Tame Secant Varieties and Group Actions

Abstract

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with respect to forms of bidegree $(1,d)$, our results are related to the simultaneous rank of degree d forms in $n+1$ variables.

1. Introduction

The dimensions of the secant varieties of a variety embedded in a complex projective space are a classical algebro-geometric topic [1] which was resurrected because, for certain X, they are important for applications. For instance, they give the dimension of the sets of tensors with fixed format and tensor rank [2,3]. Recently, the case of partially symmetric tensors saw strong results which, just one year before, would seem out of reach [4,5,6], and which used a key observation made by A. T. Blomenhofer and A. Casarotti [4], which helped them to give better statements for the dimensions of the secant varieties of all known homogeneous varieties used by the Applied Mathematics community. In this paper, to extend the range of their applications, we use algebro-geometric tools.

As far as we know, all these applications arise in the following way. Let Y be a projective manifold, which is homogeneous for the action of the connected algebraic group G, and $\mathcal{L}$ a very ample line bundle on Y, which is G-equivariant. Since $\mathcal{L}$ is very ample, it induces an embedding $j:Y\to \mathbb{P}\left(H^0{(Y,\mathcal{L})}^{\vee }\right)$. Set $X:=j\left(Y\right)$, $n:=\dim X$ and $r:=\dim H^0(Y,\mathcal{L})-1$. We obtain an embedding $X\subset {\mathbb{P}}^r$ with X irreducible and non-degenerate and G acting on ${\mathbb{P}}^r$ with $GX=X$.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety defined over an algebraically closed field $\mathbb{K}$ of characteristic 0 (for the case of the real numbers, see Remark 10). For any integer $i\ge 1$, let ${\sigma }_i\left(X\right)\subset {\mathbb{P}}^r$ denote the i-th secant variety of X. The set ${\sigma }_i\left(X\right)$ is an irreducible closed subvariety of ${\mathbb{P}}^r$ and $\dim {\sigma }_i\left(X\right)\le \min \{r,i(n+1)-1\}$ [1,2,3,7]. The integer $\min \{r,i(n+1)-1\}$ is the expected dimension of ${\sigma }_i\left(X\right)$, and X is said to be defective or secant-defective if $\dim {\sigma }_i\left(X\right)<\min \{r,i(n+1)-1\}$ for some $i\ge 2$. Set $a_1\left(X\right)=n+1$. For all $i\ge 2$, set $a_i\left(X\right):=\dim {\sigma }_i\left(X\right)-\dim {\sigma }_{i-1}\left(X\right)$. It is easy to check, and is known by the “classics”, that, for all $i\ge 2$, we have $a_i\left(X\right)\le a_{i+1}\left(X\right)$ ([7], Prop. 2.1(i)). Obviously, $a_i\left(X\right)=0$ if ${\sigma }_{i-1}\left(X\right)={\mathbb{P}}^r$. It is also known that curves are not secant-defective, i.e., that $\dim {\sigma }_i\left(X\right)=\min \{2i-1,r\}$ if $n=1$, and that if $a_i\left(X\right)=1$, then ${\sigma }_{i+1}\left(X\right)={\mathbb{P}}^r$, i.e., 1 may appear, at most, once in the sequence $\left\{a_i\left(X\right)\right\}$. There are examples that show that no other restriction occurs, i.e., that for all $n\ge 2$ and $r\ge n+1$, there are sequences ${\left\{a_i\right\}}_{i\ge 1}$ of non-negative integers with $a_1=n+1$, $a_{i+1}\le a_i$ for all i, $a_i=1$ for at most one i, and ${\sum }_{i\ge 1}{a}_i=r$, and there is a smooth n-dimensional projective variety and a non-degenerate embedding $X\subset {\mathbb{P}}^r$ with $a_i\left(X\right)=a_i$ for all i [8,9]. The catch in these examples is that there are arbitrarily larger strings of consecutive integers $a_j\left(X\right)$, $i\le j\le x$ with $2\le a_i\left(X\right)=a_x\left(X\right)\le n$. When either X is not secant-defective, i.e., $a_i\left(X\right)=n+1$ for all i is such that ${\sigma }_{i+1}\left(X\right)\ne {\mathbb{P}}^r$ or $a_i\left(X\right)>a_{i+1}$ if $n\ge a_i\left(X\right)>0$, then a very strong result holds (see Remark 1). Since it was used in many important cases to show secant non-defectivity [4,5,6,10], here we give a formal definition of it.

Definition 1. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim X$. We say that the pair $({\mathbb{P}}^r,X)$ satisfies ♣ if for all $i\ge 2$ such that ${\sigma }_i\left(X\right)\ne {\mathbb{P}}^r$ either $a_i\left(X\right)=n+1$ or $a_{i+1}\left(X\right)<a_i\left(X\right)$.

Remark 1. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim X$. Assume that X is secant-defective, but that $({\mathbb{P}}^r,X)$ satisfies ♣, and call $c\left(X\right)$ the first integer c such that $\dim {\sigma }_c\left(X\right)\le c(n+1)-2$. Let $g\left(X\right)$ be the first integer such that ${\sigma }_g\left(X\right)={\mathbb{P}}^r$. Set $c:=c\left(X\right)$, $g:=g\left(X\right)$ and $k:=g-c$. Then:

1. 

We have $0\le k\le n$.

2. 

We have $a_{i+1}\left(X\right)<a_i\left(X\right)$ for all $i=c-1,\dots ,g-2$.

3. 

We have $c\ge {\displaystyle \frac{r+1}{n+1}}-{\displaystyle \frac{nk}{n+1}}+{\displaystyle \frac{k(k-1)}{2n+2}}$.

4. 

We have $g\le {\displaystyle \frac{r+1}{n+1}}+k-{\displaystyle \frac{k(k-1)}{2n+2}}$.

Let $c_1\left(X\right)$ be be the maximal positive integer x such that $\dim {\sigma }_x\left(X\right)=x(n+1)-1$ with $n:=\dim X$. We have $c_1\left(X\right)\le \lfloor (r+1)/(n+1)\rfloor$. If X is not secant-defective, then $c_1\left(X\right)=\lfloor (r+1)/(n+1)\rfloor$. If X is secant-defective, then $c_1\left(X\right)=c\left(X\right)-1$. If $c_1\left(X\right)=\lfloor (r+1)/(n+1)\rfloor$, X is not secant-defective if, and only if, $g\left(X\right)=\lceil (r+1)/(n+1)\rceil$, and this is the case if, and only if, either $(r+1)/(n+1)\in \mathbb{Z}$ or $g\left(X\right)=c_1\left(X\right)$.

If $a_2\left(X\right)\le n$ and X is smooth, then far stronger inequalities for the sequence ${\left\{a_i\left(X\right)\right\}}_{i\ge 1}$ are due to F.L. Zak ([11], Th. V.1.8, [12]), so strong that, for a smooth X, the inequality $a_2\left(X\right)\le n$ implies that either ${\sigma }_{i+1}\left(X\right)={\mathbb{P}}^r$ or $a_{i+1}\le a_i\left(X\right)-n-1+a_2\left(X\right)$.

We say that $({\mathbb{P}}^r,X)$ satisfies ♠ if either it is not secant-defective or if it satisfies the conditions of Remark 1. We say that $({\mathbb{P}}^r,X)$ satisfies ♡ if no secant variety ${\sigma }_i\left(X\right)⊊{\mathbb{P}}^r$ is a cone. By [7], Prop. 2.1(ii), if $({\mathbb{P}}^r,X)$ satisfies ♡, then it satisfies ♣, and hence it satisfies ♠.

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $\mathrm{Aut}({\mathbb{P}}^r,X):=\{f\in \mathrm{Aut}\left({\mathbb{P}}^r\right)\mid f\left(X\right)=f\}$. The set $\mathrm{Aut}({\mathbb{P}}^r,X)$ is a closed algebraic subgroup of the algebraic group $PGL(n+1)=GL(n+1)/{\mathbb{K}}^{*}{\mathrm{Id}}_{n+1,n+1}$. Let $\eta :SL(n+1)\to PGL(n+1)$ denote the quotient map. Set $SL(n+1,X):={\eta }^{-1}\left(\mathrm{Aut}({\mathbb{P}}^r,X)\right)$, i.e., let $SL(n+1,X)$ denote the inverse image of $\mathrm{Aut}({\mathbb{P}}^r,X)$ in $SL(n+1)$. If ${\mathbb{K}}^{n+1}$ is irreducible representation of $SL(n+1,X)$, then X satisfies the thesis of Remark 1 ([4], Th. 1.1, has a weaker statement, but essentially the same proof). One of the aims of this paper is the class of embeddings $X\subset {\mathbb{P}}^r$ for which ♣ holds. One of our results, Theorem 5, drops the irreducibility condition, only requiring that X is G-homogeneous, a condition satisfied in the applications so far [4,5,6].

Take an integral and non-degenerate variety $X\subset {\mathbb{P}}^r$ and an algebraic group $G\subseteq \mathrm{Aut}({\mathbb{P}}^r,X)$. Let $\mathcal{F}(X,G)$ denote the set of all linear subspaces $E⊊{\mathbb{P}}^r$ such that $E\ne \varnothing$ and $g\left(E\right)=E$ for all $g\in G$.

Theorem 1. 

Let G be an algebraic group acting linearly on ${\mathbb{K}}^{r+1}$ and hence ${\mathbb{P}}^r$. Let $\mathcal{F}(X,G)$ be the set of projectivizations of proper G-invariant subspaces of ${\mathbb{K}}^{n+1}$. Let $X\subset {\mathbb{P}}^r$ be a non-degenerate G-embedding. Set $n:=\dim X$. The pair $({\mathbb{P}}^r,X)$ satisfies ♡, and hence it satisfies ♣ and ♠ if for each $F\in \mathcal{F}(X,G)$ one of the following conditions is satisfied:

1. 

F induces a base point free morphism to a projective space;

2. 

F has a base locus $B:=X\cap F$ on X, but the rational map on $X\setminus X\cap B$ induced by F has image of dimension n;

3. 

F has a base locus B on X, but the rational map on $X\setminus X\cap B$ induced by F has image of dimension $>n-\dim B$.

Remark 2. 

Note that we allow the case $G=\left\{1\right\}$. Hence, we may apply Theorem 1 to an arbitrary non-degenerate variety $X\subset {\mathbb{P}}^r$. Of course, $\mathcal{F}(X,\{1\left\}\right)$ is the union of all Grassmannians of ${\mathbb{P}}^r$, and hence it is too big to be used for any test.

If ${\mathbb{K}}^{r+1}$ is a direct sum of pairwise not isomorphic representations, then $\mathcal{F}(X,G)$ is a finite set, and hence there are only finitely many tests to be conducted to prove that ♡ holds.

Let $\mathcal{G}(X,G)$ denote the set of all linear subspaces $E\in \mathcal{F}(X,G)$ such that $E\cap X\ne \varnothing$ and $\dim {\ell }_E(X\setminus X\cap E)<n$.

Remark 3. 

If $\mathcal{G}(X,G)=\varnothing$, then $({\mathbb{P}}^r,X)$ satisfies ♡ by Theorem 1. Assume that $\mathcal{G}(X,G)\ne \varnothing$ and that $\mathcal{G}(X,G)$ is finite. This is the case if ${\mathbb{P}}^r$ is a direct sum of finitely many non-isomorphic irreducible representations. In this case, one could, in principle, check these cases (see Section 5).

All the necessary conditions for the failure of ♡ are true even if there is no action of any group. This is the content of Section 3 (see Proposition 1). Call $\mathcal{G}\left(X\right)$ the set of all linear subspaces $F\subset {\mathbb{P}}^r$ for which none of the conditions in Theorem 1 are satisfied. Without any action of G, we may prove the following result.

Theorem 2. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Assume the existence of a secant variety ${\sigma }_j\left(X\right)\ne {\mathbb{P}}^r$ which is a cone. Let $E\subset {\mathbb{P}}^r$ be a linear subspace contained in the vertex of ${\sigma }_j\left(X\right)$. Set $e:=\dim E$ and let ${\ell }_E:{\mathbb{P}}^r\setminus E\to {\mathbb{P}}^{r-1-e}$ denote the linear projection from E. Let $Y\subset {\mathbb{P}}^{r-e-1}$ denote the closure of ${\ell }_E(X\setminus X\cap E)$. Set $m:=\dim Y$. Then $g\left(X\right)=g\left(Y\right)\ge \lceil (r-e)/(m+1)\rceil$. If $m<n$, then $c_1\left(X\right)\le \lfloor (e+1)/(n-m+1)\rfloor$.

The assumption $n>m$ in the second part of Theorem 2 is satisfied in the cases we are interested in (Remark 8). If $F_1\subset F⊊{\mathbb{P}}^r$ and $F_1\in \mathcal{F}\left(X\right)$, then $F\in \mathcal{F}\left(X\right)$. Hence, the minimal (by inclusion) elements of $\mathcal{F}\left(X\right)$ are potentially very interesting. When they are simultaneously minimal and maximal, applying Theorem 2, we obtain many numerical conditions such that, if at least one of them fail, then ♡ holds. In Section 4, we give several cases involving a group G proving that X satisfies ♡. Then we discuss some conceptual tests (Section 5). In Section 6, we consider the Segre product of a G-invariant pair $({\mathbb{P}}^r,X)$ and an H-invariant pair $({\mathbb{P}}^s,Y)$ with respect to the group $G\times H$ (Proposition 4).

The next result shows that varieties contained in certain cones are secant-defective and do not satisfy ♠. This is not a new observation, and, in the case $n=2$, there are even “if and only if” clauses [13].

Theorem 3. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Let $E\subset {\mathbb{P}}^r$ be a linear subspace such that $m:=\dim {\ell }_E(X\setminus X\cap E)<n$. Set $e:=\dim E$.

(a) 

If $\lceil (r-e)/(m+1)\rceil >\lceil (r+1)/(n+1)\rceil$, then X is secant-defective.

(b) 

If $\lceil (r-e)/(m+1)\rceil >\lfloor {\displaystyle \frac{r+1}{n+1}}+{\displaystyle \frac{n+2}{2}}\rfloor$, then ♠ fails.

Remark 4. 

Assume ${\mathbb{P}}^r=\mathbb{P}\left(V\right)$ with V a direct sum of pair-wise non-isomorphic irreducible G-representations with G an algebraic group acting on X. In this case. to apply the test given by Theorems 1 and 3. we only need to finitely test many linear spaces $E\subset {\mathbb{P}}^r$. Note that we allow for the case in which G is a finite group.

There are examples showing that even having a codimension 1 orbit B while G acts transitively on $X\setminus B$ is not sufficient to be sure that $({\mathbb{P}}^r,X)$ satisfies ♣ [8,9]. Note that, in the set-up of Theorem 1, all base loci B are G-invariants.

If $({\mathbb{P}}^r,X)$ satisfies ♡, then for all $j>0$ and any integral variety $Y\subset {\mathbb{P}}^r$, the dimension of the join $[{\sigma }_j\left(X\right):Y]$ of ${\sigma }_j\left(X\right)$ and Y is “as large as possible, i.e., $\dim [{\sigma }_j\left(Y\right):Y]=\min \{r,\dim {\sigma }_j\left(X\right)+\dim Y+1\}$ ([7], Prop. 2.1). We explored an abstract version of this result in [10].

For each $n\ge 2$, there are examples of pairs $({\mathbb{P}}^r,X)$ with ♠, but not ♣ (Example 1).

In Section 7, we consider the case of toric varieties. The aims are somewhat different. We give some test (Theorems 6 and 7 and Corollary 1) to see if ♣ holds. These are geometrical conditions, but we do not know if there are efficient.

The Segre–Veronese embeddings of ${\mathbb{P}}^m\times {\mathbb{P}}^n$ are known to satisfy ♣, and this was used in [5,6]. We prove a new theorem for the embeddings with respect to the line bundle ${\mathcal{O}}_{{\mathbb{P}}^m\times {\mathbb{P}}^n}(1,d)$ (Theorem 8). These embeddings are related to the simultaneous rank of degree d forms in $n+1$ variables.

2. Preliminary Results

We work over an algebraically closed field with characteristic 0 (see Remark 10 for the case of the real numbers).

We recall that, for any linear subspace $E⊊{\mathbb{P}}^r$, ${\ell }_E:{\mathbb{P}}^r\setminus E\to {\mathbb{P}}^{r-e-1}$, $e:=\dim E$, is the linear projection from E. Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $t\left(E\right):=\dim X\cap E$ and $m\left(E\right):=\dim {\ell }_E(X\setminus X\cap E)$. Of course, the integers $t\left(E\right)$ and $m\left(E\right)$ depend on the variety X.

For all integral subvarieties A, B of a projective space ${\mathbb{P}}^m$, let $[A;B]\subseteq {\mathbb{P}}^m$ denote the join of A and B, i.e., set $\left[\right\{p\};\{p\left\}\right]=\left\{p\right\}$ case $A=B=\left\{p\right\}$, while, in all other cases, let $[A;B]$ denote the closure in ${\mathbb{P}}^r$ of the union of all lines $\langle \{p,q\}\rangle$ spanned by a point of p of A and a different point q of B. The set $[A;B]$ is an integral variety and $\dim [A;B]\le \min \{m,\dim A+\dim B+1\}$ if A and B are not the same point. The secant variety ${\sigma }_i\left(A\right)$ is the join of i copies of A.

Remark 5. 

We recall that $♡\Rightarrow ♣\Rightarrow ♠$.

Remark 6. 

Let $X\subset {\mathbb{P}}^r$ be an integral n-dimensional variety which satisfies ♣. If X is not secant-defective, then $g\left(X\right)=\lceil (r+1)/(n+1)\rceil$. Now assume that X is secant-defective. Fix an integer $n\ge 2$. Set $f_n\left(x\right)=x-(x^2-x)/(2n+2)$. The real-valued real function $f_n\left(x\right)$ has derivative $f_n^{\prime }\left(x\right)=(2x-1)(2n+2)$. Hence, $f_n\left(n\right)$ is the maximum of $f_n\left(x\right)$ in the closed interval $[0,n]$. Note that $f_n\left(n\right)={\displaystyle \frac{n^2+3n}{2n+2}}\le {\displaystyle \frac{n+2}{2}}$. Hence, $g\left(X\right)\le \lfloor {\displaystyle \frac{r+1}{n+1}}+{\displaystyle \frac{n+2}{2}}\rfloor$. Since X is assumed to be secant-defective, $g\left(X\right)\ge (r+2)/(n+1)$.

Remark 7. 

Note that if $({\mathbb{P}}^r,X)$ satisfies ♣, then $(n+1)c_1\left(X\right)\ge r+1-n(n+1)/2$. Obviously, $(n+1)c_1\left(X\right)\le r+1$. Moreover, the pair $({\mathbb{P}}^r,X)$ is not secant-defective if, and only if, $c_1\left(X\right)=\lfloor (r+1)/(n+1)\rfloor$ and $g\left(X\right)=\lceil (r+1)/(n+1)\rceil$.

Remark 8. 

Let $W,X\subset {\mathbb{P}}^r$ be varieties with W a cone. Call E the vertex of W and write $W=[E;C]$ as a join. Since $[W;X]=\left[\right[E;C]:X]=[E;[C;Y\left]\right]$, $[W:X]$ is a cone with vertex containing E. Hence, if ${\sigma }_j\left(X\right)$ is a cone with vertex E, then ${\sigma }_{j+1}\left(X\right)$ is a cone with vertex containing E.

Example 1. 

Fix integers $n\ge 2$ and r such that $r\equiv 4(\mathrm{mod}n+1)$ and $r\ge 2n+6$. Set $j:=(r-4)/(n+1)$. Both [8,9] gave examples of pairs $({\mathbb{P}}^r,X)$ with $\dim X=n$, $a_i\left(X\right)=n+1$ for all $i<j$, $a_j=a_{j+1}=2$, $a_{j+2}=1$ and $g\left(X\right)=j+3$.

Remark 9. 

Take an integral and non-degenerate variety $X\subset {\mathbb{P}}^r$. If no proper secant variety of X is a cone, then no proper secant variety of ${\sigma }_j\left(X\right)$ is a cone. Take any algebraic group $G\subset \mathrm{Aut}\left({\mathbb{P}}^r\right)$ such that $G\left(X\right)=X$. If no proper secant variety of X is a G-invariant cone, then no proper secant variety of ${\sigma }_j\left(X\right)$ is a G-invariant cone. Hence, if a group G shows that X satisfies ♡, then ${\sigma }_j\left(X\right)$ satisfies ♡. Note that ${\sigma }_j\left(X\right)$ is singular, and hence not homogeneous, if $j\ge 2$ and ${\sigma }_j\left(X\right)\ne {\mathbb{P}}^r$.

Lemma 1. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Let $E⊊{\mathbb{P}}^r$ be a linear subspace. Set $e:=\dim E$ and $Y:=\overline{{\ell }_E(X\setminus X\cap E)}\subseteq {\mathbb{P}}^{r-e-1}$. Then for all $j\ge 1$ the variety ${\sigma }_j\left(Y\right)$ is the closure in ${\mathbb{P}}^{r-e-1}$ of the variety ${\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$.

Proof. 

Since X is non-degenerate, ${\sigma }_j\left(X\right)$ is non-degenerate. Hence, ${\sigma }_j\left(X\right)⊈E$. Hence, we have $X\setminus X\cap E\ne \varnothing$ and ${\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E\ne \varnothing$. Thus, Y and ${\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$ are well-defined irreducible varieties. Hence, to prove that they are equal it is sufficient to prove that they have a common non-empty Zariski open subset. Take a general $u\in {\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$, say $u={\ell }_E\left(v\right)$ with v general in ${\sigma }_j\left(X\right)$. There are $a_1,\dots ,a_j\in X$ such that $v\in \langle \{a_1,\dots ,a_j\}\rangle$ (we are not assuming that $a_i\ne a_h$ for all $i\ne h$ or that $a_1,\dots ,a_j$ are linearly independent). For a general v we may assume $\{a_1,\dots ,a_j\}\cap E=\varnothing$. Hence, each ${\ell }_E\left(a_i\right)$, $i=1,\dots ,j$, is well-defined. Since ${\ell }_E$ is a linear projection, $u\in \langle \{{\ell }_E\left(a_1\right),\dots ,{\ell }_E\left(a_j\right)\rangle$. Hence, $u\in {\sigma }_j\left(Y\right)$.

Since ${\ell }_E(X\setminus X\cap E)$ is a constructible ([14], Ex. II.3.18, Ex. II.3.19), it contains a Zariski open subset U of Y, $U\ne \varnothing$. Take a general $w\in {\sigma }_j\left(Y\right)$. There are $b_1,\dots ,b_j\in U$ such that $w\in \langle \{b_1,\dots ,b_j\rangle$. Since $U\subseteq {\ell }_E(X\setminus X\cap E)$, $w\in {\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$. □

The next remark shows that the generic rank $g\left(X\right)$ of the variety $X\subset {\mathbb{P}}^r$ is useful even if we are interested in real solutions of the problem related to the X-rank of X, e.g., the real tensor rank of a real tensor.

Remark 10. 

Suppose the algebraically closed base field is the field $\mathbb{C}$ of complex numbers, but that the homogeneous equations defining X are defined over $\mathbb{R}$. In this case, the embedded variety X is defined over $\mathbb{R}$, and we write $X\left(\mathbb{R}\right)$ for its real points and $X\left(\mathbb{C}\right)$ for its complex points. The set $X\left(\mathbb{R}\right)$ is a real projective variety. Since $\mathbb{C}$ is algebraically closed, $g\left(X\right(\mathbb{C}\left)\right)=g\left(X\right)$. Let $X_{\mathrm{reg}}$ denote the set of all smooth points of X. Assume $X_{\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$. With this assumption $X_{\mathrm{reg}}\left(\mathbb{R}\right)$ with the euclidean topology is a finite union of n-dimensional differential manifolds. With this assumption, $2g\left(X\right)$ is an upper bound for the $X\left(\mathbb{R}\right)$-rank of a point of ${\mathbb{P}}^r\left(\mathbb{R}\right)$ [15].

3. General Results

Proposition 1. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim X$. Assume the existence of an integer $i\ge 2$ such that $a_i\left(X\right)=a_{i+1}\left(X\right)\le n$. Then ${\sigma }_{i-1}\left(X\right)$ is a cone. Call E the vertex of ${\sigma }_{i-1}\left(X\right)$. Set $e:=\dim E$. Then $E\cap X\ne \varnothing$, $t\left(E\right):=\dim E\cap X\ge n-a_i\left(X\right)$ and the linear projection ${\ell }_E:{\mathbb{P}}^r\setminus E\to {\mathbb{P}}^{r-e-1}$ induces a morphism $\ell :X\setminus X\cap E\to {\mathbb{P}}^{r-e-1}$ with $\dim \overline{\ell (X\setminus X\cap E)}=a_i\left(X\right)-1$. Moreover, $t\left(E\right)\ge n-a_i\left(X\right)$.

Proof. 

The fact that ${\sigma }_{i-1}\left(X\right)$ is a cone is known ([7], Cor. 2.2), but we need the proof given by B. Ådlandsvik ([7], Prop. 2.1 (ii)) to obtain the other assertions. Since X is secant-defective, $n\ge 2$ ([1], Remark 1.6).

Fix a general $(z,z^{\prime })\in {\sigma }_{i-1}{\left(X\right)}^2$ and a general $y\in X$. Since E is the vertex of ${\sigma }_{i-1}\left(X\right)$, $T_z{\sigma }_{i-1}\left(X\right)\cap T_{z^{\prime }}{\sigma }_{i-1}\left(X\right)\supset E$. Using Terracini’s Lemma, he proved that $T_{y}X\cap T_z{\sigma }_{i-1}\left(X\right)$ has at least dimension $n+1-a_i\left(X\right)$. Taking $z^{\prime }$ he got $T_{y}X\cap E\ne \varnothing$ and the inequality $\dim T_{y}X\cap E\ge n-a_i\left(X\right)$. Hence, the differential of ${\ell }_{E|X\setminus X\cap E}\left(X\right)$ has, at most, rank $a_i\left(X\right)-1$ at y. Since y is general in X and we are in characteristic 0, $\dim \overline{\ell (X\setminus X\cap E)}\le a_i\left(X\right)-1$. Since ${\sigma }_{i-1}\left(X\right)$ is a cone with vertex i, ${\sigma }_i\left(X\right)$ is a cone with vertex containing E (Remark 8), and we see that ${\sigma }_i\left(X\right)$ is the cone with vertex E and a base the join J of $\overline{\ell (X\setminus X\cap E)}$ and ${\ell }_E({\sigma }_{i-1}\left(X\right)\setminus E)$. Since $a_i\left(X\right)=\dim {\sigma }_i\left(X\right)-\dim {\sigma }_{i-1}\left(X\right)$ and J has, at most, dimension $\dim {\sigma }_{i-1}\left(X\right)-e-1+\dim \overline{\ell (X\setminus X\cap E)}+1$, we obtain $\dim \overline{\ell (X\setminus X\cap E)}=a_i\left(X\right)-1$.

Assume that either $X\cap E=\varnothing$ or $\dim E\cap X\le n-a_i\left(X\right)-1$. Take a general linear section C of X of codimension $n-a_i\left(X\right)$. The theorem of Bertini suggests that C is irreducible and (since E is a single fixed linear subspace) $C\cap E=\varnothing$. Hence, ${\ell }_{E|C}:C\to {\mathbb{P}}^{n-e-1}$ is a morphism. By the definition of linear projection, ${\ell }_{E|C}:C\to {\mathbb{P}}^{n-e-1}$ is a finite map, and hence $\dim {\ell }_E\left(C\right)=\dim C$. Since $\overline{\ell (X\setminus X\cap E)}\supseteq {\ell }_E\left(C\right)$, $\dim \overline{\ell (X\setminus X\cap E)}\ge a_i\left(X\right)$, a contradiction. □

Proposition 2. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim X$. Assume the existence of an integer $i\ge 2$ such that $a_i\left(X\right)=a_{i+1}\left(X\right)\le n$. Then ${\sigma }_{i-1}\left(X\right)$ is a cone. Call E the vertex of ${\sigma }_{i-1}\left(X\right)$. Assume the existence of an integer $j\ge i+1$ such that $a_j\left(X\right)=a_{j+1}\left(X\right)<a_i\left(X\right)$ and ${\sigma }_{j+1}\left(X\right)\ne {\mathbb{P}}^r$. Then ${\sigma }_j\left(X\right)$ is a cone. Let F its vertex. Then $F⊋E$, $n\ge 3$, and $\dim {\ell }_F(X\setminus X\cap F)=a_j\left(X\right)-1<\dim {\ell }_E(X\setminus X\cap E)=a_i\left(X\right)-1$.

Proof. 

Set $e:=\dim E$. Set $Y:=\overline{{\ell }_E(X\setminus X\cap E)}\subset {\mathbb{P}}^{r-e-1}$. Apply Proposition 1 to the pair $({\mathbb{P}}^{r-e-1},Y)$. □

Remark 11. 

It is obvious how to extend Proposition 2 to the case in which there are 3 or more strings like $a_i\left(X\right)=a_{i+1}\left(Y\right)$. This case occurs in the examples in [8,9].

Remark 12. 

Proposition 1 gives $m\left(E\right):=\dim {\ell }_E(X\setminus X\cap E)=a_i\left(X\right)-1<n$ and $t\left(E\right):=\dim X\cap E\ge n-m\left(E\right)-1$. Hence, knowing the integer $a_i\left(X\right)$ without knowing anything else, it is sufficient to obtain a key numerical integer, $m\left(E\right)$, of the pair $({\mathbb{P}}^r,X)$. Note that $e\left(E\right):=\dim E\ge t\left(E\right)$ with equality if, and only if, $t\left(E\right)$ is a linear space and $X\cap E=E$, i.e., if, and only if, $i=2$ and X is a cone with vertex E.

Proof of Theorem 1. 

The theorem is a consequence of Proposition 1. □

Proof of Theorem 2. 

For all $j\ge 1$, the variety ${\sigma }_j\left(Y\right)$ is the closure in ${\mathbb{P}}^{r-e-1}$ of the variety ${\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$ (Lemma 1). By assumption, $g\left(X\right)>j$. Fix an integer, $h\ge j$. Since ${\sigma }_h\left(X\right)$ is a cone with vertex containing E (Remark 8), we have $\dim {\sigma }_h\left(X\right)=e+1+\dim {\sigma }_h\left(Y\right)$. Hence, $g\left(Y\right)=g\left(X\right)$. Set $m:=\dim Y$. The definition of linear projection gives that, for each $p\in X_{\mathrm{reg}}$ the linear space, $T_{p}X\cap E$ has dimension at least $n-m$. Take a positive integer, x. If $x>\lfloor (e+1)/(n-m+1)\rfloor$, then x$(n-m)$-dimensional linear subspaces of E cannot be linearly independent. Hence, Theorem 2 follows from the Terracini Lemma ([1], Cor. 1.10). □

Remark 13. 

The assumption $n>m$ in the second part of Theorem 2 is satisfied in the cases we are interested in, i.e., the ones quoting [1], Prop. 2.1, by Proposition 1.

Proof of Theorem 3. 

Set $Y:=\overline{{\ell }_E(X\setminus X\cap E)}\subset {\mathbb{P}}^{r-e-1}$. Obviously, $g\left(Y\right)\ge \lceil (r-e)/(m+1)\rceil$. Lemma 1 gives $g\left(Y\right)\le g\left(X\right)$. If $\lceil (r-e)/(m+1)\rceil >\lceil (r+1)/(n+1)\rceil$, then X is secant-defective. If $\lceil (r-e)/(m+1)\rceil >\lfloor {\displaystyle \frac{r+1}{n+1}}+(n+2)/2\rfloor$, then ♠ fails. □

The following result may be used to prove that a certain cone does not exist.

Theorem 4. 

Let $X\subset {\mathbb{P}}^r$ such that there is an integer $i\ge 3$ such that $0<a_i\left(X\right)=a_{i+1}\left(X\right)\le n$. Assume that $i-1$ is the minimal integer j, such that ${\sigma }_j\left(X\right)$ is cone. Let E be the vertex of ${\sigma }_{i-1}\left(X\right)$. Set $e:=\dim E$ and let Y denote the closure of the irreducible constructible set ${\ell }_E(X\setminus X\cap E)$ in ${\mathbb{P}}^{n-e-1}$. We have $X\cap E\ne \varnothing$ and $g\left(X\right)=g\left(Y\right)$. Set $m:=\dim Y$. If $({\mathbb{P}}^{r-e-1},Y)$ satisfies ♡, then $g\left(Y\right)\le \lfloor {\displaystyle \frac{r-e}{m+1}}+(m+2)/2\rfloor$.

(i) 

Assume $\dim {\sigma }_{i-1}\left(X\right)=(n+1)(i-1)-1$ and $\dim {\sigma }_{i-1}\left(Y\right)=(i-1)(m+1)-1$. Then $(i-1)(n-m)=e+1$.

(ii) 

Assume $\dim {\sigma }_{i-1}\left(X\right)=(n+1)(i-1)-1$ and $\dim {\sigma }_{i-1}\left(Y\right)\le (i-1)(m+1)-2$. Then $(i-1)(n-m)\ge e+2$.

(iii) 

Assume $\dim {\sigma }_{i-1}\left(X\right)\le (n+1)(i-1)-2$. Then $a_i\left(X\right)\le n-1$ and $m\le n-2$.

Proof. 

Set $m:=m\left(E\right)$. Recall that $m=a_i\left(X\right)-1<n$ and that $g\left(X\right)=g\left(Y\right)$ (Proposition 1). Since $a_{i+1}\left(X\right)=a_i\left(X\right)>0$, ${\sigma }_i\left(X\right)\le {\mathbb{P}}^r$. Recall that $a_x\left(X\right)\ge a_{x+1}\left(X\right)$ for all x.

For all $j\ge 1$, the variety ${\sigma }_j\left(Y\right)$ is the closure in ${\mathbb{P}}^{r-e-1}$ of the irreducible constructible set ${\ell }_E({\sigma }_j\left(X\right)\setminus {\sigma }_j\left(X\right)\cap E)$ (Lemma 1). Since ${\sigma }_h\left(X\right)$ is a cone with vertex containing E, for all $h\ge i-1$, we have $\dim {\sigma }_h\left(X\right)=e+1+\dim {\sigma }_h\left(Y\right)$. Hence, $g\left(Y\right)=g\left(X\right)\ge i+1$. Now assume that $({\mathbb{P}}^{r-e-1},Y)$ satisfies ♡. In this case, $g\left(Y\right)\le \lfloor (r-e)/(m+1)+(m+2)/2\rfloor$ (Remark 6).

(a)

Assume $\dim {\sigma }_{i-1}\left(X\right)=(n+1)(i-1)-1$. First assume that ${\sigma }_{i-1}\left(Y\right)$ has the expected dimension, i.e., assume $\dim {\sigma }_{i-1}\left(Y\right)=(i-1)(m+1)-1$. We obtain $(i-1)(n-m)=e+1$. Now assume $\dim {\sigma }_{i-1}\left(Y\right)\le (i-1)(m+1)-2$. We obtain $(i-1)(n-m)\ge e+2$.

(b)

Assume $\dim {\sigma }_{i-1}\left(X\right)\le (n+1)(i-1)-2$. Let j be the minimal integer such that $\dim {\sigma }_j\left(X\right)\le n+1)j-2$. We have $2\le j\le i-1$. The integer j is the minimal integer such that $a_{j+1}\left(X\right)\le n$. Since the sequence ${\left\{a_h\left(X\right)\right\}}_{h\ge 2}$, satisfies $a_h\left(X\right)\ge a_{h+1}\left(X\right)$, $\dim {\sigma }_j\left(X\right)=(n+1)(j-1)+a_j\left(X\right)$. Since ${\sigma }_{i-1}\left(X\right)$ is the first secant variety which is a cone, [7], Prop. 2.1, gives $a_{j+1}\left(X\right)<a_j\left(X\right)$, and the sequence $a_h\left(X\right)$ is strictly decreasing for $j<h\le i-1$. Hence, $a_i\left(X\right)\le n-1$ and $m=a_i\left(X\right)-1\le n-2$.

□

Remark 14. 

Let $X\subset {\mathbb{P}}^r$ such that there is an integer $i\ge 3$ such that $0<a_i\left(X\right)=a_{i+1}\left(X\right)\le n$. Assume that $i-1$ is the minimal integer j such that ${\sigma }_j\left(X\right)$ is cone. Let E be the vertex of ${\sigma }_{i-1}\left(X\right)$. Set $e:=\dim E$ and let Y denote the closure of ${\ell }_E(X\setminus X\cap E)$ in ${\mathbb{P}}^{n-e-1}$. Assume $\dim Y=n$. Recall that $g\left(X\right)=g\left(Y\right)$. Since $\dim {\sigma }_{i-1}\left(X\right)=e+1+\dim {\sigma }_{i-1}\left(Y\right)$, ${\sigma }_{i-1}\left(Y\right)$ is defective.

Remark 15. 

Let $X\subset {\mathbb{P}}^r$ such that there is an integer $i\ge 3$ such that $0<a_i\left(X\right)=a_{i+1}\left(X\right)\le n$. Assume that $i-1$ is the minimal integer j such that ${\sigma }_j\left(X\right)$ is cone. Let E be the vertex of ${\sigma }_{i-1}\left(X\right)$. Set $e:=\dim E$ and let Y denote the closure of ${\ell }_E(X\setminus X\cap E)$ in ${\mathbb{P}}^{n-e-1}$. We have $X\cap E\ne \varnothing$ and $g\left(X\right)=g\left(Y\right)$. Set $m:=\dim Y$. Assume that $({\mathbb{P}}^{r-e-1},Y)$ satisfies ♡ and that there is an integer $x<i$ such that $n\ge a_x\left(X\right)>a_i\left(X\right)$ and call j the first such an integer. Since $i-1$ is the first integer such that ${\sigma }_{i-1}\left(X\right)$ is a cone and $a_j\left(X\right)\le n$, the function $a_x\left(X\right)$ is strictly decreasing for the integers $j\le x\le i$, we obtain $a_i\left(X\right)\le n-i+j$ and, hence, $m\le n-j+i$.

4. A Group Acts

Take an algebraic group, $G\subseteq \mathrm{Aut}({\mathbb{P}}^r,X)$. Let $\mathcal{F}(X,G)$ denote the set of all linear subspaces $E⊊{\mathbb{P}}^r$ such that $E\ne \varnothing$ and $g\left(E\right)=E$ for all $g\in G$. Let $\mathcal{G}(X,G)$ denote the set of all $E\in \mathcal{F}(X,G)$ such that $E\cap X\ne \varnothing$ and $\dim {\ell }_E(X\setminus X\cap E)<n$. If $\mathcal{G}(X,G)=\varnothing$, then $({\mathbb{P}}^r,X)$ satisfies ♡ by Theorem 1.

From now on, we assume $\mathcal{G}(X,G)\ne \varnothing$. An element $E\in \mathcal{G}(X,E)$ is said to be very critical if it is a minimal element of $\mathcal{G}(X,G)$ and $m\left(F\right)=m\left(E\right)$ for all $F\in \mathcal{G}(X,G)$ containing E.

Remark 16. 

Take $E,F\in \mathcal{G}(X,G)$ such that $F\supset E$. We have $m\left(E\right)>m\left(F\right)$ by Proposition 1. Hence, if a minimal element M of $\mathcal{G}(X,G)$ has $m\left(M\right)=n-1$, then M is very critical.

Lemma 2. 

Take a very critical $E\in \mathcal{G}(X,E)$. Set $e:=e\left(E\right)$. Let $Y\subset {\mathbb{P}}^{r-e-1}$ denote the closure of ${\ell }_E(X\setminus X\cap E)$. Then $({\mathbb{P}}^{r-e-1},Y)$ satisfies ♡.

Proof. 

Set $m:=m\left(E\right)$. Assume that $({\mathbb{P}}^{r-e-1},Y)$ does not satisfy ♡. Hence, there is an integer $i\ge 2$ such that $a_i\left(Y\right)=a_{i+1}\left(Y\right)\le m$ and ${\sigma }_{i+1}\left(Y\right)\le m$. We may take as i the minimum of such an integer. By [7], Prop. 2.1(ii), there is a linear space $M\subset {\mathbb{P}}^{r-e-1}$ such that ${\sigma }_{i-1}\left(Y\right)$ is a cone with vertex M. Let $F\supset E$ denote the linear subspace such that ${\ell }_E(F\setminus E)=M$. Since $\dim {\ell }_F(Y\setminus Y\cap M)<m$, $\dim {\ell }_F(X\setminus X\cap F)=\dim {\ell }_F(Y\setminus Y\cap M)<m$. Since ${\ell }_E$ commutes with the action of G, $G\subseteq \mathrm{Aut}({\mathbb{P}}^{r-e-1},Y)$. Hence $GF=F$, contradicting the assumption that E is very critical. □

Remark 17. 

Take ${\mathbb{P}}^r$, X and G such that $\mathcal{G}(X,G)\ne \varnothing$. Since $GE=E$ and $GX=X$, $G(X\cap E)=X\cap E$, i.e., $X\cap E$ is a union of orbits. From the examples described in [8,9], we see that not much can be said if there is $E\in \mathcal{G}(X,G)$ with $t\left(E\right)=n-1$. Recall that $m\left(E\right)\ge n-1-t\left(E\right)$ for all $E\in \mathcal{G}(X,G)$ (Proposition 1). There is one case in which everything is easier, when X has finitely many orbits, an open orbit while the other ones are finite sets (remember that G may be not connected). In this case, $m\left(E\right)=n-1$ for every $E\in \mathcal{G}(X,G)$. Remark 16 gives that every element of $\mathcal{G}(X,G)$ is very critical. Hence, we may apply Proposition 1 and Lemma 2 to each element of $\mathcal{G}(X,G)$.

Remark 18. 

Assume that ${\mathbb{P}}^r$ is direct sum of 2 irreducible projective representation. If $\mathcal{G}(X,G)=\varnothing$, then $({\mathbb{P}}^r,X)$ satisfies ♣ by Theorem 1 and Remark 5. Now assume $\mathcal{G}(X,G)\ne \varnothing$. Then any element of $\mathcal{G}(X,G)$ is very critical. Hence, we may apply Proposition 1 and Lemma 2 to each element of $\mathcal{G}(X,G)$.

Remark 19. 

Take $E\in \mathcal{F}(X,G)$ and $F\in \mathcal{G}(X,G)$. If $F\subset E$, then $E\in \mathcal{G}(X,G)$.

Remark 19 shows that it is interesting to look at the minimal elements of $\mathcal{G}(X,G)$. For each $E\in \mathcal{G}(X,G)$, let $e\left(E\right)$ denote the integer $\dim {\ell }_E(X\setminus X\cap E)$.

Theorem 5. 

Let G be a connected linear group acting transitively on the projective manifold X and $X⊊{\mathbb{P}}^r$ a G-equivariant non-degenerate projective embedding. Then $({\mathbb{P}}^r,X)$ satisfies ♡ and ♣.

Proof. 

It is sufficient to prove ♡ (Remark 5). Hence, we may assume the existence of a minimal positive integer a such that ${\sigma }_a\left(X\right)\ne {\mathbb{P}}^r$ and ${\sigma }_a\left(X\right)$ is a cone. Since $X\ne {\mathbb{P}}^r$ and X is homogeneous, X is not a cone. Hence, $a\ge 2$. Let E be the vertex of ${\sigma }_a\left(X\right)$. Since E is the vertex of the G-invariant variety ${\sigma }_a\left(X\right)$, E is G-invariant. Hence, $E\cap X$ is G-invariant. Since X is non-degenerate and ${\sigma }_a\left(X\right)\ne {\mathbb{P}}^r$, $X⊈E$. Since X is homogeneous and $E\cap X$ is G-invariant, $E\cap X=\varnothing$. Proposition 1 gives a contradiction. □

Remark 20. 

Suppose that X has finitely many orbits. Let ${\tau }_1$ be the set of all proper orbits and τ the set of all finite unions of proper orbits. Each orbit A is locally closed in X and its closure $\overline{A}$ in X is the union of A and finitely many lower dimensional orbits. These sets are easy to describe. Call ${\mathcal{F}}_1$ the set of all linear spaces $\langle A\rangle$ spanned by some orbit A. Note that $\langle \overline{A}\rangle =\langle A\rangle$. Again, the set ${\mathcal{F}}_1$ is finite, and one could test all of them. However, in general, if ${\sigma }_{i-1}\left(X\right)$ is a cone with vertex M, then $M\ne \langle X\cap M\rangle$ (this is obvious if $M\cap X$ is a single point, but X is not a cone). Hence, we cannot use Proposition 1 to test the elements of ${\mathcal{F}}_1$. We apply Remark 14. The pair $({\mathbb{P}}^{r-1},Y)$ is G-invariant.

5. Conceptual Tests

First, a disclaimer. We do not have any test that, given a subvariety X inside ${\mathbb{P}}^r$ by an explicit set of equations, says whether or not a proper secant variety of X is a cone. We assume that we have to test a finite family $\mathrm{\Gamma }$ of linear subspaces of ${\mathbb{P}}^r$. We know that this the case if G acts on X and ${\mathbb{P}}^r=\mathbb{P}\left(V\right)$ with V a direct sum of pairwise irreducible representations (Remark 18).

At the minimum (even without knowing anything about X, except that it has dimension n), we must be able to know all integers $\dim F$, $F\in \mathrm{\Gamma }$. The set $\mathrm{\Gamma }$ becomes a partially ordered set using the inclusion for a partial order. Each vertex $F\in \mathrm{\Gamma }$ is equipped with a number, the integer $\dim F$. We write $(\mathrm{\Gamma },\preccurlyeq )$ for this “decorated” partially ordered set. We say that a chain $F_1⊊F_2⊊\dots ⊊F_s$ of different elements has a length s and that it is maximal if $(\mathrm{\Gamma },\preccurlyeq )$ has no longer chains with $F_1,\dots ,F_s$ as some of its members. If $\mathrm{\Gamma }=\mathcal{G}(X,G)$ maximal chains of length 1 corresponds to elements which are both minimal and maximal ) and described as in Proposition 1 and Lemma 2 (see Remarks 17 and 18). For longer chains, use Proposition 2 and Remark 11.

If ♣ fails, then there is a unique chain associated to the flat parts of the sets $\left\{a_i\left(X\right)\right\}$ (it may not be maximal, because $\mathrm{\Gamma }$ may contains several not-relevant linear subspaces). Knowing this important chain and the integers $m\left(E\right)$ and $\dim E$ for the elements of this chain are almost equivalent to knowing all integers $\left\{a_i\left(X\right)\right\}$, as described in Remark 1, applied to all the steps (Remark 11).

Recall that, in each step, the new variety $\overline{{\ell }_E(X\setminus X\cap E)}$ has the same generic rank, $g\left(X\right)$, as X. Hence, if we may describe $\overline{{\ell }_E(X\setminus X\cap E)}$, we obtain $g\left(X\right)$.

From now on, we assume that, for each $F\in \mathrm{\Gamma }$, we are able to describe $\dim (X\cap F)$ and $\dim {\ell }_{E|X\setminus S\cap E}$. With these assumptions, we may decrease $\mathrm{\Gamma }$ and assume that all $F\in \mathrm{\Gamma }$ satisfy the 4 conditions of Theorem 1. We take a single maximal chain in $\mathrm{\Gamma }$. At each step, the generic range does not change. Assume that, at the last step (hence with a lower dimensional variety $Y_s$), we are able to compute the generic rank $g\left(Y_s\right)$, and then we know that $g\left(Y_s\right)=g\left(X\right)$ (Lemma 1). If $g\left(X\right)>\lceil (r+1)/(n+1)$, then X is secant-defective. If $g\left(X\right)>\lfloor {\displaystyle \frac{r+1}{n+1}}+{\displaystyle \frac{n+2}{2}}\rfloor$, then $({\mathbb{P}}^r,X)$ does not satisfies ♠ (Remark 6).

6. Further Remarks

Proposition 3. 

Take $({\mathbb{P}}^r,X)$ with property ♣ and take a general linear subspace $V\subset {\mathbb{P}}^r$ of codimension x with $n+1\le x\le n(n+1)/2$. Then $V\cap X=\varnothing$ and $c_1\left({\ell }_V\left(X\right)\right)=\lfloor x/(n+1)\rfloor$. If $\lfloor x/(n+1)\rfloor +1\le (r-n(n+1)/2)/(n+1)$, then ${\ell }_V\left(X\right)$ is not secant-defective.

Proof. 

Since V is general and $x>n$, $V\cap X=\varnothing$. Hence, ${\ell }_{V|X}:X\to {\ell }_V\left(X\right)$ is a finite morphism. The Terracini Lemma implies that, if $(i+1)n\le x$, the variety ${\sigma }_i\left({\ell }_V\left(X\right)\right)$ has the expected dimension $(n+1)i-1$ if, and only if, $\dim {\sigma }_i\left(X\right)=(n+1)i-1$ has the expected dimension. Since $({\mathbb{P}}^r,X)$ satisfies ♣, $(n+1)c_1\left(X\right)\ge r-n(n+1)/2$ (Remark 7). Hence, $c_1\left(Y\right)=\lfloor x/(n+1)\rfloor$. Hence, X is not secant-defective if, and only if, ${\sigma }_{c_1\left(Y\right)+1}\left(Y\right)={\mathbb{P}}^{x-1}$. By the generality of V, this is the case if, and only if, $\dim {\sigma }_{c_1\left(Y\right)+1}\left(X\right)\ge x$. This inequality is true if $c_1\left(Y\right)+1\le c_1\left(X\right)$, which is true if $\lfloor x/(n+1)\rfloor +1\le (r-n(n+1)/2)/(n+1)$ (Remark 7), because the left hand of the inequality is an integer. □

Remark 21. 

To test the thesis of Proposition 3, it is sufficient to test one single codimension x linear space V with the only restriction that $\dim \left({\ell }_V(X\setminus X\cap V)\right)=n$.

Proposition 4. 

Fix integral and non-degenerate varieties $X\subset {\mathbb{P}}^r=\mathbb{P}\left(V\right)$, $Y\subset {\mathbb{P}}^s=\mathbb{P}\left(W\right)$ and algebraic groups $G\subseteq SL\left(V\right)$, $GX=X$, $H\subseteq SL\left(W\right)$, $HY=Y$. Let $\nu :{\mathbb{P}}^r\times {\mathbb{P}}^s\to {\mathbb{P}}^{rs+r+s}$ denote the Segre embedding. See $G\times H$ as an algebraic group acting on $X\times Y$, ${\mathbb{P}}^{rs+r+s}$ and on $\nu (X\times Y)$. Assume that one of the following set of conditions is satisfied:

1. 

V is an irreducible G-representation and W an irreducible H-representation.

2. 

X is G-homogeneous and Y is H-homogeneous.

3. 

V an irreducible G-representation, G and H are reductive, and Y is H-homogeneous.

4. 

G and H are reductive, V is an irreducible representation of G, and every $F\in \mathcal{F}(Y,H)$ is base-point-free.

Set $U:=\nu (X\times Y)$. Then $({\mathbb{P}}^{rs+r+s},U)$ satisfies ♡ and ♣.

Proof. 

Let ${\pi }_2:{\mathbb{P}}^r\times {\mathbb{P}}^s\to {\mathbb{P}}^s$ denote the projection onto the second factor. By Remark 5, it is sufficient to prove the first statement. Assume that a proper secant variety of U is a cone. Call E the vertex of this cone and i the first integer such that ${\sigma }_{i-1}\left(U\right)$ is a cone with vertex E. Since $G\times H\left(U\right)=U$, we have $G\times H\left(E\right)=E$.

If X is G-homogeneous and Y is H-homogeneous, then $\nu (X\times Y)$ is $G\times H$-homogeneous. In this case, we use Theorem 5.

Note that, in the Segre embedding, ${\mathbb{P}}^{rs+r+s}=\mathbb{P}(V\otimes W)$. If V is an irreducible G-representation and W is an irreducible H-representation, then $V\otimes W$ is an irreducible representation for the group $G\times H$, and, hence, it is sufficient to use [4].

Now assume that V is an irreducible G-representation, that Y is H-homogeneous, and that W is a finite sum of irreducible representations, which is always true if H is reductive. Write $E:=\mathbb{P}\left(E_1\right)$ with $E_1$ a linear subspace of $V\otimes W$. Since V is irreducible and W is a finite direct sum of finitely many irreducible representations, $E_1=V\otimes F_1$ with $F_1$ an H-invariant linear subspace of W. Set $F:=\mathbb{P}\left(F_1\right)$. By Proposition 1, we have $U\cap E\ne \varnothing$. Take $\nu (a,b)\in E\cap U$. Since $G\times H(E\cap U)=E\cap U)$, $\nu \left(\right\{a\}\times Y)\subseteq E\cap U$. Since Y is non-degenerate, $Y⊈F$ is a contradiction.

Now assume that V is an irreducible representation, that G and H are reductive, and that each $F\in \mathcal{F}(Y,H)$ satisfies $F\cap Y=\varnothing$. Write $E=\mathbb{P}\left(E_1\right)$ with $E_1$ a linear subspace of $V\otimes W$. The group $G\times H$ is reductive. Thus, $V\otimes W$ is a direct sum of irreducible representations. Since V is irreducible, $E_1=V\otimes F_1$ for some linear subspace $F_1\subset W$. Set $F:=\mathbb{P}\left(F_1\right)$. Since $G\times H\left(E\right)=E$, $F\in \mathcal{F}(Y,H)$. Since $F\cap Y=\varnothing$, $E\cap U=\varnothing$, contradicting Theorem 1. □

The following example shows that there are pairs $({\mathbb{P}}^r,X)$, $({\mathbb{P}}^s,Y)$ homogeneous with respect to a connected and semisimple groups, with embedding associated with irreducible representations, X not secant-defective, Y not secant-defective, but with their Segre product secant-defective. Of course, their Segre embedding satisfies ♣, e.g., by [4] or by Proposition 21.

Example 2. 

Take $r=s=3$, $X=Y={\mathbb{P}}^1\times {\mathbb{P}}^1$, $G=H=SL\left(2\right)\times SL\left(2\right)$ and ${\mathbb{P}}^1\times {\mathbb{P}}^1$ embedded by Segre embedding. X and Y are not defective, but $X\times Y$ is defective ([16,17], Th. 3.1 ).

We explain our main reason to consider the set-up of Proposition 4.

Remark 22. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Fix an integer $s>0$ and set $U:=X\times {\mathbb{P}}^s$ embedded in ${\mathbb{P}}^{rs+r+s}$ by the composition of the inclusion $X\subset {\mathbb{P}}^r$ and the Segre embedding $\nu :{\mathbb{P}}^r\times {\mathbb{P}}^s\to {\mathbb{P}}^{rs+r+s}$. There is an equality between the U-rank (resp. border U-rank) of a certain element of ${\mathbb{P}}^{rs+r+s}$ and the simultaneous X-rank (resp. simultaneous border X-rank) of s elements of ${\mathbb{P}}^r$ ([18], Th. 2.5, [19], Lemma 2.4). Write ${\mathbb{P}}^s=\mathbb{P}\left(W\right)$ with W an irreducible representation of the group $H:=SL(s+1)$. Hence, if $({\mathbb{P}}^r,X)$ is G-invariant, we may apply Proposition 4 with $H=SL(s+1)$, Y homogeneous and W an irreducible H-representation.

7. Toric Varieties

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate toric variety [20]. Set $n:=\dim X$ and assume $n\ge 2$. Set $G:={\left({\mathbb{C}}^{*}\right)}^n$. G acts on X with finitely may orbits, and it acts on ${\mathbb{P}}^r$. Since G is abelian, each finite-dimensional complex representation of G is a direct sum of irreducible one-dimensional representations. Most of the results for the secant varieties of toric varieties are only for ${\sigma }_2\left(X\right)$ [21] and study more refined results, e.g., their degrees and homogeneous ideals [21,22]. There are examples for which ♣ fails [8,9]. We give some suggestions for testing if ♣ holds.

Let $T_X$ be the set of all closed subsets T of X with $\dim \langle T\rangle \le r-n-1$, which are a union of finitely many orbits, i.e., which are finite unions of closures of G-orbits. Fix a positive integer $a<n$. We say that $T\in T_X$ satisfies $A\left(a\right)$ if, for each 0-dimensional orbit $u\in X\setminus T$, there is a closure C of an a-dimensional orbit, such that $u\in C$ and $\dim C\cap T=a-1$. Let $T_X\left(a\right)$ denote the set of all $T\in T_X$ satisfying $A\left(a\right)$. Set $T_X(\star ):={\cup }_{a=1}^{n-1}{T}_X\left(a\right)$.

Theorem 6. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate toric variety for which ♣ fails. Fix $E\in \mathcal{E}\left(X\right)$. Then there is $T\in T_X(★)$, such that $\langle T\rangle \subseteq E$. Moreover, if $\dim {\ell }_E(X\setminus X\cap E)=n-b$, then $T\in {\cap }_{a=1}^b{T}_X\left(a\right)$.

Proof. 

Set $e:=\dim E$. Since $GX=X$, $GE=E$. By Theorem 1, X intersects E and $\dim {\ell }_E(X\setminus X\cap E)<n$. Hence, $b\ge 1$. Set $T:=X\cap E$ (set-theoretic intersection). Since $GE=E$ and $GX=X$, $GT=T$, i.e., T is a finite union of open orbits of X. Since X and E are closed, T is closed in ${\mathbb{P}}^r$, and hence it is a finite union of closures of G-orbits. Set $F:=\langle T\rangle$. Since $GF=F$ and $F\subseteq E$, to prove the theorem, it is sufficient to prove that $F\in T_X\left(a\right)$ for all $1\le a\le b$. Take any 0-dimensional G-orbit $u\in X\setminus T$. Let ${\ell }_E:{\mathbb{P}}^r\setminus E\to {\mathbb{P}}^{r-e-1}$ denote the linear projection from E. Let $Y\subset {\mathbb{P}}^{r-e-1}$ denote the closure of ${\ell }_E(X\setminus T)$. The restriction of ${\ell }_E$ to $X\setminus T$ induces a morphism $\mu :X\setminus T\to Y$. Since $GE=E$, Y is toric and $\mu$ is a toric morphism. By the definitions of E and b we have $\dim Y=n-b$. Since $u\notin T$, ${\ell }_L\left(u\right)=\mu \left(u\right)$ is a well-defined point of Y. Since a general fiber of $\mu$ has dimension $n-b$, every irreducible component of ${\mu }^{-1}\left(\mu \left(u\right)\right)$ containing u has dimensions at least $n-b$ ([14], Ex. II.3.22). Let $C_1\subseteq X\setminus T$ be an irreducible component of ${\mu }^{-1}\left(\mu \left(u\right)\right)$ containing u. Let C be the closure of $C_1$ in X. The set $C_1$ is a quasi-projective toric variety, and C is a closed toric subvariety of X.

(a)

In this step, we prove that $\dim C\cap T=b-1$, i.e., $T\in T_X\left(b\right)$. Assume $c:=\dim C\cap T\le b-2$. Take a general linear subspace $V\subset {\mathbb{P}}^r$ containing u such that $\dim V=r-c+1$. The generality of V and the irreducibility of C gives that each irreducible component of $C\cap V$ has dimension 1. Let J be an irreducible component of $C\cap V$ containing u. Since $u\notin C\cap T$ and V is general, $V\cap C\cap T=\varnothing$, and, in particular, $J\cap T=\varnothing$. Since $J\subset X$ and $X\cap E=T$, $E\cap J=\varnothing$. Hence, ${\ell }_{E|J}:J\to {\mathbb{P}}^{r-e-1}$ is a finite map, contradicting the assumption that $\mu \left(J\right)=\mu \left(u\right)$.

(b)

Step (a) proves the case $a=b$. Now assume $a<b$. Since $a>0$ by assumption, $b\ge 2$. Let $W\subset {\mathbb{P}}^r$ be a general codimension $b-a$ linear subspace of ${\mathbb{P}}^r$ containing u. Instead of C, we take $C\cap W$. Since $\dim C\cap T=b-1$, we have $\dim C\cap T\cap W=a-1$. Since $u\in W$, $u\in C\cap W$. Hence, we may apply the proof of step (a) to the variety $C\cap W$.

□

Obviously, Theorem 6 implies the following corollary.

Corollary 1. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate toric variety. If $T_X\left(1\right)=\varnothing$, then X satisfies ♣.

Then we fix an integer $a\ge 2$ and study the properties of a toric variety $X\subset {\mathbb{P}}^r$ such that $T_X\left(a\right)=\varnothing$ and $T_X(a-1)\ne \varnothing$. Note that $n:=\dim X>a$.

Proposition 5. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate toric variety. Set $n:=\dim X$. Assume the existence of an integer a such that $n>a\ge 2$ and $T_X\left(a\right)=\varnothing$. If there is a positive integer i such that $a_i\left(X\right)\le n+1-a$, then:

(a) 

$a_{j+1}\left(X\right)\le \max \{0,a_{j}X)-1\}$ for all $j\ge i$;

(b) 

We have $g\left(X\right)\le i+a_i\left(X\right)$ and $r-\dim {\sigma }_i\left(X\right)\le a_i\left(X\right)(a_i\left(X\right)+1)/2$.

Proof. 

Part (a) follows from Theorem 6. Part (b) follows from part (a). □

We say that X is a pseudocone if there is a closed toric subvariety $T⊊X$ such that $\dim \langle T\rangle \le r-n-1$ and every closed one-dimensional orbit $J\subset X$ meets T. If $T⊊X$ makes X a pseudocone, we say that T is a pseudovertex.

Theorem 7. 

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate toric variety for which ♣ fails. Then X is a pseudocone. Moreover, for each $E\in \mathcal{E}\left(X\right)$, there is a pseudovertex F, such that $E\supseteq F$, $E\cap X=F\cap X$ (set-theoretically) and $GE=E$.

Proof. 

By assumption, $\mathcal{E}\left(X\right)\ne \varnothing$. Set $e:=\dim E$. Since $GX=X$, $GE=E$. By === X intersects E. Set $T:=X\cap E$ (set-theoretic intersection). Since $GE=E$ and $GX=X$, $GT=T$, i.e., T is a finite union of open orbits of X. Since X and E are closed, T is closed in ${\mathbb{P}}^r$, and, hence, it is a finite union of closures of G-orbits. Set $F:=\langle T\rangle$. Since $GF=F$ and $F\subseteq E$, to prove the theorem, it is sufficient to prove that each one-dimensional orbit has a closure C intersecting T. Theorem 6 gives $T\in T_X\left(1\right)$, and, hence, $C\cap T\ne \varnothing$. □

Proposition 6. 

Take two different coordinates, say $x_i$ and $x_j$, with $i\ne j$. Then $x_i$ and $x_j$ correspond to non-isomorphic irreducible representations of G.

Proof. 

The variables $x_i$ and $x_j$ correspond to different characters, say ${\chi }_1$ and ${\chi }_2$, of $M\otimes \mathbb{C}$ by the definition of toric embedding ([20], Prop. 4.3.2). Every character of a group G is an irreducible representation of G because it has dimension 1. The identity and the inverse $t\mapsto t^{-1}$ are the only automorphism of ${\mathbb{C}}^{*}$ as an algebraic variety. Hence, to prove the lemma, it is sufficient to prove that ${\chi }_2\ne {\chi }_1^{-1}$. This is true, because, at most, one among ${\chi }_1$ and ${\chi }_2$ is the trivial character, ${\mathcal{O}}_X\left(1\right)$ is a very ample line bundle on the projective variety X, and X is embedded in ${\mathbb{P}}^r$ by the complete linear system $|{\mathcal{O}}_X\left(1\right)|$. □

8. How to Use ♣ with the Differential Horace Lemma

We know ♣ for the Segre–Veronese embeddings of all multiprojective spaces because their embeddings are associated to irreducible representations [4,6]. These varieties are homogeneous spaces and, hence, one could quote Theorem 5. In this section, we show how two use ♣ to obtain results not covered in [5,6]. Remark 22, i.e., the simultaneous rank and simultaneous border rank of degree d forms in $n+1$ variables, gives one of the motivations for the following result.

Theorem 8. 

Fix integers $m>0$, $n\ge 3$ and $d\ge 3$. Set $X={\mathbb{P}}^m\times {\mathbb{P}}^n$ and take $H\in |{\mathcal{O}}_X(0,1)|$. Set $e:=\lfloor (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right)/(m+n)\rfloor$ and $f:=(m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right)-(m+n)e$. Assume that ${\mathcal{O}}_H(1,d)$ is not defective and that the following numerical conditions hold:

$$(2n+2m-1)(n+m)\le (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-2}{n-1}}\right);$$

(1)

$$(n+m)\left[n+m+(n+m+1)(n+m+2)/2\right]\le (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right);$$

(2)

$$\begin{array}{cc}\hfill & (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d}{n}}\right)-n-m-(n+m+1)(e+f)\ge \hfill \\ \hfill & (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-2}{n}}\right)+(n+m+1)(n+m)/2.\hfill \end{array}$$

(3)

Then ${\mathcal{O}}_X(1,d)$ is not secant-defective.

Note that, in the statement of Theorem 8, we have $0\le f\le m+n-1$, while $e\sim (m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right)/(n+m)$. Hence, for fixed n and m, the left hand side of (3) has order $d^n$, while the right hand side has order $d^{n-2}$. For any smooth variety M and $p\in M$, let $(2p,M)$ denote the closed subscheme of M with ${\left({\mathcal{I}}_{p,M}\right)}^2$ as its ideal sheaf. The scheme $(2p,M)$ is zero-dimensional, $\mathrm{deg}\left(\right(2p,M\left)\right)=\dim M+1$ and ${(2p,M)}_{\mathrm{red}}=\left\{p\right\}$. For any finite set $S\subset M$, set $(2S,M):={\cup }_{p\in S}(2p,M)$. For any positive integer x, let $S(M,x)$ denote the set of all subsets of M with cardinality x. The set $S(M,x)$ is an irreducible quasi-projective variety. If $M\subset {\mathbb{P}}^r$ and S is general in $S(M,x)$, then the Terracini Lemma gives $\dim {\sigma }_x\left(M\right)=\dim \langle (2S,M)\rangle$ ([1], Cor. 1.11). Write $2p$ and $2S$ instead of $(2p,X)$ and $(2S,X)$, respectively.

Remark 23. 

Let $Z\subset W\subset {\mathbb{P}}^r$ be zero-dimensional schemes. We have $\langle Z\rangle \subseteq \langle W\rangle$ and $\dim \langle W\rangle \le \dim \langle Z\rangle +\mathrm{deg}\left(W\right)-\mathrm{deg}\left(Z\right)$. Hence, if $\dim \langle W\rangle =\mathrm{deg}\left(W\right)-1$, i.e., W is linearly independent, then $\dim \langle Z\rangle =\mathrm{deg}\left(Z\right)-1$, i.e., Z is linearly independent. Obviously, if $\langle Z\rangle ={\mathbb{P}}^r$, then $\langle W\rangle ={\mathbb{P}}^r$.

The following lemma is the form of the Differential Horace Lemma [2,23], which we use in the proof of Theorem 8.

Lemma 3. 

Let X be an integral projective variety, $\mathcal{L}$ a line bundle on X, H an effective Cartier divisor such that $h^1\left(\mathcal{L}\right)=h^1\left(\mathcal{L}(-H)\right)=h^1\left(\mathcal{L}(-2H)\right)=0$, and E a zero-dimensional scheme such that $E\cap H=\varnothing$. Set $c:=\dim X$. Take integers $x\ge 0$ and $y\ge 0$. Take a general $(S^{\prime },S^{″})\in S(H,x)\times S(H,x)$ and a general $S\subset X$ such that $\#S=x+y$.

(a) 

To prove that $h^0({\mathcal{I}}_{E\cup 2S}\otimes \mathcal{L})=h^0({\mathcal{I}}_E\otimes \mathcal{L})-(c+1)x$, it is sufficient to prove that $h^0(H,{\mathcal{I}}_{(2S^{\prime },H)\cup S^{″}}\otimes {\mathcal{L}}_{|H})=0$ and $h^0({\mathcal{I}}_{E\cup S^{\prime }\cup \left(2S^{\prime }H\right)}\otimes \mathcal{L}(-H))=0$.

(b) 

To prove that $h^1({\mathcal{I}}_{E\cup 2S}\otimes \mathcal{L})=h^1({\mathcal{I}}_E\otimes \mathcal{L})-(c+1)x$, it is sufficient to prove that $h^1(H,{\mathcal{I}}_{(2S^{\prime },H)\cup S^{″}}\otimes {\mathcal{L}}_{|H})=0$ and $h^1({\mathcal{I}}_{E\cup S^{\prime }\cup (2S^{″},H)}\otimes \mathcal{L}(-H))=0$.

Remark 24. 

In the statement of Lemma 3, there is a set $S^{\prime }$, which is general in H, but not in X. We would like to prove that it gives $\min \{\#S^{\prime },h^0({\mathcal{I}}_{E\cup (2S^{″},H)}\otimes \mathcal{L}(-H))\}$ independent conditions to the vector space $H^0({\mathcal{I}}_{E\cup (2S^{″},H)}\otimes \mathcal{L}(-H))$. Recall that $\dim H=c-1$ and, hence, the scheme $(2S^{″},H)$ has degree $yc$. We have ${\mathrm{Res}}_H(E\cup S^{\prime }\cup (2S^{″},H))=E$. Hence, it is sufficient to prove the inequality $h^0({\mathcal{I}}_E\otimes \mathcal{L}(-2H))\le \max \{0,h^0({\mathcal{I}}_E\otimes \mathcal{L}(-H))-\#S^{\prime }-yc\}$ ([6], Lemma 2.6).

Proof of Theorem 8 

We have $h^0(H,{\mathcal{O}}_H(1,d))=(m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right)$. Thus, we have $e=\lfloor h^0(H,{\mathcal{O}}_H(1,d))/(n+m)\rfloor$. Note that $0\le f\le n+m-1$.

By Remark 23, it is sufficient to prove that a general union Z of z double-points of X satisfies either $h^0\left({\mathcal{I}}_Z(1,d)\right)=0$ or $h^1\left({\mathcal{I}}_Z(1,d)\right)=0$, where

$$z\in \{\lfloor h^0\left({\mathcal{O}}_X(1,d)\right)/(n+m+1)\rfloor ,\lceil h^0\left({\mathcal{O}}_X(1,d)\right)/(n+m+1)\rceil \}.$$

We have $h^0\left({\mathcal{O}}_X(1,d)\right)=(m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d}{n}}\right)$ and $h^0\left({\mathcal{O}}_H(1,d)\right)=(m+1)\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-1}{n-1}}\right)$. Recall that $0\le f\le n+m-1$. Take a general $(S,S^{\prime })\in S(H,e)\times S(H,f)$. Since ${\mathcal{O}}_H(1,d)$ is not secant-defective, $h^i(H,{\mathcal{I}}_{(2S,H)\cup S^{\prime }}(1,d))=0$, $i=0,1$. Hence, the part of the assumptions of Lemma 3 related to H are satisfied.

Claim 1. 

write Claim 1 as originally done BALLICO $h^1(H,{\mathcal{I}}_{(2S^{\prime },H)}(d-1))=0$.

Proof of Claim 1. 

Since H is homogeneous for the group $SL(m+1)\times SL\left(n\right)$, Theorem 5 and Remark 1 show that it is sufficient to use that $f\le n+m-1$, $\dim H=n+m-1$ and that $(2n+2m-1)(n+m)\le h^0\left({\mathcal{O}}_H(1,d-1)\right)$, i.e., we use (1).

(a) Assume $z\ge e+f$. Let $Z^{\prime }\subset X$ be a general union of $z-e-f$ double-points of X. By Lemma 3, with $E:=Z^{\prime }$, it is sufficient to prove that either $h^0\left({\mathcal{I}}_{Z^{\prime }\cup S\cup (2S^{\prime },H)}(1,d-1)\right)=0$ or $h^1\left({\mathcal{I}}_{Z^{\prime }\cup S\cup (2S^{\prime },H)}(1,d-1)\right)=0$. □

Claim 2. write Claim 2 as I originally did BALLICO We have $h^1\left({\mathcal{I}}_{Z^{\prime }\cup (2S^{\prime },H)}(d-1)\right)=0$.

Proof of Claim 2. 

Recall that $(2S^{\prime },H)$ has degree $(m+n)f$. By Claim 1, $\langle (2S^{\prime },H)\rangle$ has the expected dimension $(m+n)f-1$, with the convention $\dim \langle \varnothing \rangle =-1$ if $f=0$. By Remark 1, it is sufficient to have

$$\mathrm{deg}\left(Z^{\prime }\right)+(m+n)f\le h^0\left({\mathcal{O}}_X(1,d-1)\right)-(n+m+1)(n+m+2)/2.$$

We have $(n+m)e+f=h^0\left({\mathcal{O}}_X\left(d\right)\right)-h^0\left({\mathcal{O}}_X(1,d)\right)$ and $(n+m+1)z\le h^0\left({\mathcal{O}}_X\left(d\right)\right)+n+m$. Since $\mathrm{deg}\left(Z^{\prime }\right)+(m+n)f=(n+m+1)z-(n+m+1)e+f$, it is sufficient to assume $e\ge n+m+(n+m+1)(n+m+2)/2$, i.e., the inequality (2). □

Claim 2 gives $h^0\left({\mathcal{I}}_{Z^{\prime }\cup (2S^{\prime },H}(d-1)\right)=h^0\left({\mathcal{O}}_X(1,d-1)\right)-\mathrm{deg}\left(Z^{\prime }\right)-(n+m)f$. Since $Z^{\prime }$ is general, ${\mathrm{Res}}_H(Z^{\prime }\cup (2S^{\prime },H)\cup S^{\prime })=Z^{\prime }$. Since $d\ge 3$, to get $h^0\left({\mathcal{I}}_{Z^{\prime }}(1,d-2)\right)=0$, it is sufficient use Remark 1 and (3). Claim 2, the generality of $S^{\prime }$, and Remark 24 prove that either $h^0\left({\mathcal{I}}_{Z^{\prime }\cup S\cup (2S^{\prime },H)}(1,d-1)\right)=0$ or $h^1\left({\mathcal{I}}_{Z^{\prime }\cup S\cup (2S^{\prime },H)}(1,d-1)\right)=0$.

(b) Assume $z\le e+f-1$. If $z\le e$, we use the proof of step (a) with $Z^{\prime }=\varnothing$, $S^{\prime }=\varnothing$, and $\#S=z$. If $e+1\le z\le e+f-1$, we use the proof of step (a) with $Z^{\prime }=\varnothing$, the same S, and, instead of $S^{\prime }$, a general element of $S(H,z-e)$. □

9. Discussion

Let $X\subset {\mathbb{P}}^r\left(\mathbb{C}\right)$ be a complex n-dimensional projective variety. For all positive integer i, let ${\sigma }_i\left(X\right)$ be the i-th secant variety of X. For certain embedded varieties X, the integers $\dim {\sigma }_i\left(X\right)$, $i\ge 1$, are important for real-life applications. For instance, if X is a Segre embedding of a multiprojective space, then they give the dimension of all tensors with a fixed format and of tensor rank i, while if $X\subset {\mathbb{P}}^r$ is the d-Veronese embedding of ${\mathbb{P}}^n$, then $\dim {\sigma }_i\left(X\right)$ is the dimension of the set of degree d forms in $n+1$ variables with additive rank i. We have $\dim {\sigma }_i\left(X\right)\le \min \{r,i(n+1)-1\}$, and the right hand side of this inequality shows the expected value of this dimension.

B. Ådlandsvik, A. T. Blomenhofer, and A. Casarotti proved that, if no secant variety of X is a cone, then almost all (all except, at most, $n+1$) have the expected dimension, and the other ones have dimensions very near to the expected one. Blomenhofer and Casarotti used this observation to shatter the previous records on the number of ${\sigma }_i\left(X\right)$ with the expected dimensions when (as in the case of tensors and partially symmetric tensors) ${\mathbb{P}}^r=\mathbb{P}\left(V\right)$ with G being an irreducible representation. Their insight was used by several other mathematicians to tackle the secant varieties associated with partially symmetric tensors.

In this paper, we extend the cases, e.g., for all homogenous embedding and for Segre products of 2 embeddings with respect to different algebraic groups. The proofs use algebraic geometry.

We say that ♡ is true if no proper secant variety of X is a cone. We prove that if ♡ fails, then we have finitely many linear projections to a smaller embedded variety $Y\subset {\mathbb{P}}^{r_1}\left(\mathbb{C}\right)$. A key invariant of the embedded variety $X\subset {\mathbb{P}}^r\left(\mathbb{C}\right)$ is its generic X-rank, the first integer such that ${\sigma }_{g\left(X\right)}\left(X\right)={\mathbb{P}}^r\left(\mathbb{C}\right)$. This is known as the generic X-rank in the cases used in the applications (the generic tensor rank for a fixed X). In each linear projection from a linear space E, we obtain an embedding $Y\subset {\mathbb{P}}^{r-e-1}\left(\mathbb{C}\right)$, e the dimension of E, with $g\left(Y\right)=g\left(X\right)$. Thus, if we are able to check $g\left(Y\right)$, we see if ♡ holds and/or how much X is secant-defective.

We consider toric varieties and the Segre–Veronese embeddings of ${\mathbb{P}}^m\times {\mathbb{P}}^n$ with respect to the line bundle ${\mathcal{O}}_{{\mathbb{P}}^m\times {\mathbb{P}}^n}(1,d)$, which is related to the simultaneous rank of degree d forms in $n+1$ variables.

If X is defined over $\mathbb{R}$, a result of Blekherman and Teitler, knowing $g\left(X\right)$ over $\mathbb{C}$ also suggests that the maximal $X\left(\mathbb{R}\right)$-rank is at most $2g\left(X\right)$.