q}^{\prime}$ is a conic, which is singular at ${R}^{\prime},{Q}^{\prime}$. Since the double line $\langle {R}^{\prime},{Q}^{\prime}\rangle $ is the only plane conic that is singular at ${R}^{\prime},{Q}^{\prime}$, we can say that $dim({T}_{P}({\sigma}_{2}({X}_{2,2})))=4<5$; hence, ${\sigma}_{2}({X}_{2,2})$ is defective with defect equal to zero.
Since the twoVeronese surface is defined by the complete linear system of quadrics, Corollary 1 allows us to rephrase the defectivity of ${\sigma}_{2}({X}_{2,2})$ in terms of the number of conditions imposed by twofat points to forms of degree two; i.e., we say that
two twofat points of ${\mathbb{P}}^{2}$ do not impose independent conditions on ternary quadrics.
As we have recalled above, imposing the vanishing at the twofat point means to impose the annihilation of all partial derivatives of first order. In ${\mathbb{P}}^{2}$, these are three linear conditions on the space of quadrics. Since we are considering a scheme of two twofat points, we have six linear conditions to impose on the sixdimensional linear space of ternary quadrics; in this sense, we expect to have no plane cubic passing through two twofat points. However, since the double line is a conic passing doubly thorough the two twofat points, we have that the six linear conditions are not independent. We will come back in the next sections on this relation between the conditions imposed by a scheme of fat points and the defectiveness of secant varieties.
Corollary 1 can be generalized to noncomplete linear systems on X.
Remark 6. Let D be any divisor of an irreducible projective variety X. With $D$, we indicate the complete linear system defined by D. Let $V\subset D$ be a linear system. We use the notation:for the subsystem of divisors of V passing through the fixed points ${P}_{1},\dots ,{P}_{s}$ with multiplicities at least ${m}_{1},\dots ,{m}_{s}$ respectively. When the multiplicities
${m}_{i}$ are equal to two, for
$i=1,\dots ,s$, since a twofat point in
${\mathbb{P}}^{n}$ gives
$n+1$ linear conditions, in general, we expect that, if
$dim(X)=n$, then:
Suppose that
V is associated with a morphism
${\phi}_{V}:{X}_{0}\to {\mathbb{P}}^{r}$ (if
$dim(V)=r$), which is an embedding on a dense open set
${X}_{0}\subset X$. We will consider the variety
$\overline{{\phi}_{V}({X}_{0})}$.
The problem of computing $dim(V(2{P}_{1},\dots ,2{P}_{s}))$ is equivalent to that one of computing the dimension of the sth secant variety to $\overline{{\phi}_{V}({X}_{0})}$.
Proposition 2. Let X be an integral scheme and V be a linear system on X such that the rational function ${\phi}_{V}:X\u290f{\mathbb{P}}^{r}$ associated with V is an embedding on a dense open subset ${X}_{0}$ of X. Then, ${\sigma}_{s}\left(\overline{{\phi}_{V}({X}_{0})}\right)$ is defective if and only if for general points, we have ${P}_{1},\dots ,{P}_{s}\in X$: This statement can be reformulated via apolarity, as we will see in the next section.
2.1.4. Apolarity
This section is an exposition of inverse systems techniques, and it follows [
54].
As already anticipated at the end of the proof of Terracini’s lemma, the whole Alexander and Hirshowitz technique to compute the dimensions of secant varieties of Veronese varieties is based on the computation of the dual space to the tangent space to ${\sigma}_{s}({\nu}_{d}({\mathbb{P}}^{n}))$ at a generic point. Such a duality is the apolarity action that we are going to define.
Definition 9 (Apolarity action)
. Let $S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ and $R=\mathbb{k}[{y}_{0},\dots ,{y}_{n}]$ be polynomial rings and consider the action of ${R}_{1}$ on ${S}_{1}$ and of ${R}_{1}$ on ${S}_{1}$ defined by:i.e., we view the polynomials of ${R}_{1}$ as “partial derivative operators” on ${S}_{1}$. Now, we extend this action to the whole rings
R and
S by linearity and using properties of differentiation. Hence, we get the apolarity action:
where:
for
$\alpha ,\beta \in {\mathbb{N}}^{n+1}$,
$\alpha =({\alpha}_{0},...,{\alpha}_{n})$,
$\beta =({\beta}_{0},\dots ,{\beta}_{n})$, where we use the notation
$\alpha \le \beta $ if and only if
${a}_{i}\le {b}_{i}$ for all
$i=0,\dots ,n$, which is equivalent to the condition that
${x}^{\alpha}$ divides
${x}^{\beta}$ in
S.
Remark 7. Here, are some basic remarks on apolarity action:
Definition 10. Let I be a homogeneous ideal of R. The inverse system ${I}^{1}$ of I is the Rsubmodule of S containing all the elements of S, which are annihilated by I via the apolarity action.
Remark 8. Here are some basic remarks about inverse systems:
Now, we need to recall a few facts on Hilbert functions and Hilbert series.
Let $X\subset {\mathbb{P}}^{n}$ be a closed subscheme whose defining homogeneous ideal is $I:=I(X)\subset S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$. Let $A=S/I$ be the homogeneous coordinate ring of X, and ${A}_{d}$ will be its degree d component.
Definition 11. The Hilbert function
of the scheme X is the numeric function:The Hilbert series
of X is the generating power series: In the following, the importance of inverse systems for a particular choice of the ideal I will be given by the following result.
Proposition 3. The dimension of the part of degree d of the inverse system of an ideal $I\subset R$ is the Hilbert function of $R/I$ in degree d: Remark 9. If $V\times W\to \mathbb{k}$ is a nondegenerate bilinear form and U is a subspace of V, then with ${U}^{\perp}$, we denote the subspace of W given by:With this definition, we observe that: if we consider the bilinear map in (9) and an ideal $I\subset R$, then: ${({I}^{1})}_{d}\cong {I}_{d}^{\perp}.\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$
moreover, if I is a monomial ideal, then:
${I}_{d}^{\perp}=\langle \mathrm{monomials}\mathrm{of}{\mathrm{R}}_{\mathrm{d}}\mathrm{that}\mathrm{are}\mathrm{not}\mathrm{in}{\mathrm{I}}_{\mathrm{d}}\rangle $;
for any two ideals $I,J\subset R$: ${(I\cap J)}^{1}={I}^{1}+{J}^{1}.$
If
$I={\wp}_{1}^{{m}_{1}+1}\cap \cdots \cap {\wp}_{s}^{{m}_{s}+1}\subset R=\mathbb{k}[{y}_{0},\dots ,{y}_{n}]$ is the defining ideal of the scheme of fat points
${m}_{1}{P}_{1}+\cdots +{m}_{s}{P}_{s}\in {\mathbb{P}}^{n}$, where
${P}_{i}=[{p}_{{i}_{0}}:{p}_{{i}_{1}}:\dots :{p}_{in}]\in {\mathbb{P}}^{n}$, and if
${L}_{{P}_{i}}={p}_{{i}_{0}}{x}_{0}+{p}_{{i}_{1}}{x}_{1}+\cdots +{p}_{{i}_{n}}{x}_{n}\in S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$, then:
and also:
This last result gives the following link between the Hilbert function of a set of fat points and ideals generated by sums of powers of linear forms.
Proposition 4. Let $I={\wp}_{1}^{{m}_{1}+1}\cap \cdots \cap {\wp}_{s}^{{m}_{s}+1}\subset R=\mathbb{k}[{y}_{0},\dots ,{y}_{n}]$, then the inverse system ${({I}^{1})}_{d}\subset {S}_{d}=\mathbb{k}{[{x}_{0},\dots ,{x}_{n}]}_{d}$ is the dth graded part of the ideal $({L}_{{P}_{1}}^{d{m}_{1}},\dots ,{L}_{{P}_{s}}^{d{m}_{s}})\subset S$, for $d\ge max\{{m}_{i}+1,\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,s\}$.
Finally, the link between the big Waring problem (Problem 2) and inverse systems is clear. If in (
11), all the
${m}_{i}$’s are equal to one, the dimension of the vector space
$\langle {L}_{{P}_{1}}^{d1}{S}_{1},\dots ,{L}_{{P}_{s}}^{d1}{S}_{1}\rangle $ is at the same time the Hilbert function of the inverse system of a scheme of
s double fat points and the rank of the differential of the application
$\varphi $ defined in (
2).
Proposition 5. Let ${L}_{1},\dots ,{L}_{s}$ be linear forms of $S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ such that:and let ${P}_{1},\dots ,{P}_{s}\in {\mathbb{P}}^{n}$ such that ${P}_{i}=[{a}_{{i}_{0}},\dots ,{a}_{{i}_{n}}].$ Let ${\wp}_{i}\subset R=\mathbb{k}[{y}_{0},\dots ,{y}_{n}]$ be the prime ideal associated with ${P}_{i}$, for $i=1,\dots ,s$, and let:with ${\varphi}_{s,d}({L}_{1},\dots ,{L}_{s})={L}_{1}^{d}+\cdots +{L}_{s}^{d}.$ Then,Moreover, by (10), we have: Now, it is quite easy to see that:
Therefore, putting together Terracini’s lemma and Proposition 5, if we assume the
${L}_{i}$’s (hence, the
${P}_{i}$’s) to be generic, we get:
Example 3. Let $P\in {\mathbb{P}}^{n}$, $\wp \subset S$ be its representative prime ideal and $f\in S$. Then, the order of all partial derivatives of f vanishing in P is almost t if and only if $f\in {\wp}^{t+1}$, i.e., P is a singular point of $V(f)$ of multiplicity greater than or equal to $t+1$. Therefore,It is easy to conclude that one tfat point of ${\mathbb{P}}^{n}$ has the same Hilbert function of $\left(\genfrac{}{}{0pt}{}{t1+n}{n}\right)$ generic distinct points of ${\mathbb{P}}^{n}$. Therefore, $dim({X}_{n,d})=\mathrm{HF}(S/{\wp}^{2},d)1=(n+1)1$. This reflects the fact that Veronese varieties are never onedefective, or, equivalently, since ${X}_{n,d}={\sigma}_{1}({X}_{n,d})$, that Veronese varieties are never defective: they always have the expected dimension $1\xb7n+11$. Example 4. Let ${P}_{1},{P}_{2}$ be two points of ${\mathbb{P}}^{2}$, ${\wp}_{i}\subset S=\mathbb{k}[{x}_{0},{x}_{1},{x}_{2}]$ their associated prime ideals and ${m}_{1}={m}_{2}=2$, so that $I={\wp}_{1}^{2}\cap {\wp}_{2}^{2}$. Is the Hilbert function of I equal to the Hilbert function of six points of ${\mathbb{P}}^{2}$ in general position? No; indeed, the Hilbert series of six general points of ${\mathbb{P}}^{2}$ is $1+3z+6{\sum}_{i\ge 2}{z}^{i}$. This means that I should not contain conics, but this is clearly false because the double line through ${P}_{1}$ and ${P}_{2}$ is contained in I. By (12), this implies that ${\sigma}_{2}({\nu}_{2}({\mathbb{P}}^{2}))\subset {\mathbb{P}}^{5}$ is defective, i.e., it is a hypersurface, while it was expected to fill all the ambient space. Remark 10 (Fröberg–Iarrobino’s conjecture)
. Ideals generated by powers of linear forms are usually called power ideals. Besides the connection with fat points and secant varieties, they are related to several areas of algebra, geometry and combinatorics; see [55]. Of particular interest is their Hilbert function and Hilbert series. In [56], Fröberg gave a lexicographic inequality for the Hilbert series of homogeneous ideals in terms of their number of variables, number of generators and their degrees. That is, if $I=({G}_{1},\dots ,{G}_{s})\subset S=\mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ with $deg({G}_{i})={d}_{i}$, for $i=1,\dots ,s$,where $\u2308\xb7\u2309$ denotes the truncation of the power series at the first nonpositive term. Fröberg conjectured that equality holds generically, i.e., it holds on a nonempty Zariski open subset of $\mathbb{P}{S}_{{d}_{1}}\times \dots \times \mathbb{P}{S}_{{d}_{s}}$. By semicontinuity, fixing all the numeric parameters $(n;{d}_{1},\dots ,{d}_{s})$, it is enough to exhibit one ideal for which the equality holds in order to prove the conjecture for those parameters. In [57] (Main Conjecture 0.6), Iarrobino suggested to look to power ideals and asserted that, except for a list of cases, their Hilbert series coincides with the righthandside of (14). By (11), such a conjecture can be translated as a conjecture on the Hilbert function of schemes of fat points. This is usually referred to as the Fröberg–Iarrobino conjecture; for a detailed exposition on this geometric interpretation of Fröberg and Iarrobino’s conjectures, we refer to [58]. As we will see in the next section, computing the Hilbert series of schemes of fat points is a very difficult and largely open problem. Back to our problem of giving the outline of the proof of Alexander and Hirshowitz Theorem (Theorem 2): Proposition 5 clearly shows that the computation of ${T}_{Q}({\sigma}_{s}({\nu}_{d}({\mathbb{P}}^{n})))$ relies on the knowledge of the Hilbert function of schemes of double fat points. Computing the Hilbert function of fat points is in general a very hard problem. In ${\mathbb{P}}^{2}$, there is an extremely interesting and still open conjecture (the SHGH conjecture). The interplay with such a conjecture with the secant varieties is strong, and we deserve to spend a few words on that conjecture and related aspects.
2.2. Fat Points in the Plane and SHGH Conjecture
The general problem of determining if a set of generic points
${P}_{1},\dots ,{P}_{s}$ in the plane, each with a structure of
${m}_{i}$fat point, has the expected Hilbert function is still an open one. There is only a conjecture due first to B. Segre in 1961 [
59], then rephrased by B. Harbourne in 1986 [
60], A. Gimigliano in 1987 [
61], A. Hirschowitz in 1989 [
62] and others. It describes how the elements of a sublinear system of a linear system
$\mathcal{L}$ formed by all divisors in
$\mathcal{L}$ having multiplicity at least
${m}_{i}$ at the points
${P}_{1},\dots ,{P}_{s}$, look when the linear system does not have the expected dimension, i.e., the sublinear system depends on fewer parameters than expected. For the sake of completeness, we present the different formulations of the same conjecture, but the fact that they are all equivalent is not a trivial fact; see [
63,
64,
65,
66].
Our brief presentation is taken from [
63,
64], which we suggest as excellent and very instructive deepening on this topic.
Let
X be a smooth, irreducible, projective, complex variety of dimension
n. Let
$\mathcal{L}$ be a complete linear system of divisors on
X. Fix
${P}_{1},\dots ,{P}_{s}$ distinct points on
X and
${m}_{1},\dots ,{m}_{s}$ positive integers. We denote by
$\mathcal{L}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ the sublinear system of
$\mathcal{L}$ formed by all divisors in
$\mathcal{L}$ having multiplicity at least
${m}_{i}$ at
${P}_{i}$,
$i=1,\dots ,s$. Since a point of multiplicity
m imposes
$\left(\genfrac{}{}{0pt}{}{m+n1}{n}\right)$ conditions on the divisors of
$\mathcal{L}$, it makes sense to define the expected dimension of
$\mathcal{L}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ as:
If
$\mathcal{L}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ is a linear system whose dimension is not the expected one, it is said to be a special linear system. Classifying special systems is equivalent to determining the Hilbert function of the zerodimensional subscheme of
${\mathbb{P}}^{n}$ given
s general fat points of given multiplicities.
A first reduction of this problem is to consider particular varieties X and linear systems $\mathcal{L}$ on them. From this point of view, the first obvious choice is to take $X={\mathbb{P}}^{n}$ and $\mathcal{L}={\mathcal{L}}_{n,d}:={\mathcal{O}}_{{\mathbb{P}}^{n}}(d)$, the system of all hypersurfaces of degree d in ${\mathbb{P}}^{n}$. In this language, ${\mathcal{L}}_{n,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ are the hypersurfaces of degree d in $n+1$ variables passing through ${P}_{1},\dots ,{P}_{s}$ with multiplicities ${m}_{1},\dots ,{m}_{s}$, respectively.
The SHGH conjecture describes how the elements of ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ look when not having the expected dimension; here are two formulations of this.
Conjecture 1 (Segre, 1961 [
59])
. If ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ is a special linear system, then there is a fixed double component for all curves through the scheme of fat points defined by ${\wp}_{1}^{{m}_{1}}\cap \cdots \cap {\wp}_{s}^{{m}_{s}}$. Conjecture 2 (Gimigliano, 1987 [
61,
67])
. Consider ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i}).$ Then, one has the following possibilities:the system is nonspecial, and its general member is irreducible;
the system is nonspecial; its general member is nonreduced, reducible; its fixed components are all rational curves, except for at most one (this may occur only if the system is zerodimensional); and the general member of its movable part is either irreducible or composed of rational curves in a pencil;
the system is nonspecial of dimension zero and consists of a unique multiple elliptic curve;
the system is special, and it has some multiple rational curve as a fixed component.
This problem is related to the question of what selfintersections occur for reduced irreducible curves on the surface
${X}_{s}$ obtained by blowing up the projective plane at the
s points. Blowing up the points introduces rational curves (infinitely many when
$s>8$) of selfintersection
$1$. Each curve
$\mathcal{C}\subset {X}_{s}$ corresponds to a curve
${D}_{\mathcal{C}}\subset {\mathbb{P}}^{2}$ of some degree
d vanishing to orders
${m}_{i}$ at the
s points:
and the selfintersection
${\mathcal{C}}^{2}$ is
${d}^{2}{m}_{1}^{2}\cdots {m}_{s}^{2}$ if
${D}_{\mathcal{C}}\in {\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$.
Example 5. An example of a curve ${D}_{\mathcal{C}}$ corresponding to a curve $\mathcal{C}$ such that ${\mathcal{C}}^{2}=1$ on ${X}_{s}$ is the line through two of the points; in this case, $d=1$, ${m}_{1}={m}_{2}=1$ and ${m}_{i}=0$ for $i>2$, so we have ${d}^{2}{m}_{1}^{2}\cdots {m}_{s}^{2}=1$.
According to the SHGH conjecture, these $(1)$curves should be the only reduced irreducible curves of negative selfintersection, but proving that there are no others turns out to be itself very hard and is still open.
Definition 12. Let ${P}_{1},\dots ,{P}_{s}$ be s points of ${\mathbb{P}}^{n}$ in general position. The expected dimension
of $\mathcal{L}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ is:where:is the virtual dimension
of $\mathcal{L}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$. Consider the blowup
$\pi :{\tilde{\mathbb{P}}}^{2}\u290f{\mathbb{P}}^{2}$ of the plane at the points
${P}_{1},\dots ,{P}_{s}$. Let
${E}_{1},\dots ,{E}_{s}$ be the exceptional divisors corresponding to the blownup points
${P}_{1},\dots ,{P}_{s}$, and let
H be the pullback of a general line of
${\mathbb{P}}^{2}$ via
$\pi $. The strict transform of the system
$\mathcal{L}:={\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ is the system
$\tilde{\mathcal{L}}=dH{\sum}_{i=1}^{s}{m}_{i}{E}_{i}$. Consider two linear systems
$\mathcal{L}:={\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ and
${\mathcal{L}}^{\prime}:={\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}^{\prime}{P}_{i})$. Their intersection product is defined by using the intersection product of their strict transforms on
${\tilde{\mathbb{P}}}^{2}$, i.e.,
Furthermore, consider the anticanonical class
$K:={K}_{{\tilde{\mathbb{P}}}^{2}}$ of
${\tilde{\mathbb{P}}}^{2}$ corresponding to the linear system
${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{P}_{i}$), which, by abusing notation, we also denote by
$K$. The adjunction formula tells us that the arithmetic genus
${p}_{a}(\tilde{\mathcal{L}})$ of a curve in
$\tilde{\mathcal{L}}$ is:
which one defines to be the geometric genus of
$\mathcal{L}$, denoted
${g}_{\mathcal{L}}$.
This is the classical Clebsch formula. Then, Riemann–Roch says that:
because clearly,
${h}^{2}({\tilde{\mathbb{P}}}^{2},\tilde{\mathcal{L}})=0$. Hence,
Now, we can see how, in this setting, special systems can naturally arise. Let us look for an irreducible curve
$\mathcal{C}$ on
${\tilde{\mathbb{P}}}^{2}$, corresponding to a linear system
$\mathcal{L}$ on
${\mathbb{P}}^{2}$, which is expected to exist, but, for example, its double is not expected to exist. It translates into the following set of inequalities:
which is equivalent to:
and it has the only solution:
which makes all the above inequalities equalities. Accordingly,
$\mathcal{C}$ is a rational curve, i.e., a curve of genus zero, with selfintersection
$1$, i.e., a
$(1)$curve. A famous theorem of Castelnuovo’s (see [
68] (p. 27)) says that these are the only curves that can be contracted to smooth points via a birational morphism of the surface on which they lie to another surface. By abusing terminology, the curve
$\mathsf{\Gamma}\subset {\mathbb{P}}^{2}$ corresponding to
$\mathcal{C}$ is also called a
$(1)$curve.
More generally, one has special linear systems in the following situation. Let
$\mathcal{L}$ be a linear system on
${\mathbb{P}}^{2}$, which is not empty; let
$\mathcal{C}$ be a
$(1)$curve on
${\mathbb{P}}^{2}$ corresponding to a curve
$\mathsf{\Gamma}$ on
${\mathbb{P}}^{2}$, such that
$\tilde{\mathcal{L}}\xb7\mathcal{C}=N<0$. Then,
$\mathcal{C}$ (respectively,
$\mathsf{\Gamma}$) splits off with multiplicity
N as a fixed component from all curves of
$\tilde{\mathcal{L}}$ (respectively,
$\mathcal{L}$), and one has:
where
$\tilde{\mathcal{M}}$ (respectively,
$\mathcal{M}$) is the residual linear system. Then, one computes:
and therefore, if
$N\ge 2$, then
$\mathcal{L}$ is special.
Example 6. One immediately finds examples of special systems of this type by starting from the $(1)$curves of the previous example. For instance, consider $\mathcal{L}:={\mathcal{L}}_{2,2d}({\sum}_{i=1}^{5}d{P}_{i})$, which is not empty, consisting of the conic ${\mathcal{L}}_{2,2}({\sum}_{i=1}^{d}{P}_{i})$ counted d times, though it has virtual dimension $\left(\genfrac{}{}{0pt}{}{d}{2}\right)$.
Even more generally, consider a linear system
$\mathcal{L}$ on
${\mathbb{P}}^{2}$, which is not empty,
${\mathcal{C}}_{1},\dots ,{\mathcal{C}}_{k}$ some
$(1)$curves on
${\tilde{\mathbb{P}}}^{2}$ corresponding to curves
${\mathsf{\Gamma}}_{1},\dots ,{\mathsf{\Gamma}}_{k}$ on
${\mathbb{P}}^{2}$, such that
$\tilde{\mathcal{L}}\xb7{\mathcal{C}}_{i}={N}_{i}<0$,
$i=1,\dots ,k$. Then, for
$i=1,\dots ,k$,
As before, $\mathcal{L}$ is special as soon as there is an $i=1,\dots ,k$ such that ${N}_{i}\ge 2$. Furthermore, ${\mathcal{C}}_{i}\xb7{\mathcal{C}}_{j}={\delta}_{i,j}$, because the union of two meeting $(1)$curves moves, according to the Riemann–Roch theorem, in a linear system of positive dimension on ${\tilde{\mathbb{P}}}^{2}$, and therefore, it cannot be fixed for $\tilde{\mathcal{L}}$. In this situation, the reducible curve $\mathcal{C}:={\sum}_{i=1}^{k}{\mathcal{C}}_{i}$ (respectively, $\mathsf{\Gamma}:={\sum}_{i=1}^{k}{N}_{i}{\mathsf{\Gamma}}_{i}$) is called a $(1)$configuration on ${\tilde{\mathbb{P}}}^{2}$ (respectively, on ${\mathbb{P}}^{2}$).
Example 7. Consider $\mathcal{L}:={\mathcal{L}}_{2,d}({m}_{0}{P}_{0}{\sum}_{i=1}^{s}{m}_{i}{P}_{i})$, with ${m}_{0}+{m}_{i}=d+{N}_{i}$, ${N}_{i}\ge 1$. Let ${\mathsf{\Gamma}}_{i}$ be the line joining ${P}_{0}$, ${P}_{i}$. It splits off ${N}_{i}$ times from $\mathcal{L}$. Hence:If we require the latter system to have nonnegative virtual dimension, e.g., $d\ge {\sum}_{i=1}^{s}{m}_{i}$, if ${m}_{0}=d$ and some ${N}_{i}>1$, we have as many special systems as we want. Definition 13. A linear system $\mathcal{L}$ on ${\mathbb{P}}^{2}$ is$(1)$reducible if $\tilde{\mathcal{L}}={\sum}_{i=1}^{k}{N}_{i}{\mathcal{C}}_{i}+\tilde{\mathcal{M}}$, where $\mathcal{C}={\sum}_{i=1}^{k}{\mathcal{C}}_{i}$ is a $(1)$configuration, $\tilde{\mathcal{M}}\xb7{\mathcal{C}}_{i}=0$, for all $i=1,\dots ,k$ and $vir.dim(\mathcal{M})\ge 0$. The system $\mathcal{L}$ is called$(1)$special if, in addition, there is an $i=1,\dots ,k$ such that ${N}_{i}>1$.
Conjecture 3 (Harbourne, 1986 [
60], Hirschowitz, 1989 [
62])
. A linear system of plane curves ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ with general multiple base points is special if and only if it is $(1)$special, i.e., it contains some multiple rational curve of selfintersection $1$ in the base locus. No special system has been discovered except $(1)$special systems.
Eventually, we signal a concise version of the conjecture (see [
67] (Conjecture 3.3)), which involves only a numerical condition.
Conjecture 4. A linear system of plane curves ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ with general multiple base points and such that ${m}_{1}\ge {m}_{2}\ge \dots \ge {m}_{s}\ge 0$ and $d\ge {m}_{1}+{m}_{2}+{m}_{3}$ is always nonspecial.
The idea of this conjecture comes from Conjecture 3 and by working on the surface $X=\tilde{{\mathbb{P}}^{2}}$, which is the blow up of ${\mathbb{P}}^{2}$ at the points ${P}_{i}$; in this way, the linear system ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ corresponds to the linear system $\tilde{\mathcal{L}}=d{E}_{0}{m}_{1}{E}_{1}\dots {E}_{s}$ on X, where $({E}_{0},{E}_{1},\dots ,{E}_{s})$ is a basis for $\mathrm{Pic}(X)$, and ${E}_{0}$ is the strict transform of a generic line of ${\mathbb{P}}^{2}$, while the divisors ${E}_{1},\dots ,{E}_{s}$ are the exceptional divisors on ${P}_{1},\dots ,{P}_{s}$. If we assume that the only special linear systems ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ are those that contain a fixed multiple ($1$)curve, this would be the same for $\tilde{\mathcal{L}}$ in $\mathrm{Pic}(X)$, but this implies that either we have ${m}_{s}<1$, or we can apply Cremona transforms until the fixed multiple ($1$)curve becomes of type ${m}_{i}^{\prime}{E}_{i}^{\prime}$ in $\mathrm{Pic}(X)$, where the ${E}_{i}^{\prime}$’s are exceptional divisors in a new basis for $\mathrm{Pic}(X)$. Our conditions in Conjecture 4 prevent these possibilities, since the ${m}_{i}$ are positive and the condition $d\ge {m}_{1}+{m}_{2}+{m}_{3}$ implies that, by applying a Cremona transform, the degree of a divisor with respect to the new basis cannot decrease (it goes from d to ${d}^{\prime}=2d{m}_{i}{m}_{j}{m}_{k}$, if the Cremona transform is based on ${P}_{i}$, ${P}_{j}$ and ${P}_{k}$), hence cannot become of degree zero (as ${m}_{i}^{\prime}{E}_{i}^{\prime}$ would be).
One could hope to address a weaker version of this problem. Nagata, in connection with his negative solution of the fourteenth Hilbert problem, made such a conjecture.
Conjecture 5 (Nagata, 1960 [
69])
. The linear system ${\mathcal{L}}_{2,d}({\sum}_{i=1}^{s}{m}_{i}{P}_{i})$ is empty as soon as $s\ge 10$ and $d\le \sqrt{s}$. Conjecture 5 is weaker than Conjecture 3, yet still open for every nonsquare $n\ge 10$. Nagata’s conjecture does not rule out the occurrence of curves of selfintersection less than $1$, but it does rule out the worst of them. In particular, Nagata’s conjecture asserts that ${d}^{2}\ge s{m}^{2}$ must hold when $s\ge 10$, where $m=({m}_{1}+\cdots +{m}_{s})/s$. Thus, perhaps there are curves with ${d}^{2}{m}_{1}^{2}\cdots {m}_{s}^{2}<0$, such as the $(1)$curves mentioned above, but ${d}^{2}{m}_{1}^{2}\cdots {m}_{s}^{2}$ is (conjecturally) only as negative as is allowed by the condition that after averaging the multiplicities ${m}_{i}$ for $n\ge 10$, one must have ${d}^{2}s{m}^{2}\ge 0$.
Now, we want to find a method to study the Hilbert function of a zerodimensional scheme. One of the most classical methods is the socalled Horace method ([
8]), which has also been extended with the Horace differential technique and led J. Alexander and A. Hirschowitz to prove Theorem 2. We explain these methods in
Section 2.2.1 and
Section 2.2.2, respectively, and we resume in
Section 2.2.3 the main steps of the Alexander–Hirschowitz theorem.
2.2.1. La Méthode D’Horace
In this section, we present the socalled Horace method. It takes this name from the ancient Roman legend (and a play by Corneille: Horace, 1639) about the duel between three Roman brothers, the “Orazi”, and three brothers from the enemy town of Albalonga, the “Curiazi”. The winners were to have their town take over the other one. After the first clash among them, two of the Orazi died, while the third remained alive and unscathed, while the Curiazi were all wounded, the first one slightly, the second more severely and the third quite badly. There was no way that the survivor of the Orazi could beat the other three, even if they were injured, but the Roman took to his heels, and the three enemies pursued him; while running, they got separated from each other because they were differently injured and they could run at different speeds. The first to reach the Orazio was the healthiest of the Curiazi, who was easily killed. Then, came the other two who were injured, and it was easy for the Orazio to kill them one by one.
This idea of “killing” one member at a time was applied to the three elements in the exact sequence of an ideal sheaf (together with the ideals of a residual scheme and a “trace”) by A. Hirschowitz in [
45] (that is why now, we keep the french version “Horace” for Orazi) to compute the postulation of multiple points and count how many conditions they impose.
Even if the following definition extends to any scheme of fat points, since it is the case of our interest, we focus on the scheme of twofat points.
Definition 14. We say that a scheme Z of r twofat points, defined by the ideal ${I}_{Z}$, imposes independent conditions on the space of hypersurfaces of degree d in $n+1$ variable ${\mathcal{O}}_{{\mathbb{P}}^{n}}(d)$ if ${\mathrm{codim}}_{\mathbb{k}}\left({I}_{Z}{)}_{d}\right)$ in ${S}^{d}V$ is $min\left\{\left(\genfrac{}{}{0pt}{}{n+d}{d}\right),r(n+1)\right\}$.
This definition, together with the considerations of the previous section and (
12) allows us to reformulate the problem of finding the dimension of secant varieties to Veronese varieties in terms of independent conditions imposed by a zerodimensional scheme of double fat points to forms of a certain degree.
Corollary 2. The sth secant variety ${\sigma}_{s}({X}_{n,d})$ of a Veronese variety has the expected dimension if and only if a scheme of s generic twofat points in ${\mathbb{P}}^{n}$ imposes independent conditions on ${\mathcal{O}}_{{\mathbb{P}}^{n}}(d)$.
Example 8. The linear system $\mathcal{L}:={\mathcal{L}}_{n,2}({\sum}_{i=1}^{s}2{P}_{i})$ is special if $s\le n$. Actually, quadrics in ${\mathbb{P}}^{n}$ singular at s independent points ${P}_{1},\dots ,{P}_{s}$ are cones with the vertex ${\mathbb{P}}^{s1}$ spanned by ${P}_{1},\dots ,{P}_{s}$. Therefore, the system is empty as soon as $s\ge n+1$, whereas, if $s\le n$, one easily computes:Therefore, by (12), this equality corresponds to the fact that ${\sigma}_{s}({\nu}_{2}({\mathbb{P}}^{n}))$ are defective for all $s\le n$; see Theorem 2 (1). We can now present how Alexander and Hirschowitz used the Horace method in [
8] to compute the dimensions of the secant varieties of Veronese varieties.
Definition 15. Let $Z\subset {\mathbb{P}}^{n}$ be a scheme of twofat points whose ideal sheaf is ${\mathcal{I}}_{Z}$. Let $H\subset {\mathbb{P}}^{n}$ be a hyperplane. We define the following:
Example 9. Let $Z=2{P}_{0}\subset {\mathbb{P}}^{n}$ be the twofat point defined by ${\wp}^{2}={({x}_{1},\dots ,{x}_{n})}^{2}$, and let H be the hyperplane $\{{x}_{n}=0\}$. Then, the residue ${\mathrm{Res}}_{H}(Z)\subset {\mathbb{P}}^{n}$ is defined by:hence, it is a simple point of ${\mathbb{P}}^{n}$; the trace ${\mathrm{Tr}}_{H}(Z)\subset H\simeq {\mathbb{P}}^{n1}$ is defined by:where the ${\overline{x}}_{i}$’s are the coordinate of the ${\mathbb{P}}^{n1}\simeq H$, i.e., ${\mathrm{Tr}}_{H}(Z)$ is a twofat point in ${\mathbb{P}}^{n1}$ with support at ${P}_{0}\in H$. The idea now is that it is easier to compute the conditions imposed by the residue and by the trace rather than those imposed by the scheme
Z; in particular, as we are going to explain in the following, this gives us an inductive argument to prove that a scheme
Z imposes independent conditions on hypersurfaces of certain degree. In particular, for any
d, taking the global sections of the restriction exact sequence:
we obtain the socalled Castelnuovo exact sequence:
from which we get the inequality:
Let us assume that the supports of
Z are
r points such that
t of them lie on the hyperplane
H, i.e.,
${\mathrm{Res}}_{H}(Z)$ is the union of
$rt$ many twofat points and
t simple points in
${\mathbb{P}}^{n}$ and
${\mathrm{Tr}}_{H}(Z)$ is a scheme of
t many twofat points in
${\mathbb{P}}^{n1}$ i.e., with the notation of linear systems introduced above,
Assuming that:
${\mathrm{Res}}_{H}(Z)$ imposes independent conditions on
${\mathcal{O}}_{{\mathbb{P}}^{n}}(d1)$, i.e.,
and
${\mathrm{Tr}}_{H}(Z)$ imposes independent conditions on
${\mathcal{O}}_{{\mathbb{P}}^{n1}}(d)$, i.e.,
then, by (
16) and since the expected dimension (Definition 12) is always a lower bound for the actual dimension, we conclude the following.
Theorem 3 (Brambilla–Ottaviani [
36])
. Let Z be a union of r many twofat points in ${\mathbb{P}}^{n}$, and let $H\subset {\mathbb{P}}^{n}$ be a hyperplane such that t of the r points of Z have support on H. Assume that ${\mathrm{Tr}}_{H}({Z}_{r})$ imposes independent conditions on ${\mathcal{O}}_{H}(d)$ and that ${\mathrm{Res}}_{H}{Z}_{r}$ imposes independent conditions on ${\mathcal{O}}_{{\mathbb{P}}^{n}}(d1)$. If one of the pairs of the following inequalities occurs:$tn\le \left(\genfrac{}{}{0pt}{}{d+n1}{n1}\right)$ and $r(n+1)tn\le \left(\genfrac{}{}{0pt}{}{d+n1}{n}\right)$,
$tn\ge \left(\genfrac{}{}{0pt}{}{d+n1}{n1}\right)$ and $r(n+1)tn\ge \left(\genfrac{}{}{0pt}{}{d+n1}{n}\right)$,
then Z imposes independent conditions on the system ${\mathcal{O}}_{{\mathbb{P}}^{n}}(d)$.
The technique was used by Alexander and Hirschowitz to compute the dimension of the linear system of hypersurfaces with double base points, and hence, the dimension of secant varieties of Veronese varieties is mainly the Horace method, via induction.
The regularity of secant varieties can be proven as described above by induction, but nonregularity cannot. Defective cases have to be treated case by case. We have already seen that the case of secant varieties of Veronese surfaces (Example 4) and of quadrics (Example 8) are defective, so we cannot take them as the first step of the induction.
Let us start with
${\sigma}_{s}({X}_{3,3})\subset {\mathbb{P}}^{19}$. The expected dimension is
$4s1$. Therefore, we expect that
${\sigma}_{5}({X}_{3,3})$ fills up the ambient space. Now, let
Z be a scheme of five many twofat points in general position in
${\mathbb{P}}^{3}$ defined by the ideal
${I}_{Z}={\wp}_{1}^{2}\cap \dots \cap {\wp}_{5}^{2}$. Since the points are in general position, we may assume that they are the five fundamental points of
${\mathbb{P}}^{3}$ and perform our computations for this explicit set of points. Then, it is easy to check that:
Hence,
${\sigma}_{5}({X}_{3,3})={\mathbb{P}}^{19}$, as expected. This implies that:
Indeed, as a consequence of the following proposition, if the
sth secant variety is regular, so it is the
$(s1)$th secant variety.
Proposition 6. Assume that X is sdefective and that ${\sigma}_{s+1}(X)\ne {\mathbb{P}}^{N}$. Then, X is also $(s+1)$defective.
Proof. Let ${\delta}_{s}$ be the sdefect of X. By assumptions and by Terracini’s lemma, if ${P}_{1},\cdots ,{P}_{s}\in X$ are general points, then the span ${T}_{{P}_{1},\dots ,{P}_{s}}:=\langle {T}_{{P}_{1}}X,\dots ,{T}_{{P}_{s}}X\rangle $, which is the tangent space at a general point of ${\sigma}_{s}(X)$, has projective dimension $min(N,sn+s1){\delta}_{s}$. Hence, adding one general point ${P}_{s+1}$, the space ${T}_{{P}_{1},\cdots ,{P}_{s},{P}_{s+1}}$, which is the span of ${T}_{{P}_{1},\cdots ,{P}_{s}}$ and ${T}_{{P}_{s+1}}X$, has dimension at most $min\{N,sn+s1\}{\delta}_{s}+n+1$. This last number is smaller than N, while it is clearly smaller than $(s+1)n+s$. Therefore, X is $(s+1)$defective. □
In order to perform the induction on the dimension, we would need to study the case of $d=4$, $s=8$ in ${\mathbb{P}}^{3}$, i.e., ${\sigma}_{8}({X}_{3,4})$. We need to compute ${\mathrm{HF}}_{Z}(4)=\mathrm{HF}(\mathbb{k}[{x}_{0},\dots ,{x}_{3}]/({\wp}_{1}^{2}\cap \cdots \cap {\wp}_{8}^{2}),4)$. In order to use the Horace lemma, we need to know how many points in the support of scheme Z lie on a given hyperplane. The good news is upper semicontinuity, which allows us to specialize points on a hyperplane. In fact, if the specialized scheme has the expected Hilbert function, then also the general scheme has the expected Hilbert function (as before, this argument cannot be used if the specialized scheme does not have the expected Hilbert function: this is the main reason why induction can be used to prove the regularity of secant varieties, but not the defectiveness). In this case, we choose to specialize four points on H, i.e., $Z=2{P}_{1}+\dots +2{P}_{8}$ with ${P}_{1},\dots ,{P}_{4}\in H$. Therefore,
${\mathrm{Res}}_{H}(Z)={P}_{1}+\cdots +{P}_{4}+2{P}_{5}+\cdots +2{P}_{8}\subset {\mathbb{P}}^{3}$;
${\mathrm{Tr}}_{H}({Z}_{8})=2\tilde{{P}_{1}}+\cdots +2\tilde{{P}_{4}}\subset H$, where $2\tilde{{P}_{i}}$’s are twofat points in ${\mathbb{P}}^{2}$
Consider Castelunovo Inequality (
16). Four twofat points in
${\mathbb{P}}^{3}$ impose independent conditions to
${\mathcal{O}}_{{\mathbb{P}}^{3}}(3)$ by (
17), then adding four simple general points imposes independent conditions; therefore,
${\mathrm{Res}}_{H}Z$ imposes the independent condition on
${\mathcal{O}}_{{\mathbb{P}}^{3}}(3)$. Again, assuming that the supports of
${\mathrm{Tr}}_{H}(Z)$ are the fundamental points of
${\mathbb{P}}^{2}$, we can check that it imposes the independent condition on
${\mathcal{O}}_{{\mathbb{P}}^{2}}(4)$. Therefore,
In conclusion, we have proven that
Now, this argument cannot be used to study ${\sigma}_{9}({X}_{3,4})$ because it is one of the defective cases, but we can still use induction on d.
In order to use induction on the degree
d, we need a starting case, that is the case of cubics. We have done
${\mathbb{P}}^{3}$ already; see (
17). Now,
$d=3$,
$n=4$,
$s=7$ corresponds to a defective case. Therefore, we need to start with
$d=3$ and
$n=5$. We expect that
${\sigma}_{10}({X}_{5,3})$ fills up the ambient space. Let us try to apply the Horace method as above. The hyperplane
H is a
${\mathbb{P}}^{4}$; one twofat point in
${\mathbb{P}}^{4}$ has degree five, so we can specialize up to seven points on
H (in
${\mathbb{P}}^{4}$, there are exactly
$35=7\times 5$ cubics), but seven twofat points in
${\mathbb{P}}^{4}$ are defective in degree three; in fact, if
$Z=2{P}_{1}+\dots +2{P}_{7}\subset {\mathbb{P}}^{4}$, then
${dim}_{\mathbb{k}}{\left({I}_{Z}\right)}_{3}=1$. Therefore, if we specialize seven twofat points on a generic hyperplane
H, we are “not using all the room that we have at our disposal”, and (
16) does not give the correct upper bound. In other words, if we want to get a zero in the trace term of the Castelunovo exact sequence, we have to “add one more condition on
H”; but, to do that, we need a more refined version of the Horace method.
2.2.2. La méthode d’Horace Differentielle
The description we are going to give follows the lines of [
70].
Definition 16. An ideal I in the algebra of formal functions $\mathbb{k}[[x,y]]$, where $x=({x}_{1},\dots ,{x}_{n1})$, is called a verticallygraded
(with respect to y) ideal if:where, for $i=0,\dots ,m1$, ${I}_{i}\subset \mathbb{k}[[x]]$ is an ideal. Definition 17. Let Q be a smooth ndimensional integral scheme, and let D be a smooth irreducible divisor on Q. We say that $Z\subset Q$ is a verticallygraded subscheme of Q with base D and support $z\in D$, if Z is a zerodimensional scheme with support at the point z such that there is a regular system of parameters $(x,y)$ at z such that $y=0$ is a local equation for D and the ideal of Z in ${\widehat{\mathcal{O}}}_{Q,z}\cong \mathbb{k}[[x,y]]$ is vertically graded.
Definition 18. Let $Z\in Q$ be a verticallygraded subscheme with base D, and let $p\ge 0$ be a fixed integer. We denote by ${\mathrm{Res}}_{D}^{p}(Z)\in Q$ and ${\mathrm{Tr}}_{D}^{p}(Z)\in D$ the closed subschemes defined, respectively, by the ideals sheaves: In ${\mathrm{Res}}_{D}^{p}(Z)$, we remove from Z the $(p+1)$th “slice” of Z, while in ${\mathrm{Tr}}_{D}^{p}(Z)$, we consider only the $(p+1)$th “slice”. Notice that for $p=0$, this recovers the usual trace ${\mathrm{Tr}}_{D}(Z)$ and residual schemes ${\mathrm{Res}}_{D}(Z)$.
Example 10. Let $Z\subset {\mathbb{P}}^{2}$ be a threefat point defined by ${\wp}^{3}$, with support at a point $P\in H$ lying on a plane $H\subset {\mathbb{P}}^{3}$. We may assume $\wp =({x}_{1},{x}_{2})$ and $H=\{{x}_{2}=0\}$. Then, $3P$ is vertically graded with respect to H: Now, we compute all residues (white dots) and traces (black dots) as follows: Finally, let
${Z}_{1},\dots ,{Z}_{r}\in Q$ be verticallygraded subschemes with base
D and support
${z}_{i}$; let
$Z={Z}_{1}\cup \cdots \cup {Z}_{r}$, and set
$\mathbf{p}=({p}_{1},\dots ,{p}_{r})\in {\mathbb{N}}^{r}$. We write:
We are now ready to formulate the Horace differential lemma.
Proposition 7 (Horace differential lemma, [
71] (Proposition 9.1))
. Let H be a hyperplane in ${\mathbb{P}}^{n}$, and let $W\subset {\mathbb{P}}^{n}$ be a zerodimensional closed subscheme. Let ${Y}_{1},\dots ,{Y}_{r},\phantom{\rule{0.166667em}{0ex}}{Z}_{1},\dots ,{Z}_{r}$ be zerodimensional irreducible subschemes of ${\mathbb{P}}^{n}$ such that ${Y}_{i}\cong {Z}_{i}$, $i=1,\dots ,r$, ${Z}_{i}$ has support on H and is vertically graded with base H, and the supports of $Y={Y}_{1}\cup \cdots \cup {Y}_{r}$ and $Z={Z}_{1}\cup \cdots \cup {Z}_{r}$ are generic in their respective Hilbert schemes. Let $\mathbf{p}=({p}_{1},\dots ,{p}_{r})\in {\mathbb{N}}^{r}$. Assume:${H}^{0}({\mathcal{I}}_{T{r}_{H}W\cup T{r}_{H}^{p}(Z),H}(d))=0$ and
${H}^{0}({\mathcal{I}}_{Re{s}_{H}W\cup Re{s}_{H}^{p}(Z)}(d1))=0$;
For twofat points, the latter result can be rephrased as follows.
Proposition 8 (Horace differential lemma for twofat points)
. Let $H\subset {\mathbb{P}}^{n}$ be a hyperplane, ${P}_{1},\dots ,{P}_{r}\in {\mathbb{P}}^{n}$ be generic points and $W\subset {\mathbb{P}}^{n}$ be a zerodimensional scheme. Let $Z=2{P}_{1}+\cdots +2{P}_{r}\subset {\mathbb{P}}^{n}$, and let ${Z}^{\prime}=2{P}_{1}^{\prime}+\dots +2{P}_{r}^{\prime}$ such that the ${P}_{i}^{\prime}$’s are generic points on H. Let ${D}_{2,H}({P}_{i}^{\prime})=2{P}_{i}^{\prime}\cap H$ be zerodimensional schemes in ${\mathbb{P}}^{n}$. Hence, let:Then, if the following two conditions are satisfied:then, ${dim}_{\mathbb{k}}{({I}_{W+Z})}_{d}=0$. Now, with this proposition, we can conclude the computation of
${\sigma}_{10}({X}_{5,3})$. Before
Section 2.2.2, we were left with the problem of computing the Hilbert function in degree three of a scheme
$Z=2{P}_{1}+\cdots +2{P}_{10}$ of ten twofat points with generic support in
${\mathbb{P}}^{5}$: since a twofat point in
${\mathbb{P}}^{5}$ imposes six conditions, the expected dimension of
${({I}_{Z})}_{3}$ is zero. In this case, the “standard” Horace method fails, since if we specialize seven points on a generic hyperplane, we lose one condition that we miss at the end of the game. We apply the Horace differential method to this situation. Let
${P}_{1}^{\prime},\dots ,{P}_{8}^{\prime}$ be generic points on a hyperplane
$H\simeq {\mathbb{P}}^{4}\subset {\mathbb{P}}^{5}$. Consider:
Now,
dime is satisfied because we have added on the trace exactly the one condition that we were missing. It is not difficult to prove that
degue is also satisfied: quadrics through
$\overline{Z}$ are cones with the vertex the line between
${P}_{9}$ and
${P}_{10}$; hence, the dimension of the corresponding linear system equals the dimension of a linear system of quadrics in
${\mathbb{P}}^{4}$ passing through a scheme of seven simple points and two twofat points with generic support. Again, such quadrics in
${\mathbb{P}}^{4}$ are all cones with the vertex the line passing through the support of the two twofat points: hence, the dimension of the latter linear system equals the dimension of a linear system of quadrics in
${\mathbb{P}}^{3}$ passing through a set of eight simple points with general support. This has dimension zero, since the quadrics of
${\mathbb{P}}^{3}$ are ten. In conclusion, we obtain that the Hilbert function in degree three of a scheme of ten twofat points in
${\mathbb{P}}^{5}$ with generic support is the expected one, i.e., by (
12), we conclude that
${\sigma}_{10}({X}_{5,3})$ fills the ambient space.
2.2.3. Summary of the Proof of the Alexander–Hirshowitz Theorem
We finally summarize the main steps of the proof of Alexander–Hirshowitz theorem (Theorem 2):
The dimension of ${\sigma}_{s}({X}_{n,d})$ is equal to the dimension of its tangent space at a general point Q;
By Terracini’s lemma (Lemma 1), if
Q is general in
$\langle {P}_{1},\dots ,{P}_{s}\rangle $, with
${P}_{1},\dots ,{P}_{s}\in X$ general points, then:
By using the apolarity action (see Definition 9), one can see that:
where
${\wp}_{1}^{2}\cap \cdots \cap {\wp}_{s}^{2}$ is the ideal defining the scheme of twofat points supported by
${\mathbb{P}}^{n}$ corresponding to the
${P}_{i}$’s via the
dth Veronese embedding;
Nonregular cases, i.e., where the Hilbert function of the scheme of twofat points is not as expected, have to be analyzed case by case; regular cases can be proven by induction:
 (a)
The list of nonregular cases corresponds to defective Veronese varieties and is very classical; see
Section 2.1.3, page 11 and [
36] for the list of all papers where all these cases were investigated. We explained a few of them in Examples 2, 4 and 8;
 (b)
The proof of the list of nonregular cases classically known is complete and can be proven by a double induction procedure on the degree d and on the dimension n (see Theorem 3 and Proposition 6):
2.3. Algorithms for the SymmetricRank of a Given Polynomial
The goal of the second part of this section is to compute the symmetricrank of a given symmetric tensor. Here, we have decided to focus on algorithms rather than entering into the details of their proofs, since most of them, especially the more advanced ones, are very technical and even an idea of the proofs would be too dispersive. We believe that a descriptive presentation is more enlightening on the difference among them, the punchline of each one and their weaknesses, rather than a precise proof.
2.3.1. On Sylvester’s Algorithm
In this section, we present the socalled Sylvester’s algorithm (Algorithm 1). It is classically attributed to Sylvester, since he studied the problem of decomposing a homogeneous polynomial of degree
d into two variables as a sum of
dth powers of linear forms and solved it completely, obtaining that the decomposition is unique for general polynomials of odd degree. The first modern and available formulation of this algorithm is due to Comas and Seiguer; see [
27].
Despite the “age” of this algorithm, there are modern scientific areas where it is used to describe very advanced tools; see [
14] for the measurements of entanglement in quantum physics. The following description follows [
28].
If
V is a twodimensional vector space, there is a wellknown isomorphism between
${\bigwedge}^{dr+1}({S}^{d}V)$ and
${S}^{dr+1}({S}^{r}V)$; see [
72]. In terms of projective algebraic varieties, this isomorphism allows us to view the
$(dr+1)$th Veronese embedding of
${\mathbb{P}}^{r}\simeq \mathbb{P}{S}^{r}V$ as the set of
$(r1)$dimensional linear subspaces of
${\mathbb{P}}^{d}$ that are
rsecant to the rational normal curve. The description of this result, via coordinates, was originally given by Iarrobino and Kanev; see [
25]. Here, we follow the description appearing in [
73] (Lemma 2.1). We use the notation
$G(k,W)$ for the Grassmannian of
kdimensional linear spaces in a vector space
W and the notation
$\mathbb{G}(k,n)$ for the Grassmannian of
kdimensional linear spaces in
${\mathbb{P}}^{n}$.
Lemma 2. Consider the map ${\varphi}_{r,dr+1}:{\mathbb{P}}^{}({S}^{r}V)\to G(dr+1,{S}^{d}V)$ that sends the projective class of $F\in {S}^{r}V$ to the $(dr+1)$dimensional subspace of ${S}^{d}V$ made by the multiples of F, i.e.,Then, the following hold: the image of ${\varphi}_{r,dr+1}$, after the Pl ucker embedding of $G(dr+1,{S}^{d}V)$ inside $\mathbb{P}({\bigwedge}^{dr+1}{S}^{d}V)$, is the $(dr+1)$th Veronese embedding of $\mathbb{P}{S}^{r}V$;
identifying $G(dr+1,{S}^{d}V)$ with $\mathbb{G}(r1,\mathbb{P}{S}^{d}{V}^{\ast})$, the above Veronese variety is the set of linear spaces rsecant to a rational normal curve ${\mathcal{C}}_{d}\subset \mathbb{P}{S}^{d}{V}^{\ast}$.
For the proof, we follow the constructive lines of [
28], which we keep here, even though we take the proof as it is, since it is short and we believe it is constructive and useful.
Proof. Let
$\{{x}_{0},{x}_{1}\}$ be the variables on
V. Then, write
$F={u}_{0}{x}_{0}^{r}+{u}_{1}{x}_{0}^{r1}{x}_{1}+\cdots +{u}_{r}{x}_{1}^{r}\in {S}^{r}V$. A basis of the subspace of
${S}^{d}V$ of forms of the type
$FH$ is given by:
The coordinates of these elements with respect to the standard monomial basis
$\{{x}_{0}^{d},{x}_{0}^{d1}{x}_{1},\cdots ,{x}_{1}^{d}\}$ of
${S}^{d}V$ are thus given by the rows of the following
$(r+1)\times (d+1)$ matrix:
The standard Plücker coordinates of the subspace
${\varphi}_{r,dr+1}([F])$ are the maximal minors of this matrix. It is known (see for example [
74]) that these minors form a basis of
$\mathbb{k}{[{u}_{0},\cdots ,{u}_{r}]}_{dr+1}$, so that the image of
${\varphi}_{r,dr+1}([F])$ is indeed a Veronese variety, which proves (1).
To prove (2), we recall some standard facts from [
74]. Consider homogeneous coordinates
${z}_{0},\cdots ,{z}_{d}$ in
${\mathbb{P}}^{}({S}^{d}{V}^{\ast})$, corresponding to the dual basis of the basis
$\{{x}_{0}^{d},{x}_{0}^{d1}{x}_{1},\cdots ,{x}_{1}^{d}\}$. Consider
${\mathcal{C}}_{d}\subset {\mathbb{P}}^{}({S}^{d}{V}^{\ast})$, the standard rational normal curve with respect to these coordinates. Then, the image of
$[F]$ by
${\varphi}_{r,dr+1}$ is precisely the
rsecant space to
${\mathcal{C}}_{d}$ spanned by the divisor on
${\mathcal{C}}_{d}$ induced by the zeros of
F. This completes the proof of (2). □
The rational normal curve
${\mathcal{C}}_{d}\subset {\mathbb{P}}^{d}$ is the
dth Veronese embedding of
$\mathbb{P}V\simeq {\mathbb{P}}^{1}$ inside
$\mathbb{P}{S}^{d}V\simeq {\mathbb{P}}^{d}$. Hence, a symmetric tensor
$F\in {S}^{d}V$ has symmetricrank
r if and only if
r is the minimum integer for which there exists a
${\mathbb{P}}^{r1}\simeq \mathbb{P}W\subset {\mathbb{P}}^{}{S}^{d}V$ such that
$F\in \mathbb{P}W$ and
$\mathbb{P}W$ is
rsecant to the rational normal curve
${\mathcal{C}}_{d}\subset {\mathbb{P}}^{}({S}^{d}V)$ in
r distinct points. Consider the maps:
Clearly, we can identify
$\mathbb{P}{S}^{d}{V}^{\ast}$ with
$\mathbb{P}{S}^{d}V$; hence, the Grassmannian
$\mathbb{G}(r1,\mathbb{P}{S}^{d}{V}^{\ast})$ can be identified with
$\mathbb{G}(r1,\mathbb{P}{S}^{d}V)$. Now, by Lemma 2, a projective subspace
$\mathbb{P}W$ of
$\mathbb{P}{S}^{d}{V}^{\ast}\simeq \mathbb{P}{S}^{d}V\simeq {\mathbb{P}}^{d}$ is
rsecant to
${\mathcal{C}}_{d}\subset \mathbb{P}{S}^{d}V$ in
r distinct points if and only if it belongs to
$\mathrm{Im}({\alpha}_{r,dr+1}\circ {\varphi}_{r,dr+1})$ and the preimage of
${\mathbb{P}}^{}W$ via
${\alpha}_{r,dr+1}\circ {\varphi}_{r,dr+1}$ is a polynomial with
r distinct roots. Therefore, a symmetric tensor
$F\in {S}^{d}V$ has symmetricrank
r if and only if
r is the minimum integer for which the following two conditions hold:
F belongs to some ${\mathbb{P}}^{}W\in \mathrm{Im}({\alpha}_{r,dr+1}\circ {\varphi}_{r,dr+1})\subset \mathbb{G}(r1,\mathbb{P}{S}^{d}V)$,
there exists a polynomial $F\in {S}^{r}V$ that has r distinct roots and such that ${\alpha}_{r,dr+1}({\varphi}_{r,dr+1}([F]))={\mathbb{P}}^{}(W)$.
Now, let
$\mathbb{P}U$ be a
$(dr)$dimensional linear subspace of
$\mathbb{P}{S}^{d}V$. The proof of Lemma 2 shows that
$\mathbb{P}U$ belongs to the image of
${\varphi}_{r,dr+1}$ if and only if there exist
${u}_{0},\dots ,{u}_{r}\in \mathbb{k}$ such that
$U=\langle {F}_{1},\dots ,{F}_{dr+1}\rangle $, where, with respect to the standard monomial basis
$\mathcal{B}=\{{x}_{0}^{d},{x}_{0}^{d1}{x}_{1},\dots ,{x}_{1}^{d}\}$ of
${S}^{d}V$, we have:
Let
${\mathcal{B}}^{\ast}=\{{z}_{0},\dots ,{z}_{d}\}$ be the dual basis of
$\mathcal{B}$ with respect to the apolar pairing. Therefore, there exists a
$W\subset {S}^{d}V$ such that
$\mathbb{P}W={\alpha}_{r,dr+1}(\mathbb{P}U)$ if and only if
$W={H}_{1}\cap \cdots \cap {H}_{dr+1}$, and the
${H}_{i}$’s are as follows:
This is sufficient to conclude that
$F\in {\mathbb{P}}^{}{S}^{d}V$ belongs to an
$(r1)$dimensional projective subspace of
${\mathbb{P}}^{}{S}^{d}V$ that is in the image of
${\alpha}_{r,dr+1}\circ {\varphi}_{r,dr+1}$ defined in (
19) if and only if there exist
${H}_{1},\dots ,{H}_{dr+1}$ hyperplanes in
${S}^{d}V$ as above, such that
$F\in {H}_{1}\cap \dots \cap {H}_{dr+1}$. Now, given
$F\in {S}^{d}V$ with coordinates
$({a}_{0},\dots ,{a}_{d})$ with respect to the dual basis
${\mathcal{B}}^{\ast}$, we have that
$F\in {H}_{1}\cap \dots \cap {H}_{dr+1}$ if and only if the following linear system admits a nontrivial solution in the
${u}_{i}$’s
If
$dr+1<r+1$, this system admits an infinite number of solutions. If
$r\le d/2$, it admits a nontrivial solution if and only if all the maximal
$(r+1)$minors of the following catalecticant matrix (see Definition 1) vanish:
Remark 11. The dimension of ${\sigma}_{r}({\mathcal{C}}_{d})$ is never defective, i.e., it is the minimum between $2r1$ and d. Actually, ${\sigma}_{r}({\mathcal{C}}_{d})\u228a\mathbb{P}{S}^{d}V$ if and only if $1\le r<\u2308\frac{d+1}{2}\u2309$. Moreover, an element $[F]\in \mathbb{P}{S}^{d}V$ belongs to ${\sigma}_{r}({\mathcal{C}}_{d})$ for $1\le r<\u2308\frac{d+1}{2}\u2309$, i.e., ${\underline{\mathrm{R}}}_{\mathrm{sym}}(F)=r$, if and only if $Ca{t}_{r,dr}(F)$ does not have maximal rank. These facts are very classical; see, e.g., [1]. Therefore, if we consider the monomial basis $\left\{{\left(\genfrac{}{}{0pt}{}{d}{i}\right)}^{1}{x}_{0}^{i}{x}_{1}^{di}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=0,\dots ,d\right\}$ of ${S}^{d}V$ and write $F={\sum}_{i=0}^{d}{\left(\genfrac{}{}{0pt}{}{d}{i}\right)}^{1}{a}_{i}{x}_{0}^{i}{x}_{1}^{di}$, then we write the $(i,di)$th catalecticant matrix of F as $Ca{t}_{i,di}(F)={\left({a}_{h+k}\right)}_{\begin{array}{c}h=0,\dots ,i\\ k=0,\dots ,di\end{array}}.$
Algorithm 1: Sylvester’s algorithm. 
 The algorithm works as follows. 
Require: A binary form $F={\sum}_{i=0}^{d}{a}_{i}{x}_{0}^{i}{x}_{1}^{di}\in {S}^{d}V$. 
Ensure: A minimal Waring decomposition $F={\sum}_{i=1}^{r}{\lambda}_{i}{L}_{i}{({x}_{0},{x}_{1})}^{d}$. 
 1:
initialize $r\leftarrow 0$;  2:
if$\mathrm{rk}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{r,dr}(F)$ is maximal then  3:
increment $r\leftarrow r+1$;  4:
end if  5:
compute a basis of $Ca{t}_{r,dr}(F)$;  6:
take a random element $G\in {S}^{r}{V}^{\ast}$ in the kernel of $Ca{t}_{r,dr}(F)$;  7:
compute the roots of G: denote them $({\alpha}_{i},{\beta}_{i})$, for $i=1,\dots ,r$;  8:
if the roots are not distinct then  9:
go to Step 2;  10:
else  11:
compute the vector $\lambda =({\lambda}_{1},\dots ,{\lambda}_{r})\in {\mathbb{k}}^{r}$ such that:
 12:
end if  13:
construct the set of linear forms $\{{L}_{i}={\alpha}_{i}{x}_{0}+{\beta}_{i}{x}_{1}\}\subset {S}^{1}V$;  14:
return the expression ${\sum}_{i=1}^{r}{\lambda}_{i}{L}_{i}^{d}$.

Example 11. Compute the symmetricrank and a minimal Waring decomposition of the polynomialWe follow Sylvester’s algorithm. The first catalecticant matrix with rank smaller than the maximal is:in fact, $\mathrm{rk}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{2,2}(F)=2$. Now, let $\{{y}_{0},{y}_{1}\}$ the dual basis of ${V}^{\ast}$. We get that $kerCa{t}_{2,2}(F)=\langle 2{y}_{0}^{2}{y}_{0}{y}_{1}{y}_{1}^{2}\rangle $. We factorize:Hence, we obtain the roots $\{(1,1),(1,2)\}$. Then, it is direct to check that:hence, a minimal Waring decomposition is given by: The following result was proven by Comas and Seiguer in [
27]; see also [
28]. It describes the structure of the stratification by symmetricrank of symmetric tensors in
${S}^{d}V$ with
$dimV=2$. This result allows us to improve the classical Sylvester algorithm (see Algorithm 2).
Theorem 4. Let ${\mathcal{C}}_{d}=\{[F]\in \mathbb{P}{S}^{d}V\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{R}}_{\mathrm{sym}}(F)=1\}=\{[{L}^{d}]\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}L\in {S}^{1}V\}\subset {\mathbb{P}}^{d}$ be the rational normal curve of degree d parametrizing decomposable symmetric tensors. Then,where we write: Algorithm 2: Sylvester’s symmetric (border) rank algorithm [28]. 
 The latter theorem allows us to get a simplified version of the Sylvester algorithm, which computes the symmetricrank and the symmetricborder rank of a symmetric tensor, without computing any decomposition. Notice that Sylvester’s Algorithm 1 for the rank is recursive: it runs for any r from one to the symmetricrank of the given polynomial, while Theorem 4 shows that once the symmetric border rank is computed, then the symmetricrank is either equal to the symmetric border rank or it is $dr+2$, and this allows us to skip all the recursive process. 
Require: A form $F\in {S}^{d}V$, with $dimV=2$. 
Ensure: the symmetricrank ${\mathrm{R}}_{\mathrm{sym}}(F)$ and the symmetricborder rank ${\underline{\mathrm{R}}}_{\mathrm{sym}}(F)$. 
 1:
$r:=\mathrm{rk}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{\lfloor \frac{d}{2}\rfloor ,\lceil \frac{d}{2}\rceil}(F)$  2:
${\underline{\mathrm{R}}}_{\mathrm{sym}}(F)=r$;  3:
choose an element $G\in kerCa{t}_{r,dr}(F)$;  4:
ifG has distinct roots then  5:
${\mathrm{R}}_{\mathrm{sym}}(F)=r$  6:
else  7:
${\mathrm{R}}_{\mathrm{sym}}(F)=dr+2$;  8:
end if  9:
return${\mathrm{R}}_{\mathrm{sym}}(F)$

Example 12. Let $F=5{x}_{0}^{5}{x}_{1}$, and let $\{{y}_{0},{y}_{1}\}$ be the dual basis to $\{{x}_{0},{x}_{1}\}$. The smallest catalecticant without full rank is:which has rank two. Therefore $[F]\in {\sigma}_{2}({C}_{6})$. Now, $ker\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{2,3}(F)=\langle {y}_{1}^{2}\rangle $, which has a double root. Hence, $[F]\in {\sigma}_{2,6}({C}_{6})$. Remark 12. When a form $F\in \mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ can be written using less variables, i.e., $F\in \mathbb{k}[{L}_{0},\dots ,{L}_{m}]$, for ${L}_{j}\in \mathbb{k}{[{x}_{0},\dots ,{x}_{n}]}_{1}$, with $m<n$, we say that F has m essential variables (in the literature, it is also said that F is mconcise). That is, $F\in {S}^{d}W$, where $W=\langle {L}_{0},\dots ,{L}_{m}\rangle \subset V$. Then, the rank of $[F]$ with respect to ${X}_{n,d}$ is the same one as the one with respect to ${\nu}_{d}(\mathbb{P}W)\subset {X}_{n,d}$; e.g., see [75,76]. As recently clearly described in [77] (Proposition 10) and more classically in [25], the number of essential variables of F coincides with the rank of the first catalecticant matrix $Ca{t}_{1,d1}(F)$. In particular, when $[F]\in {\sigma}_{r}({X}_{n,d})\subset \mathbb{P}({S}^{d}V)$ with $dim(V)=n+1$, then, if $r<n+1$, there is a subspace $W\subset V$ with $dim(W)=r$ such that $[F]\in \mathbb{P}{S}^{d}W$, i.e., F can be written with respect to r variables. Let now
V be (
$n+1$)dimensional, and consider the following construction:
where the map
$\varphi $ in (
20) sends a zerodimensional scheme
Z with
$deg(Z)=r$ to the vector space
${({I}_{Z})}_{d}$ (it is defined in the open set formed by the schemes
Z, which impose independent conditions to forms of degree
d) and where the last arrow is the identification, which sends a linear space to its perpendicular.
As in the case $n=1$, the final image from the latter construction gives the $(r1)$spaces, which are rsecant to the Veronese variety in ${\mathbb{P}}^{N}\cong \mathbb{P}{(\mathbb{k}{[{x}_{0},\dots ,{x}_{n}]}_{d})}^{\ast}$. Moreover, each such space cuts the image of Z via the Veronese embedding.
Notation 1. From now on, we will always use the notation ${\mathsf{\Pi}}_{Z}$ to indicate the projective linear subspace of dimension $r1$ in $\mathbb{P}{S}^{d}V$, with $dim(V)=n+1$, generated by the image of a zerodimensional scheme $Z\subset {\mathbb{P}}^{n}$ of degree r via the Veronese embedding, i.e., ${\mathsf{\Pi}}_{Z}=\langle {\nu}_{d}(Z)\rangle \subset \mathbb{P}{S}^{d}V$.
Theorem 5. Any $[F]\in {\sigma}_{2}({X}_{n,d})\subset \mathbb{P}{S}^{d}V$, with $dim(V)=n+1$ can only have symmetricrank equal to 1, 2 or d. More precisely:more precisely, ${\sigma}_{2,d}({X}_{n,d})=\tau ({X}_{n,d})\setminus {X}_{n,d}$, where $\tau ({X}_{n,d})$ denotes the tangential variety of ${X}_{n,d}$, i.e., the Zariski closure of the union of the tangent spaces to ${X}_{n,d}$. Proof. This is actually a quite direct consequence of Remark 12 and of Theorem 4, but let us describe the geometry in some detail, following the proof of [
28]. Since
$r=2$, every
$Z\in {\mathrm{Hilb}}_{2}({\mathbb{P}}^{n})$ is the complete intersection of a line and a quadric, so the structure of
${I}_{Z}$ is well known, i.e.,
${I}_{Z}=({L}_{0},\dots ,{L}_{n2},Q)$, where
${L}_{i}$’s are linearly independent linear forms and
Q is a quadric in
${S}^{2}V\setminus {({L}_{0},\dots ,{L}_{n2})}_{2}$.
If
$F\in {\sigma}_{2}({X}_{n,d})$, then we have two possibilities: either
${\mathrm{R}}_{\mathrm{sym}}(F)=2$ or
${\mathrm{R}}_{\mathrm{sym}}(T)>2$, i.e.,
F lies on a tangent line
${\mathsf{\Pi}}_{Z}$ to the Veronese, which is given by the image of a scheme
$Z\subset \mathbb{P}V$ of degree 2, via the maps (
20). We can view
F in the projective linear space
$H\cong {\mathbb{P}}^{d}$ in
$\mathbb{P}({S}^{d}V)$ generated by the rational normal curve
${\mathcal{C}}_{d}\subset {X}_{n,d}$, which is the image of the line
ℓ defined by the ideal
$({L}_{0},\dots ,{L}_{n2})$ in
$\mathbb{P}V$, i.e.,
$\ell \subset {\mathbb{P}}^{n}$ is the unique line containing
Z. Hence, we can apply Theorem 4 in order to get that
${\mathrm{R}}_{\mathrm{sym}}(F)\le d$. Moreover, by Remark 12, we have
${\mathrm{R}}_{\mathrm{sym}}(F)=d$. □
Remark 13. Let us check that ${\sigma}_{2}({X}_{n,d})$ is given by the annihilation of the $(3\times 3)$minors of the first two catalecticant matrices, $Ca{t}_{1,d1}(V)$ and $Ca{t}_{2,d2}(V)$ (see Definition 1); actually, such minors are the generators of ${I}_{{\sigma}_{2}({\nu}_{d}({\mathbb{P}}^{n}))}$; see [78]. Following the construction above (20), we can notice that the coefficients of the linear spaces defined by the forms ${L}_{i}\in {V}^{\ast}$ in the ideal ${I}_{Z}$ are the solutions of a linear system whose matrix is given by the catalecticant matrix $Ca{t}_{1,d1}(V)$; since the space of solutions has dimension $n1$, we get $\mathrm{rk}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{1,d1}(V)=2$. When we consider the quadric Q in ${I}_{Z}$, instead, the analogous construction gives that its coefficients are the solutions of a linear system defined by the catalecticant matrix $Ca{t}_{2,d2}(V)$, and the space of solutions give Q and all the quadrics in ${({L}_{0},\dots ,{L}_{n2})}_{2}$, which are $\left(\genfrac{}{}{0pt}{}{n}{2}\right)+2n1$, hence: Therefore, we can write down an algorithm (Algorithm 3) to test if an element $[F]\in {\sigma}_{2}({X}_{n,d})$ has symmetric rank two or d.
Algorithm 3: An algorithm to compute the symmetricrank of an element lying on ${\sigma}_{2}({X}_{n,d})$. 
 Require: A from $F\in {S}^{d}V$, where $dimV=n+1$. 
Ensure: If $[F]\in {\sigma}_{2}({X}_{n,d})$, returns the ${\mathrm{R}}_{\mathrm{sym}}(F)$. 
 1:
compute the number of essential variables $m=\mathrm{rk}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{1,d1}(F)$;  2:
if$m=1$then  3:
print $F\in {X}_{n,d}$;  4:
else if$m>2$then  5:
print $F\notin {\sigma}_{2}({X}_{n,d})$;  6:
else  7:
let $W={(kerCa{t}_{1,d1}(F))}^{\perp}$ and view $F\in {S}^{d}W$;  8:
end if  9:
return apply Algorithm 2 to F.

Example 13. Compute the symmetricrank ofFirst of all, note that $({y}_{0}{y}_{1})\circ F=0$; in particular, $kerCa{t}_{1,3}(F)=\langle {y}_{0}{y}_{1}\rangle $. Hence, F has two essential variables. This can also be seen by noticing that $F={({x}_{0}+{x}_{1})}^{3}{x}_{2}$. Therefore, if we write ${z}_{0}={x}_{0}+{x}_{1}$ and ${z}_{1}={x}_{2}$, then $F={z}_{0}^{3}{z}_{1}\in \mathbb{k}[{z}_{0},{z}_{1}]$. Hence, we can apply Algorithms 1 and 2 to compute the symmetricrank, symmetricborder rank and a minimal decompositions of F. In particular, we write:which has rank two, as expected. Again, as in Example 12, the kernel of $Ca{t}_{2,2}(F)$ defines a polynomial with a double root. Hence, ${\underline{\mathrm{R}}}_{\mathrm{sym}}(F)=2$ and ${\mathrm{R}}_{\mathrm{sym}}(F)=4$. If we are interested in finding a minimal decomposition of F, we have to consider the first catalecticant whose kernel defines a polynomial with simple roots. In this case, we should get to:whose kernel is $\langle (1,0,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)\rangle .$ If we let $\{{w}_{0},{w}_{1}\}$ be the variables on ${W}^{\ast}$, we take a polynomial in this kernel, as for example $G={w}_{0}^{4}+{w}_{0}^{2}{w}_{1}^{2}+{w}_{0}{w}_{1}^{3}+{w}_{1}^{4}$. Now, if we compute the roots of G, we find four complex distinct roots, i.e.,where:Hence, if we write ${L}_{i}={\alpha}_{i}{z}_{0}+{\beta}_{i}{z}_{1}$, for $i=1,\dots ,4$, we can find suitable ${\lambda}_{i}$’s to write a minimal decomposition $F={\sum}_{i=1}^{4}{\lambda}_{i}{L}_{i}^{4}$. Observe that any hyperplane through $[F]$ that does not contain the tangent line to ${\mathcal{C}}_{4}$ at $[{z}_{0}^{4}]$ intersects ${C}_{4}$ at four distinct points, so we could have chosen also another point in $\langle (1,0,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)\rangle $, and we would have found another decomposition of F. Everything that we have done in this section does not use anything more than Sylvester’s algorithm for the twovariable case. In the next sections, we see what can be done if we have to deal with more variables and we cannot reduce to the binary case like in Example 13.
Sylvester’s algorithm allows us to compute the symmetricrank of any polynomial in two essential variables. It is mainly based on the fact that equations for secant varieties of rational normal curves are well known and that there are only two possibilities for the symmetricrank of a given binary polynomial with fixed border rank (Theorem 4). Moreover, those two cases are easily recognizable by looking at the multiplicity of the roots of a generic polynomial in the kernel of the catalecticant.
The first ideas that were exploited to generalize Sylvester’s result to homogeneous polynomials in more than two variables were:
a good understanding of the inverse system (and therefore, of the scheme defined by the kernels of catalecticant matrices and possible extension of catalecticant matrices, namely Hankel matrices); we will go into the details of this idea in
Section 2.3.3;
a possible classification of the ranks of polynomials with fixed border rank; we will show the few results in this direction in
Section 2.3.2.
2.3.2. Beyond Sylvester’s Algorithm Using ZeroDimensional Schemes
We keep following [
28]. Let us start by considering the case of a homogeneous polynomial with three essential variables.
If $[F]\in {\sigma}_{3}({\nu}_{d}({\mathbb{P}}^{n}))\setminus {\sigma}_{2}({\nu}_{d}({\mathbb{P}}^{n}))$, then we will need more than two variables, but actually, three are always sufficient. In fact, if $[F]\in {\sigma}_{3}({\nu}_{d}({\mathbb{P}}^{n}))$, then there always exists a zerodimensional scheme ${\nu}_{d}(Z)$ of length three contained in ${\nu}_{d}({\mathbb{P}}^{n})$, whose span contains $[F]$; the scheme $Z\subset {\mathbb{P}}^{n}$ itself spans a ${\mathbb{P}}^{2}$, which can be seen as $\mathbb{P}({({L}_{1},{L}_{2},{L}_{3})}_{1})$ with ${L}_{i}$’s linear forms. Therefore, F can be written in three variables. The following theorem computes the symmetricrank of any polynomial in $[F]\in {\sigma}_{3}({\nu}_{d}({\mathbb{P}}^{n}))\setminus {\sigma}_{2}({\nu}_{d}({\mathbb{P}}^{n}))$, and the idea is to classify the symmetricrank by looking at the structure of the zerodimensional scheme of length three, whose linear span contains $[F]$.
Theorem 6 ([
28] (Theorem 37))
. Let $d\ge 3$, ${X}_{n,d}\subset \mathbb{P}({\mathbb{k}}^{n+1})$. Then,while, for $d\ge 4$, We do not give here all the details of the proof since they can be found in [
28]; they are quite technical, but the main idea is the one described above. We like to stress that the relation between the zerodimensional scheme of length three spanning
F and the one computing the symmetricrank is in many cases dependent on the following Lemma 3. Probably, it is classically known, but we were not able to find a precise reference.
Lemma 3 ([
28] (Lemma 11))
. Let $Z\subset {\mathbb{P}}^{n}$, $n\ge 2$, be a zerodimensional scheme, with $deg(Z)\le 2d+1$. A necessary and sufficient condition for Z to impose independent conditions on hypersurfaces of degree d is that no line $\ell \subset {\mathbb{P}}^{n}$ is such that $deg(Z\cap \ell )\ge d+2$. Remark 14. Notice that if $deg(\ell \cap Z)$ is exactly $d+1+k$, then the dimension of the space of curves of degree d through them is increased exactly by k with respect to the generic case.
It is easy to see that Lemma 3 can be improved as follows; see [
79].
Lemma 4 ([
79])
. Let $Z\subset {\mathbb{P}}^{n}$, $n\ge 2$, be a zerodimensional scheme, with $deg(Z)\le 2d+1$. If ${h}^{1}({\mathbb{P}}^{n},{\mathcal{I}}_{Z}(d))>0$, there exists a unique line $\ell \subset {\mathbb{P}}^{n}$ such that $deg(Z\cap \ell )=d+1+{h}^{1}({\mathbb{P}}^{n},{\mathcal{I}}_{Z}(d))>0$. We can go back to our problem of finding the symmetricrank of a given tensor. The classification of symmetricranks of the elements in
${\sigma}_{4}({X}_{n,d})$ can be treated in an analogous way as we did for
${\sigma}_{3}({X}_{n,d})$, but unfortunately, it requires a very complicated analysis on the schemes of length four. This is done in [
80], but because of the long procedure, we prefer to not present it here.
It is remarkable that
${\sigma}_{4}({X}_{n,d})$ is the last
sth secant variety of Veronesean, where we can use this technique for the classification of the symmetricrank with respect to zerodimensional schemes of length
s, whose span contains the given polynomial we are dealing with; for
$s\ge 5$, there is a more intrinsic problem. In fact, there is a famous counterexample due to Buczyńska and Buczyśki (see [
81]) that shows that, in
${\sigma}_{5}({X}_{4,3})$, there is at least a polynomial for which there does not exist any zerodimensional scheme of length five on
${X}_{4,3}$, whose span contains it. The example is the following.
Example 14 (Buczyńska, Buczyński [
81,
82])
. One can easily check that the following polynomial:can be obtained as ${lim}_{\u03f5\to 0}\frac{1}{3\u03f5}{F}_{\u03f5}=F$ where:has symmetricrank five for $\u03f5>0$. Therefore, $[F]\in {\sigma}_{5}({\nu}_{3}({\mathbb{P}}^{4}))$.An explicit computation of ${F}^{\perp}$ yields the Hilbert series for ${\mathrm{HS}}_{R/{F}^{\perp}}(z)=1+5z+5{z}^{2}+{z}^{3}$. Let us prove, by contradiction, that there is no saturated ideal $I\subset {F}^{\perp}$ defining a zerodimensional scheme of length $\le 5$. Suppose on the contrary that I is such an ideal. Then, ${\mathrm{HF}}_{R/I}(i)\ge {\mathrm{HF}}_{R/{F}^{\perp}}(i)$ for all $i\in \mathbb{N}$. As ${\mathrm{HF}}_{R/I}(i)$ is an increasing function of $i\in \mathbb{N}$ with ${\mathrm{HF}}_{R/{F}^{\perp}}(i)\le {\mathrm{HF}}_{R/I}(i)\le 5$, we deduce that ${\mathrm{HS}}_{R/I}(t)=1+5{\sum}_{i=1}^{\infty}{z}^{i}$. This shows that ${I}_{1}=\{0\}$ and that ${I}_{2}={({F}^{\perp})}_{2}$. As I is saturated, ${I}_{2}:({x}_{0},\dots ,{x}_{4})={I}_{1}=\{0\}$, since ${\mathrm{HF}}_{R/{F}^{\perp}}(1)=5$. However, an explicit computation of ${({F}^{\perp})}_{2}:({x}_{0},\dots ,{x}_{4})$ gives $\langle {x}_{2},{x}_{3},{x}_{4}\rangle $. In this way, we obtain a contradiction, so that there is no saturated ideal of degree $\le 5$ such that $I\subset {F}^{\perp}$. Consequently, the minimal zerodimensional scheme contained in ${X}_{4,3}$ whose linear span contains $[F]$ has degree six.
In the best of our knowledge, the two main results that are nowadays available to treat these “wild” cases are the following.
Proposition 9 ([
28])
. Let $X\subset {\mathbb{P}}^{N}$ be a nondegenerate smooth variety. Let ${H}_{r}$ be the irreducible component of the Hilbert scheme of zerodimensional schemes of degree r of X containing r distinct points, and assume that for each $y\in {H}_{r}$, the corresponding subscheme Y of X imposes independent conditions on linear forms. Then, for each $P\in {\sigma}_{r}(X)$ $\setminus {\sigma}_{r}^{0}(X)$, there exists a zerodimensional scheme $Z\subset X$ of degree r such that $P\in \langle Z\rangle \cong {\mathbb{P}}^{r1}$. Conversely, if there exists $Z\in {H}_{r}$ such that $P\in \langle Z\rangle $, then $P\in {\sigma}_{r}(X)$. Obviously, five points on a line do not impose independent conditions on cubics in any ${\mathbb{P}}^{n}$ for $n\ge 5$; therefore, this could be one reason why the counterexample given in Example 14 is possible. Another reason is the following.
Proposition 10 ([
81])
. Suppose there exist points ${P}_{1},\dots ,{P}_{r}\in X$ that are linearly degenerate, that is $dim\langle {P}_{1},\dots ,{P}_{r}\rangle <r1$. Then, the join of the r tangent stars(
see [83] (Section 1.4) for a definition)
at these points is contained in ${\sigma}_{r}(X)$. In the case that X is smooth at ${P}_{1},\dots {P}_{r}$, then $\langle {T}_{{P}_{1}}X,\dots ,{T}_{{P}_{r}}X\rangle \subset {\sigma}_{r}(X)$. 2.3.3. Beyond Sylvester’s Algorithm via Apolarity
We have already defined in
Section 2.1.4 the apolarity action of
${S}^{\u2022}{V}^{\ast}\simeq \mathbb{k}[{y}_{0},\dots ,{y}_{n}]$ on
${S}^{\u2022}V\simeq \mathbb{k}[{x}_{0},\dots ,{x}_{n}]$ and inverse systems. Now, we introduce the main algebraic tool from apolarity theory to study ranks and minimal Waring decompositions: that is the apolarity lemma; see [
24,
25]. First, we introduce the apolar ideal of a polynomial.
Definition 19. Let $F\in {S}^{d}V$ be a homogeneous polynomial. Then, the apolar ideal
of F is: Remark 15. The apolar ideal is a homogeneous ideal. Clearly, ${F}_{i}^{\perp}={S}^{i}{V}^{\ast}$, for any $i>d$, namely ${A}_{F}={S}^{\u2022}{V}^{\ast}/{F}^{\perp}$ is an Artinian algebra with socle degree equal to d. Since ${dim}_{\mathbb{k}}{({A}_{F})}_{d}=1$, then it is also a Gorenstein algebra. Actually, Macaulay proved that there exists a onetoone correspondence between graded Artinian Gorenstein algebras with socle degree d and homogeneous polynomials of degree d; for details, see [24] (Theorem 8.7). Remark 16. Note that, directly by the definitions, the nonzero homogeneous parts of the apolar ideal of a homogeneous polynomial F coincide with the kernel of its catalecticant matrices, i.e., for $i=0,\dots ,d$, The apolarity lemma tells us that Waring decompositions of a given polynomial correspond to sets of reduced points whose defining ideal is contained in the apolar ideal of the polynomial.
Lemma 5 (Apolarity lemma)
. Let $Z=\{[{L}_{1}],\dots ,[{L}_{r}]\}\subset \mathbb{P}({S}^{1}V)$, then the following are equivalent:
$F={\sum}_{i=1}^{r}{\lambda}_{i}{L}_{i}^{d}$, for some ${\lambda}_{1},\dots ,{\lambda}_{r}\in \mathbb{k}$;
$I(Z)\subseteq {F}^{\perp}$.
If these conditions hold, we say that Z is a set of points apolar to F.
Proof. The fact that (1) implies (2) follows from the easy fact that, for any $G\in {S}^{d}{V}^{\ast}$, we have that $G\circ {L}^{d}$ is equal to d times the evaluation of G at the point $[L]\in \mathbb{P}V$. Conversely, if $I(Z)\subset {F}^{\perp}$, then we have that $F\in I{(Z)}_{d}^{\perp}=\langle {L}_{1}^{d},\dots ,{L}_{r}^{d}\rangle $; see Remark 9 and Proposition 4. □
Remark 17 (Yet again: Sylvester’s algorithm)
. With this lemma, we can rephrase Sylvester’s algorithm. Consider the binary form $F={\sum}_{i=0}^{d}{c}_{i}\left(\genfrac{}{}{0pt}{}{d}{i}\right){x}_{0}^{di}{x}_{1}^{i}$. Such an F can be decomposed as the sum of r distinct powers of linear forms if and only if there exists $Q={q}_{0}{y}_{0}^{r}+{q}_{1}{y}_{0}^{r1}{y}_{1}+\cdots +{q}_{r}{y}_{1}^{r}$ such that:and $Q=\mu {\mathsf{\Pi}}_{k=1}^{r}({\beta}_{k}{y}_{0}{\alpha}_{k}{y}_{1})$, for a suitable scalar $\mu \in \mathbb{k}$, where $[{\alpha}_{i}:{\beta}_{i}]$’s are different points in ${\mathbb{P}}^{1}$. In this case, there exists a choice of ${\lambda}_{1},\dots ,{\lambda}_{r}$ such that $F={\sum}_{k=1}^{r}{\lambda}_{k}{({\alpha}_{k}{x}_{0}+{\beta}_{k}{x}_{1})}^{d}$. This is possible because of the following remarks: Gorenstein algebras of codimension two are always complete intersections, i.e., Artinian Gorenstein rings have a symmetric Hilbert function, hence: If ${G}_{1}$ is squarefree, i.e., has only distinct roots, we take $Q={G}_{1}$ and ${\mathrm{R}}_{\mathrm{sym}}(F)=deg({G}_{1})$; otherwise, the first degree where we get something squarefree has to be the degree of ${G}_{2}$; in particular, we can take Q to be a generic element in ${F}_{deg({G}_{2})}^{\perp}$ and ${\mathrm{R}}_{\mathrm{sym}}(F)=deg({G}_{2})$.
By using Apolarity Theory, we can describe the following algorithm (Algorithm 4).
Algorithm 4: Iarrobino and Kanev [25]. 
 We attribute the following generalization of Sylvester’s algorithm to any number of variables to Iarrobino and Kanev: despite that they do not explicitly write the algorithm, the main idea is presented in [25]. Sometimes, this algorithm is referred to as the catalecticant method. 
Require:$F\in {S}^{d}V$, where $dimV=n+1$. 
Ensure: a minimal Waring decomposition. 
 1:
construct the most square catalecticant of F, i.e., $Ca{t}_{m,dm}(F)$ for $m=\lceil d/2\rceil ;$  2:
compute $kerCa{t}_{m,dm}(F)$;  3:
if the zeroset Z of the polynomials in $kerCa{t}_{m,dm}(F)$ is a reduced set of points, say $\{[{L}_{1}],\dots ,[{L}_{r}]\}$, then continue, otherwise the algorithm fails;  4:
solve the linear system defined by $F={\sum}_{i=1}^{s}{\lambda}_{i}{L}_{i}^{d}$ in the unknowns ${\lambda}_{i}$.

Example 15. Compute a Waring decomposition of:The most square catalecticant matrix is:Now, compute that the rank of $Ca{t}_{2,2}(F)$ is three, and its kernel is:It is not difficult to see that these three quadrics define a set of reduced points $\{[1:1:0],[1:0:1],[1:1:1]\}\subset \mathbb{P}V$. Hence, we take ${L}_{1}={x}_{0}+{x}_{1}$, ${L}_{2}={x}_{0}{x}_{2}$ and ${L}_{3}={x}_{0}{x}_{1}+{x}_{2}$, and, by the apolarity lemma, the polynomial F is a linear combinations of those forms, in particular, Clearly, this method works only if ${\mathrm{R}}_{\mathrm{sym}}(F)=\mathrm{rank}\phantom{\rule{3.33333pt}{0ex}}Ca{t}_{m,dm}(F)$, for $m=\u2308\frac{d}{2}\u2309$. Unfortunately, in many cases, this condition is not always satisfied.
Algorithm 4 has been for a long time the only available method to handle the computation of the decomposition of polynomials with more than two variables. In 2013, there was an interesting contribution due to Oeding and Ottaviani (see [
84]), where the authors used vector bundle techniques introduced in [
85] to find nonclassical equations of certain secant varieties. In particular, the very interesting part of the paper [
84] is the use of representation theory, which sheds light on the geometric aspects of this algorithm and relates these techniques to more classical results like the Sylvester pentahedral theorem (the decomposition of cubic polynomial in three variables as the sum of five cubes). For the heaviness of the representation theory background needed to understand that algorithm, we have chosen to not present it here. Moreover, we have to point out that ([
84] Algorithm 4) fails whenever the symmetricrank of the polynomial is too large compared to the rank of a certain matrix constructed with the techniques introduced in [
84], similarly as happens for the catalecticant method.
Nowadays, one of the best ideas to generalize the method of catalecticant matrices is due to Brachat, Comon, Mourrain and Tsidgaridas, who in [
29] developed an algorithm (Algorithm 5) that gets rid of the restrictions imposed by the usage of catalecticant matrices. The idea developed in [
29] is to use the socalled Hankel matrix that in a way encodes all the information of all the catalecticant matrices. The algorithm presented in [
29] to compute a Waring decomposition of a form
$F\in {S}^{d}V$ passes through the computation of an affine Waring decomposition of the dehomogenization
f of the given form with respect to a suitable variable. Let
$S=\mathbb{k}[{x}_{1},\dots ,{x}_{n}]$ be the polynomial ring in
n variables over the field
$\mathbb{k}$ corresponding to such dehomogenization.
We first need to introduce the definition of Hankel operator associated with any
$\mathsf{\Lambda}\in {S}^{\ast}$. To do so, we need to use the structure of
${S}^{\ast}$ as the
Smodule, given by:
Then, the Hankel operator associated with
$\mathsf{\Lambda}\in {S}^{\ast}$ is the matrix associated with the linear map:
Here are some useful facts about Hankel operators.
Proposition 11. $ker({H}_{\mathsf{\Lambda}})$ is an ideal.
Let ${I}_{\mathsf{\Lambda}}=ker({H}_{\mathsf{\Lambda}})$ and ${A}_{\mathsf{\Gamma}}=S/{I}_{\mathsf{\Lambda}}$.
Proposition 12. If $\mathrm{rank}({H}_{\mathsf{\Lambda}})=r<\infty $, then the algebra ${A}_{\mathsf{\Lambda}}$ is a $\mathbb{k}$vector space of dimension r, and there exist polynomials ${l}_{1},\dots {l}_{k}$ of degree one and ${g}_{1},\dots ,{g}_{k}$ of degree ${d}_{1},\dots ,{d}_{k}$, respectively, in $\mathbb{k}[{\partial}_{1},\dots ,{\partial}_{n}]$ such that:Moreover, ${I}_{\mathsf{\Lambda}}$ defines the union of affine schemes ${Z}_{1},\dots ,{Z}_{k}$ with support on the points ${l}_{1}^{\ast},\dots ,{l}_{k}^{\ast}\in {\mathbb{k}}^{n}$, respectively, and with multiplicity equal to the dimension of the vector space spanned by the inverse system generated by ${l}_{i}^{d{d}_{i}}{g}_{i}$. The original proof of this proposition can be found in [
29]; for a more detailed and expanded presentation, see [
30,
31].
Theorem 7 (Brachat, Comon, Mourrain, Tsigaridas [
29])
. An element $\mathsf{\Lambda}\in {S}^{\ast}$ can be decomposed as $\mathsf{\Lambda}={\sum}_{i=1}^{r}{\lambda}_{i}{l}_{i}^{d}$ if and only if $\mathrm{rank}{H}_{\mathsf{\Lambda}}=r$, and ${I}_{\mathsf{\Lambda}}$ is a radical ideal. Now, we consider the multiplication operators in
${A}_{\mathsf{\Lambda}}$. Given
$a\in {A}_{\mathsf{\Lambda}}$:
and,
Now,
Theorem 8. If $dim{A}_{\mathsf{\Lambda}}<\infty $, then, $\mathsf{\Lambda}={\sum}_{i=1}^{k}{l}_{i}^{d{d}_{i}}{g}_{i}$ and:
the eigenvalues of the operators ${M}_{a}$ and ${M}_{a}^{t}$ are given by $\{a({l}_{1}^{\ast}),\dots ,a({l}_{r}^{\ast})\}$;
the common eigenvectors of the operators ${({M}_{{x}_{i}}^{t})}_{1\le i\le n}$ are, up to scalar, the ${l}_{i}$’s.
Therefore, one can recover the
${l}_{i}$’s, i.e., the points
${l}_{i}^{\ast}$’s, by eigenvector computations: take
B as a basis of
${A}_{f}$, i.e., say
$B=\{{b}_{1},\dots ,{b}_{r}\}$ with
$r=\mathrm{rank}{H}_{\mathsf{\Lambda}}$, and let
${H}_{a\ast \mathsf{\Lambda}}^{B}={M}_{a}^{t}{H}_{\mathsf{\Lambda}}^{B}={H}_{\mathsf{\Lambda}}^{B}{M}_{a}$ (
${M}_{a}$ is the matrix of the multiplication by
a in the basis
B). The common solutions of the generalized eigenvalue problem:
for all
$a\in S$ yield the common eigenvectors
${H}_{\mathsf{\Lambda}}^{B}v$ of
${M}_{a}^{t}$, that is the evaluations at the points
${l}_{i}^{\ast}$’s. Therefore, these common eigenvectors
${H}_{\mathsf{\Lambda}}^{B}v$ are up to scalar the vectors
$[{b}_{i}({l}_{i}^{\ast}),\dots ,{b}_{r}({l}_{i}^{\ast})]$, for
$i=1,\dots ,r$.
If $f={\sum}_{i=1}^{r}{\lambda}_{i}{l}_{i}^{d}$, then the ${Z}_{i}$’s in Proposition 12 are simple, and one eigenvector computation is enough: in particular, for any $a\in S$, ${M}_{a}$ is diagonalizable, and the generalized eigenvectors ${H}_{\mathsf{\Lambda}}^{B}v$ are, up to scalar, the evaluations at the points ${l}_{i}^{\ast}$’s.
Now, in order to apply this algebraic tool to our problem of finding a Waring decomposition of a homogeneous polynomial $F\in \mathbb{k}[{x}_{0},\dots ,{x}_{n}]$, we need to consider its dehomogenization $f=F(1,{x}_{1},\dots ,{x}_{n})$ with respect to the variable ${x}_{0}$ (with no loss of generality, we may assume that the coefficients with respect to ${x}_{0}$ are all nonzero). Then, we associate a truncated Hankel matrices as follows.
Definition 20. Let B be a subset of monomials in S. We say that B is connected to one if $\forall \phantom{\rule{0.166667em}{0ex}}m\in B$ either $m=1$ or there exists $i\in \{1,\dots ,n\}$ and ${m}^{\prime}\in B$ such that $m={x}_{i}{m}^{\prime}$.
Let
$B,{B}^{\prime}\subset {S}_{\le d}$ be sets of monomials of degree
$\le d$, connected to one. For any
$f={\sum}_{\begin{array}{c}\alpha \in {\mathbb{N}}^{n}\\ \alpha \le d\end{array}}{c}_{\alpha}\left(\genfrac{}{}{0pt}{}{d}{d\alpha ,{\alpha}_{1},\dots ,{\alpha}_{n}}\right){\mathbf{x}}^{\alpha}\in {S}_{d}$, we consider the
Hankel matrix:
where
${h}_{\alpha}={c}_{\alpha}$ if
$\alpha \le d$, and otherwise,
${h}_{\alpha}$ is an unknown. The set of all these new variables is denoted
h. Note that, by this definition, the known parts correspond to the catalecticant matrices of
F. For simplicity, we write
${H}_{f}^{B}={H}_{f}^{B,B}$. This matrix is also called
quasiHankel [
86].
Example 16. Consider $F=4{x}_{0}{x}_{1}+2{x}_{0}{x}_{2}+2{x}_{1}{x}_{2}+{x}_{2}^{2}\in \mathbb{k}[{x}_{0},{x}_{1},{x}_{2}]$. Then, we look at the dehomogenization with respect to ${x}_{0}$ given by $f=4{x}_{1}+2{x}_{2}+2{x}_{1}{x}_{2}+{x}_{2}^{2}\in \mathbb{k}[{x}_{1},{x}_{2}]$. Then, if we consider the standard monomial basis of ${S}_{\le 2}$ given by $B=\{1,{x}_{1},{x}_{2},{x}_{1}^{2},{x}_{1}{x}_{2},{x}_{2}^{2}\}$, then we get:where the h’s are unknowns. Now, the idea of the algorithm is to find a suitable polynomial
$\overline{f}$ whose Hankel matrix extends the one of
f, has rank equal to the Waring rank of
f and the kernel gives a radical ideal. This is done by finding suitable values for the unknown part of the Hankel matrix of
f. Those
$\overline{f}$ are elements whose homogenization is in the following set:
where
${Y}_{r}^{i,di}=\{[F]\in \mathbb{P}({S}^{d}V)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{rank}Ca{t}_{i,di}(F)\le r\}$. If
$[F]\in {\mathcal{E}}_{r}^{d,0}$, we say that
f is the generalized affine decomposition of size
r.
Suppose that
${H}_{f}^{B,{B}^{\prime}}$ is invertible in
$\mathbb{k}(h)$, then we define the formal multiplication operators:
Notation 2. If B is a subset of monomials, then we write ${B}^{+}=B\cup {x}_{1}B\cup \dots \cup {x}_{n}B$. Note that, if B is connected to one, then also ${B}^{+}$ is connected to one.
The key result for the algorithm is the following.
Theorem 9 (Brachat, Comon, Mourrain, Tsigaridas [
29])
. If B and ${B}^{\prime}$ are sets of monomials connected to one, the coefficients of f are known on ${B}^{+}\times {B}^{\prime +}$, and if ${H}_{\tilde{f}}^{B,{B}^{\prime}}$ is invertible, then f extends uniquely to S if and only if: Algorithm 5: Brachat, Comon, Mourrain, Tsigaridas [29,30,31]. 
 Here is the idea of algorithm presented in [29]. In [30,31], a faster and more accurate version can be found. 
Require: Any polynomial $f\in S$. 
Ensure: An affine Waring decomposition of f. 
 1:
$r\leftarrow 1$;  2:
Compute a set B of monomials of degree $\le d$ connected to one and with $B=r$;  3:
Find parameters h such that $det({H}_{f}^{B})\ne 0$ and the operators ${M}_{i}^{B}={({H}_{f}^{B})}^{1}{H}_{{x}_{i}f}^{B}$ commute;  4:
if there is no solution then  5:
go back to 2 with $r\leftarrow r+1$;  6:
else  7:
compute the $n\xb7r$ eigenvalues ${z}_{i,j}$ and the eigenvectors ${v}_{j}$ such that ${M}_{j}{v}_{j}={z}_{i,j}{v}_{j}$, $i=1,\dots ,n$, $j=1,\dots ,r$, until one finds r different common eigenvectors;  8:
end if  9:
Solve the linear system $f={\sum}_{j=1}^{r}{\lambda}_{j}{z}_{j}^{d}$ in the ${\lambda}_{i}$’s, where the ${z}_{j}$’s are the eigenvectors found above.

For simplicity, we give the example chosen by the authors of [
29].
Example 17. We look for a decomposition of: We form a $\left(\genfrac{}{}{0pt}{}{n+d1}{d}\right)\times \left(\genfrac{}{}{0pt}{}{n+d1}{d}\right)$ matrix, the rows and the columns of which correspond to the coefficients of the polynomial with respect to the expression $f=F(1,{x}_{1},\dots ,{x}_{n})={\sum}_{\begin{array}{c}\alpha \in {\mathbb{N}}^{n}\\ \alpha \le d\end{array}}{c}_{\alpha}\left(\genfrac{}{}{0pt}{}{d}{d\alpha ,{\alpha}_{1},\dots ,{\alpha}_{n}}\right){\mathbf{x}}^{\alpha}$.
The whole $21\times 21$ matrix is the following.
$$\left(\begin{array}{ccccccccccc}& \hfill 1& \hfill {x}_{1}& \hfill {x}_{2}& \hfill {x}_{1}^{2}& \hfill {x}_{1}{x}_{2}& \hfill {x}_{2}^{2}& \hfill {x}_{1}^{3}& \hfill {x}_{1}^{2}{x}_{2}& \hfill {x}_{1}{x}_{2}^{2}& \hfill {x}_{2}^{3}\\ 1& \hfill 38& \hfill 24& \hfill 36& \hfill 1272& \hfill 288& \hfill 822& \hfill 3456& \hfill 7416& \hfill 5544& \hfill 5916\\ {x}_{1}& \hfill 24& \hfill 1272& \hfill 288& \hfill 3456& \hfill 7416& \hfill 5544& \hfill \mathrm{166,368}& \hfill 41,472& \hfill \mathrm{80,568}& \hfill 77,472\\ {x}_{2}& \hfill 36& \hfill 288& \hfill 822& \hfill 7416& \hfill 5544& \hfill 5916& \hfill 41,472& \hfill \mathrm{80,568}& \hfill 77,472& \hfill \mathrm{88,518}\\ {x}_{1}^{2}& \hfill 1272& \hfill 3456& \hfill 7416& \hfill \mathrm{166,368}& \hfill 41,472& \hfill \mathrm{80,568}& \hfill 497,664& \hfill 1,118,304& \hfill \mathrm{798,336}& \hfill 965,304\\ {x}_{1}{x}_{2}& \hfill 288& \hfill 7416& \hfill 5544& \hfill 41,472& \hfill \mathrm{80,568}& \hfill 77,472& \hfill 1,118,304& \hfill \mathrm{798,336}& \hfill 965,304& \hfill \mathrm{1,023,336}\\ {x}_{2}^{2}& \hfill 822& \hfill 5544& \hfill 5916& \hfill \mathrm{80,568}& \hfill 77,472& \hfill \mathrm{88,518}& \hfill \mathrm{798,336}& \hfill 965,304& \hfill \mathrm{1,023,336}& \hfill 1,107,804\\ {x}_{1}^{3}& \hfill 3456& \hfill \mathrm{166,368}& \hfill 41,472& \hfill 497,664& \hfill 1,118,304& \hfill \mathrm{798,336}& \hfill {h}_{6,0,0}& \hfill {h}_{5,1,0}& \hfill {h}_{4,2,0}& \hfill {h}_{3,3,0}\\ {x}_{1}^{2}{x}_{2}& \hfill 7416& \hfill 41,472& \hfill \mathrm{80,568}& \hfill 1,118,304& \hfill \mathrm{798,336}& \hfill 965,304& \hfill {h}_{5,1,0}& \hfill {h}_{4,2,0}& \hfill {h}_{3,3,0}& \hfill {h}_{2,4,0}\\ {x}_{1}{x}_{2}^{2}& \hfill 5544& \hfill \mathrm{80,568}& \hfill 77,472& \hfill \mathrm{798,336}& \hfill 965,304& \hfill \mathrm{1,023,336}& \hfill {h}_{4,2,0}& \hfill {h}_{3,3,0}& \hfill {h}_{2,4,0}& \hfill {h}_{1,5,0}\\ {x}_{2}^{3}& \hfill 5916& \hfill 77,472& \hfill \mathrm{88,518}& \hfill 965,304& \hfill \mathrm{1,023,336}& \hfill 1,107,804& \hfill {h}_{3,3,0}& \hfill {h}_{2,4,0}& \hfill {h}_{1,5,0}& \hfill {h}_{0,6,0}\end{array}\right)$$
Notice that we do not know the elements in some positions of the matrix. In this case, we do not know the elements that correspond to monomials with (total) degree higher than five.
We extract a principal minor of full rank.
We should rearrange the rows and the columns of the matrix so that there is a principal minor of full rank. We call this minor ${\Delta}_{0}$. In order to do that, we try to put the matrix in row echelon form, using elementary row and column operations.
In our example, the $4\times 4$ principal minor is of full rank, so there is no need for rearranging the matrix. The matrix ${\Delta}_{0}$ is:Notice that the columns of the matrix correspond to the set of monomials $\{1,{x}_{1},{x}_{2},{x}_{1}^{2}\}$. We compute the “shifted” matrix ${\Delta}_{1}={x}_{1}{\Delta}_{0}$.
The columns of ${\Delta}_{0}$ correspond to the set of some monomials, say $\left\{{\mathbf{x}}^{\alpha}\right\}$, where $\alpha \subset {\mathbb{N}}^{n}$. The columns of ${\Delta}_{1}$ correspond to the set of monomials $\left\{{x}_{1}\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{\alpha}\right\}$.
The shifted matrix ${\Delta}_{1}$ is: Notice that the columns correspond to the monomials $\{{x}_{1},{x}_{1}^{2},{x}_{1}{x}_{2},{x}_{1}^{3}\}$, which are just the corresponding monomials of the columns of ${\Delta}_{0}$, i.e., $\{1,{x}_{1},{x}_{2},{x}_{1}^{2}\}$, multiplied by ${x}_{1}$.
In this specific case, all the elements of the matrices ${\Delta}_{0}$ and ${\Delta}_{1}$ are known. If this is not the case, then we can compute the unknown entries of the matrix, using either necessary or sufficient conditions of the quotient algebra, e.g., it holds that the ${M}_{{x}_{i}}{M}_{{x}_{j}}{M}_{{x}_{j}}{M}_{{x}_{i}}=0$, for any $i,j\in \{1,\cdots ,n\}$.
We solve the equation $({\Delta}_{1}\lambda {\Delta}_{0})X=0$.
We solve the generalized eigenvalue/eigenvector problem [87]. We normalize the elements of the eigenvectors so that the first element is one, and we read the solutions from the coordinates of the normalized eigenvectors. The normalized eigenvectors of the generalized eigenvalue problem are: The coordinates of the eigenvectors correspond to the elements of the monomial basis $\{1,{x}_{1},{x}_{2},{x}_{1}^{2}\}$. Thus, we can recover the coefficients of ${x}_{1}$ and ${x}_{2}$ in the decomposition from the coordinates of the eigenvectors.
Recall that the coefficients of ${x}_{0}$ are considered to be one because of the dehomogenization process. Thus, our polynomial admits a decomposition: It remains to compute ${\lambda}_{i}$’s. We can do this easily by solving an overdetermined linear system, which we know that always has a solution, since the decomposition exists. Doing that, we deduce that ${\lambda}_{1}=3$, ${\lambda}_{2}=15$, ${\lambda}_{3}=15$ and ${\lambda}_{4}=5$.