1. Introduction
Zero-dimensional schemes, i.e., algebro-geometric generalizations of finite sets [
1,
2], entered the applied mathematic world at least from two paths:
- (a)
interpolation;
- (b)
describing the dimension of many varieties relevant to applications, e.g., the set of all tensors with fixed format and rank.
In the interpolation path, there are multivariate extensions of Hermite and Birkhoff interpolation in which, at certain nodes, one fixes the values of certain (or all up to a fixed order) partial derivatives [
3,
4]. Terracini loci correspond to taking all first-order derivatives. We recommend [
5] and references therein if one wants to extend these interpolation problems (over the reals) with partially real solutions. We think it would be a rich topic of research and we explain the connection at the end of
Section 4. These interpolation problems for multivariate polynomials are used in number theory and geometry and, thanks to the Terracini Lemma [
6] (Cor. 1.11), enter the topic described in (b) [
1,
2,
7,
8,
9], which is related to the present paper. For old and recent results on (b) related to partially symmetric tensors, see [
10,
11,
12].
Let X be a smooth and connected curve of genus g defined over an algebraically closed field of characteristic 0. Fix a base point-free , , and let be the image of X by the morphism v associated to the base point free . We assume , i.e., we assume that is the normalization map. For any zero-dimensional scheme , let denote its linear span, i.e., the intersection of the hyperplanes of containing A, with the convention if no hyperplane contains A. For any effective divisor , let be the scheme-theoretical image. For any zero-dimensional scheme , we say that or that if all connected components of Z have an even degree, and . Set .
We recall that, in the usual definitions of Terracini loci of
Y, only smooth points of
Y are considered (for very good reasons!) [
13,
14,
15]. We believe that our definition is flexible and, perhaps, may be linked to the singularities of maps and their enumerative geometry [
16,
17]. The big restriction is that we are assuming that the connected components of
Z have an even degree. In the original Terracini Lemma the points are general in
Y and so of course they are smooth points of
Y. But in the set-up of [
6], tangent spaces at singular points may be used. In the case
, there is a natural smooth variety associated to
Y, its normalization, and a base point-free
,
, inducing the normalization map. However, even when
v is the identity map with
Y smooth, our definition here is different from the standard one (see Example 2). We give the following examples to help the reader.
Example 1. Assume that the differential of the map obtained composing v with the inclusion is zero at some . Then scheme-theoretically and, hence, . In all the quoted definitions of , we have .
Example 2. Assume and Y smooth, i.e., and v the identity map. If , then by the Bezout theorem and the same is true for all quoted sets . Now assume . We have and , but is the set of all points of contact of the bitangent lines of Y, while is the union of and the non-ordinary inflectional points of Y, if any.
Example 3. Assume , and Y singular. We saw that if it has at least one cusp (or a singular point with a non-smooth branch). Fix and take such that . Assume that Y at o has at least 2 branches that are tangent at o. The scheme is an element of . Thus, if and Y has at least one non-ordinary singularity (a point whose tangent cone is not formed by distinct lines with multiplicity one), then and these schemes contributing to do not contribute to the usual Terracini locus .
See Remark 3 for the explanation of our assumption that our is assumed to be base point free. See Remark 4 for the case .
Proposition 1. Assume and that Y is singular. Then either or .
The following result extends [
15] (Proposition 7) to the case in which
Y is singular. It shows that, using our definition, we may extend and simplify previous results.
Theorem 1. Assume and odd. Then if and only if , i.e., if and only if Y is the rational normal curve of .
The proof of Theorem 1 is 2 lines: if
Y is smooth, use [
15] (Proposition 7), if
Y is singular, use Proposition 1. However, our point here is that not only the new definition allows some proofs in new and more general cases, but that it works fine in the set up of [
14,
15], e.g., it would be easy to copy the proof of [
15] (Proposition 7) in our language and then at a critical step, the case
, we use our definition and Proposition 1.
If , the proof gives that if Y is not a rational normal curve, then at least one among , and is not empty. However, the added generality is an illusion if Y is smooth (Remark 5).
The following result (the main result of the paper) shows that it is far easier to prove a nonemptiness theorem for , even when gives an embedding, than with the usual definition of Terracini loci. The drawback is that does not imply the corresponding result for the usual Terracini locus, even if we assume Y smooth.
Theorem 2. Assume r even. We have if and only if Y is either a rational normal curve or a linearly normal degree smooth curve of genus 1 or and Y is a nodal plane cubic.
In the set up of Theorem 2 for , we have and if , we are also assuming that Y is smooth and linearly normal (equivalently, smooth and not rational).
For any smooth point
o of a positive dimensional quasi-projective variety
W, let
denote the closed subscheme of
W with
as its ideal sheaf. We have
and
. For any finite set
S contained in the smooth locus of
W, set
. We can extend in several different ways the definition to an arbitrary smooth and connected projective variety. The following definition differs from both [
14,
15] in that it allows points with image singular points of
Y. However, it restricts the definition to ‘first-order data’, specifically ‘first-order zero-dimensional schemes’. For example, in the setup of Example 3, higher-order flexes at most points would not contribute according to the following definition.
Definition 1. Let X be a smooth and connected projective variety and a morphism birational onto its image with Y not contained in a hyperplane of . Let be a zero-dimensional scheme. We say that if there is a finite set such that , , and . Let denote the set of all such that . If we drop the assumption , we get a set . Set , and .
Remark 1. Note that . A classical observation due to Chandler shows that, to see if , it is sufficient to check the zero-dimensional subschemes of X whose connected components have degree [18,19], [13] (Lemma 2.8). Easy examples show that sometimes one needs to allow that some connected components have degree 1 [13] (Th. 4.2 4.2 for odd). If , we have and, hence, , where f is the morphism associated to the base point-free . Remark 2. If , Definition 1 often gives huge families of solutions (see Examples 10 and 11). When there are huge families, the point is to study the set of all solution Z, which is an algebraic set (Remark 6). Hence a good project is the study the geometry of .
We devote a full section (
Section 4) to the discussion of a few important related definitions:
The minimality for Terracini loci is very important, as stressed with many examples in [
13]. They are the building blocks for all Terracini loci.
The Terracini Lemma is true even for joins of finitely many varieties embedded in the same projective space [
6] (Cor. 1.11). We devote a full section (
Section 6) to the definitions of joins for finitely many varieties. There are two options for the notions of minimality: minimality and weak minimality.
Structure of the Paper
Section 2 contains some remarks used in the proofs later. Some of them also clarify some elementary properties of our definitions.
Section 3 contains the proofs of the proposition and the 2 theorems stated in the introduction.
Section 4 is concerned with minimality (it also introduces
weak minimality), Terracini loci with restricted support and allowable points. In the last part of the section, we discuss “restricted support” for the euclidean topology and show how it may be adapted to real solutions and partially complex solutions.
Section 5 gives examples (with
X a surface) concerning the Terracini loci.
Section 6 contains the definition of Terracini loci for joins of different varieties, proves one result (Proposition 5) and shows that joins of different varieties are easier than the one of a single variety (Example 12 for joins, Example 11 in
Section 5 for the secant variety of a variety).
4. Minimal Terracini Loci and Allowed Support
For all the notions , and of Terracini loci, there are at least 2 notions of minimality. We call any , and and, for instance, we write instead of or or . We recall that for every zero-dimensional scheme , X the smooth projective variety such that we are looking at the Terracini locus of , . Fix . We say that Z is minimal if for any . We say that Z is weakly minimal if for all such that . Call (resp. ) the set of all minimal (resp. weakly minimal) elements of .
Remark 8. Assume and let x be the first integer such that . Obviously, . Thus, in the set-up Theorems 1 and 2, we have , except in the listed cases with . Moreover, (for any X) if there is at which the differential of f is not injective, then .
The following examples show that sometimes minimality is stronger than weak minimality (easy examples also exist for all ).
Example 6. Take a plane curve with a cuspidal point, , whose tangent cone is formed by a line L (with multiplicity 2) tangent to Y at some . Let denote the normalization map. Call the linear series on X mapping X to Y. Take with images and in Y. Since , we have . We have and . Since , . Hence, is weakly minimal, but not minimal.
Example 7. Take a plane curve with 2 different cusps, and , and let γ the associated linear system on the normalization . Take with images and . Since spans a line, , , and for all i. Thus, is weakly minimal, but not minimal.
Now we discuss two important refinements of the Terracini loci:
- (1)
closed subsets of X that we are forced to avoid;
- (2)
points of X or finite subsets of X that we may allow.
Obviously, (1) is important if we may take interpolation data only outside a closed subset B of X (see the end of the section for a discussion of more general B for the euclidean topology). This is important if we cannot access a small part, B, of the database. Nonemptiness outside B means that our problem has at least one solution without points in B.
Obviously, (2) can be used to shorten the computational task, if we computed in advance the data at the allowed point (or more that one point).
Take a smooth projective variety X and a morphism , , birational onto its its image and with . We write and for any of , and , since the definitions in this section work for all definitions of Terracini loci. Take a closed subset for the Zariski topology. Take any . We say that if . Set . Since is an open subset of X, is an open subset of and (it may be empty). The same definition applies to the minimally Terracini and weakly minimal Terracini loci, and . If , then B is an arbitrary finite subset of X.
Fix . We say that p is allowed or that it is an allowed point for (or or or or or ) if there is (or or or or or ) such that . Fix a finite set , . We say that A is allowed for (or or or for or or ) if there is (or or or or or ) such that .
The exceptional case in the next proposition is described in Example 8.
Proposition 2. Let X be a smooth curve. Take a base point-free , , on X birational onto its image. Assume r is odd. Fix . Let be morphism associated to the . Set . Then p is an allowed point for , unless , , f is an embedding and the tangent line of Y at has order of contact 3 with Y at .
Proof. If the differential of f vanishes at p, then p is allowed for . Thus, we may assume that the differential of f is non-zero at p. Now assume . Since the differential of f is non-zero at p, there is such that . Thus, and, hence, . Thus, we may assume that Y is smooth at p. Let L be the tangent line of Y at . If L has a contact order of at least 4 with Y at , then . Now assume that L meets Y at some say . In this case . Let denote the linear projection from L. Since L meets only at and Y is smooth at p, extends to a morphism .
(a) Assume that L meets Y only at and that L has order of contact 2 with X at . If , we just use that any morphism of degree is ramified, because is algebraically simply connected. Thus, we may assume . We have .
(a1) Assume that is not birational onto its image. Call C the normalization of . The morphism induces a degree morphism . If u ramifies, there is a degree 2 subscheme such that . Now assume that u is unramified. Thus, C has genus . Thus, . Theorem 1 gives . Lifting it to X by the map u and adding , we get an element of .
(a2) Assume that is birational onto its image. Thus, the normalization map is a . Since , Theorem 1 gives the existence of . Since is induced by the linear projection from the tangent line of Y at , .
(b) Assume that L has order of contact 3 with Y at p. If , then we conclude as in step (a) (note that in steps (a1) and (a2) we only used Theorem 1, not the statement of Proposition 2 for the integer ). Now assume . Thus, either Y is smooth and rational, but not linearly normal, or . The first possibility is the exceptional case in the statement of the proposition. Now assume . In this case is the minimal degree of a linearly dependent zero-dimensional subscheme of Y and, hence, L cannot have order of contact with Y at the smooth point . □
Example 8. Take a smooth degree rational curve spanning , . Fix and let L be the tangent line of X at p. There are a degree rational normal curve and such that , where denote the linear projection from o. Take such that . Note that q is the unique point of C such that . Since every subscheme of degree of C is linearly independent, the tangent line of X at p has order of contact if and only if o is contained in the osculating plane of C at q. In particular, this is not the case if, for a fixed X, we take a general . Now assume that X is a general smooth degree rational curve of , i.e., assume that o is a general point of . Since o is general and , o is contained in no osculating plane of C. Thus, every point of X is allowed for the on X induced by the inclusion .
Proposition 3. Let X be a smooth curve. Take a base point free , , on X inducing a morphism birational onto its image, with nowhere vanishing differential and with . Then a general is allowable for .
Proof. Set . Since we chose p general after fixing the , is a smooth point of Y and the tangent line, L, of Y at has order of contact 2 with Y at . Assume for the moment that L meets Y at a point with . Thus . Since the differential of f does not vanish at o, and, hence, . Thus, we may assume . Thus, the linear projection from L induces a degree morphism . Since and is algebraically simply connected, there is at which ramifies. Thus, . Since f is a local embedding, and . Thus, . □
Remark 9. In the set-up of Theorem 1, one expects that has a dimension of 1. Of course, sometimes it has a higher dimension, e.g., if the morphism associated to the is ramified, say at the point p, and , then for all such that and, hence, . We believe that, for “most” ’s, has pure dimension 1. Proposition 2 and Example 8 show that has at least one component of positive dimension. There are ’s such that has isolated points. For instance, all points at which the has a cusp, i.e., all points of . Smooth plane curves of degree give other examples.
Proposition 4. Fix an integer . Let X be a smooth projective curve and a finite set. Let be a base point-free linear series such that the induced map is birational onto its image and its differential is everywhere non-zero. Then and a general is allowable for .
Proof. Write and let be the tangent lines to at . Since we are in characteristic zero, a general tangent line of contains no point of . Since we are in characteristic zero, the differential of the rational map from X to induced by the linear projection from has a non-zero differential at a general point of X. Fix a general and let the tangent line to at . We just see that for all i. If there is , , then and . Now assume that L meets only at . Since p is general, the order of contact of L and at is 2. Thus, the linear projection from L induces a degree morphism . Since , is ramified at some . Since for all i, . Thus, . □
The following example shows that Proposition 4 is not always true in .
Example 9. Take a smooth plane curve of degree and call the linear series on X associated to . Since X has bitangents, and, hence, . Since X has only finitely many bitangents and flexes, there is a finite set B such that .
Remark 10. In the definition of , we must be alert to the logical quantifier: we first fix B and then take f hopefully to prove that . If , then finding B such that is trivial.
Here we are using the Zariski topology. Now assume that the base field is the complex number field . We write for the points of X with the euclidean topology. Thus, is a compact and connected complex manifold of dimension and, hence, a compact and connected orientable -dimensional topological manifold. In the definition of , we may take as B any closed set . Proposition 2 shows that often at least one point of some solution of may be found in . For any closed set , we write for the set with the euclidean topology induced by the topology of . Even if T is very singular, is not topologically very bad (it is a compact complex analytic space and in particular it is a compact CW complex and it has a triangulation). Fix any metric d on , inducing the euclidean topology, e.g., the one induced by an embedding of in a big projective space equipped with the Fubini-Study metric. For all real numbers , let denote the set of all whose distance from is at most . Each is a compact subset of . If , then there is a real number such that for all , because X is irreducible and, hence, . Fix a Zariski closed set and assume . Take . Since is a finite set is a positive real number. Note that for all .
Now assume that the algebraically closed base field is the complex number field
and that both
X and
f are defined over
. Since
X is smooth, the set
of all real
is the union of finitely many connected components, each of them a compact differentiable manifold of dimension
n. In this case,
(or
if
B is defined over
) has an involution
induced by the complex conjugation of
. Thus, it is natural to check if the involution
has fixed points. These fix points,
, are the “real solutions”, but
may have
. For instance, take a smooth plane curve
with an ordinary bitangent tangent to
X at 2 complex conjugate points of
. Even worse,
may occur (for any genus
there is a smooth genus
g defined over
and with
[
22]. The reader is encouraged to look at “partially real” solutions in the sense of [
5] and references therein.
6. Joins of Two or Finitely Many Embedded Varieties
Fix an integer
and
s integral and non-degenerate varieties
,
. The join
of
is the closure in
of the union of all linear spans of sets
with
for all
i. In the usual definition, one allows the case
for some
, but we prefer to consider the case in which
for all
and we have positive integers
such that each
appears exactly
times in the join. Thus, we are looking at Terracini loci coming from the join of
, where
denote the
-th secant variety of
[
2]. Set
. We are assuming
for all
, but
and
are allowed to be projectively equivalent. Let
be a smooth projective variety and let
be a morphism with
and
birational. We take
as distinct abstract varieties (even if they are isomorphic) so that no point of
is a point of
for
. For all finite sets
, we write
as the union of the double points
of
,
. We assume
for all
i and take zero-dimensional schemes
,
. In the case
, we assume that each connected component of
has an even degree. We say that
contributes to
if
and
. We get the same formula if
for some
i taking
, with the only restriction that
. Set
and
. The notions of minimality and weak minimality for joins are different if we consider
or
.
Note that, in the next proposition, we allow the case in which both and are rational normal curves, we only assume as subsets of the same projective space.
Proposition 5. Fix an odd integer and irreducible and non-degenerate curves , , such that . Let denote the normalization map. Fix an integer and set . Then . Moreover, has at least 2 irreducible families of elements of dimension and, as an allowable set for , we may take the union of x general points of and general points of .
Proof. By Remark 5, it is sufficient to find integers and , , such that . Fix a general such that and set . Since , is finite. Since is general in , . Set If , then and, hence, . Now assume . If , say , the pair of schemes gives . Thus, we may assume . Let denote the morphism induced by the composition of with the linear projection . Set . Since is non-degenerate, and, hence, . If , and, hence, , we use the ramification formula to say that is ramified at some and, hence, we use the scheme . Now assume . If , one mimics step (a1) of the proof of Proposition 2. If , we may use the statement of Theorem 1, because .
Now we prove the “Moreover” assertion. We may take as first x points on x general points of . So these x points of are parametrized by a family of dimension x. If , then we proved the existence of our first irreducible family and even described it. Now assuming , instead of quoting Theorem 1, we take the proof of Proposition 2 and see that we may take general points of as part of our solution.
The other family is obtained first taking y general points of and then general points of . □
The following example (the equivalent for joins of Example 11) shows how much easier it is to get results for joins of different varieties. We hope that the readers will give many more examples.
Example 12. Take smooth and non-degenerate surfaces , , such that . Call and the inclusion of and . Fix any . We claim that p is allowable for . If , then we may take any and use that contains o (, because ). Now assume . In this case, the linear projection from induces a morphism of degree . We have , because spans . Since is algebraically simply connected, the purity of the branch locus shows that u ramifies over a curve, i.e., there is a 1-dimensional family of such that .