Equations for secant varieties of Veronese and other varieties

Introduction

Show a certain Young Flattening, $YF_{d,n}$, provides scheme-theoretic equations for a large class of cases where usual flattenings fail, see Theorem 1.2.3.

Determine cases where flattenings are sufficient to give defining equations (more precisely, scheme-theoretic equations), Theorem 3.2.1. Theorem 3.2.1 is primarily a consequence of work of A. Iarrobino and V. Kanev and Diesel .

Put our results in a larger context by providing a uniform formulation of all known equations for secant varieties via vector bundle methods. We use this perspective to prove some of our results, including a key induction Lemma 6.2.1. The discussion of vector bundle methods is postponed to the latter part of the paper to make the results on symmetric border rank more accessible to readers outside of algebraic geometry.

Here is a chart summarizing what is known about equations of secant varieties of Veronese varieties:

2. Young Flattenings

All the principal Pfaffians of size $8$ of the this matrix coincide, up to scale, with the classical Aronhold invariant. (Redundancy occurs here is because one should really work with the submodule $S_{21}V\subset V{\mathord{\otimes}}\wedge^{2}V\simeq V{\mathord{\otimes}}V^{*}$, where the second identification uses a choice of volume form. The Pfaffian of the map $S_{21}\to S_{21}$ is the desired equation.)

Now consider the inclusion $V\subset\wedge^{k}V^{*}{\mathord{\otimes}}\wedge^{k+1}V$, given by $v\in V$ maps to the map $\omega\mapsto v\wedge\omega$. In bases one obtains a matrix whose entries are the coefficients of $v$ or zero. In the special case $n+1=2a+1$ is odd and $k=a$, one obtains a square matrix $K_{n}$, which is skew-symmetric for odd $a$ and symmetric for even $a$. For example, when $n=2$, the matrix is

Finally, consider the following generalization of both the Aronhold invariant and the $K_{n}$. Let $a=\lfloor\frac{n}{2}\rfloor$ and let $d=2\delta+1$. Map $S^{d}V\rightarrow(S^{\delta}V{\mathord{\otimes}}\wedge^{a}V^{*}){\mathord{\otimes}}(S^{\delta}V{\mathord{\otimes}}\wedge^{a+1}V)$ by first performing the inclusion $S^{d}V\rightarrow S^{\delta}V{\mathord{\otimes}}S^{\delta}V{\mathord{\otimes}}V$ and then using the last factor to obtain a map $\wedge^{a}V\rightarrow\wedge^{a+1}V$. We get:

If $n+1$ is odd, the matrix representing $YF_{d,n}(\phi)$ is skew-symmetic, so we may take Pfaffians instead of minors.

For a decomposable $w^{d}\in S^{d}V$, the map is

In bases, one obtains a matrix in block form, where the blocks correspond to the entries of $K_{n}$ and the matrices in the blocks are the square catalecticants $\pm(\frac{\partial\phi}{\partial x_{i}})_{\delta,\delta}$ in the place of $\pm x_{i}$.

Just as with the Aronhold invariant above, there will be redundancies among the minors and Pfaffians of $YF_{d,n}(\phi)$. See §4 for a description without redundancies.

In the case $n=2a$ with odd $a$, $YF_{d,n}$ is skew-symmetric (for any $d$) and one may instead take the size ${{n}\choose a}r+2$ sub-pfaffians of $YF_{d,n}$.

In the case $n=2a$ with even $a$, $YF_{d,n}$ is symmetric.

The bounds given in Theorem 1.2.3 for $n=2$ are sharp (see Proposition 4.2.3).

3. Vector bundle methods

Let $E$ be a vector bundle on $X$ of rank $e$, write $L={\mathcal{O}}_{X}(1)$, so $V=H^{0}(X,L)^{*}$. Let $v\in V$ and consider the linear map

where $A^{E}_{v}(s)=A(s{\mathord{\otimes}}v)$. In examples $E$ will be chosen so that both $H^{0}(E)$ and $H^{0}(E^{*}\otimes L)$ are nonzero, otherwise our construction is vacuous. A key observation (Proposition 5.1.1) is that the size $(re+1)$ minors of $A_{v}^{E}$ give equations for $\sigma_{r}(X)$.

4. Overview

Acknowledgments

We thank P. Aluffi, who pointed out the refined Bézout theorem 2.4.1. This paper grew out of questions raised at the 2008 AIM workshop Geometry and representation theory of tensors for computer science, statistics and other areas, and the authors thank AIM and the conference participants for inspiration.

Background

If $G/P$ is a rational homogeneous variety and $E\rightarrow G/P$ is an irreducible homogeneous vector bundle, we write $E=E_{\mu}$ where $\mu$ is the highest weight of the irreducible $P$-module inducing $E$. We use the conventions of regarding roots and weights of simple Lie algebras.

2. Flattenings

Given $\phi\in S^{d}V$, write $\phi_{a,d-a}\in S^{a}V{\mathord{\otimes}}S^{d-a}V$ for the $(a,d-a)$-polarization of $\phi$. We often consider $\phi_{a,d-a}$ as a linear map $\phi_{a,d-a}:S^{a}V^{*}\rightarrow S^{d-a}V$. This notation is compatible with the more general one of Young flattenings that we will introduce in §4.

3. Inheritance

Let $V$ be a vector space of dimension greater than $r$.

4. Results related to degree

Sometimes it is possible to conclude global information from local equations if one has information about degrees.

We need the following result about excess intersection.

This is an application of the refined Bézout theorem of [16, Thm. 12.3]. ∎

We recall the following classical formulas, due to C. Segre. For a modern reference see .

5. Conormal spaces

Let $[\phi]\in Rank^{r}_{(a,d-a)}(S^{d}V)$ be a sufficiently general point, then

and equality holds if $\phi$ is sufficiently general.

Symmetric Flattenings (catalecticant minors)

The following result dates back to the work of Sylvester and Gundelfinger.

Apply Proposition 2.3.1 to Theorem 3.1.1 in the case $r=2$. ∎

C. Raicu recently proved that in Corollary 3.1.2 it is possible to replace the $(2,d-2)$ flattening with any $(i,d-i)$-flattening such that $2\leq i\leq d-2$.

The bound obtained in this theorem is the best we know of for even degree, apart from some cases of small degree listed below. Theorem 1.2.3 is an improvement of this bound in the case of odd degree.

We will have need of more than one symmetric flattening, so we make the following definitions, following , but modifying the notation:

Fix a sequence $\vec{r}:=(r_{1},\ldots,r_{\llcorner\frac{d}{2}\lrcorner})$ and let

Call a sequence $\vec{r}$ admissible if there exists $\phi\in S^{d}W$ such that ${\rm rank}(\phi_{j,d-j})=r_{j}$ for all $j=1,\ldots,\llcorner\frac{d}{2}\lrcorner$. It is sufficient to consider $\vec{r}$ that are admissible because if $\vec{r}$ is non-admissible, the zero set of $SFlat_{\vec{r}}$ will be contained in the union of admissible $SFlat$’s associated to smaller $\vec{r}$’s in the natural partial order. Note that $SFlat_{\vec{r}}(S^{d}W)\subseteq Sub_{r_{1}}(S^{d}W)$.

To remedy this, let $\vec{r}$ be admissible, and consider

In the commutative algebra literature (e.g. ), “Gor” is short for Gorenstein, see [25, Def. 1.11] for a history.

Unfortunately, defining equations for $Gor(\vec{r})$ are not known. One can test for membership of $SFlat_{\vec{r}}^{0}(S^{d}W)$ by checking the required vanishing and non-vanishing of minors.

[12, Thm 1.1] If $\operatorname{dim}W=3$, and $\vec{r}$ is admissible, then $Gor(\vec{r})$ is irreducible.

Theorem 3.1.9 combined with Theorem 3.1.4 allows one to extend the set of secant varieties of Veronese varieties defined by flattenings.

2. Consequences of Theorems 3.1.9 and 3.1.4

The following varieties are defined scheme-theoretically by minors of flattenings:

By [47, Thm 4.2], when $n\leq 2$, ${\rm rank}(\phi_{s,d-s})$ is nondecreasing in $s$ for $1\leq s\leq\lfloor\frac{d}{2}\rfloor$. We will use this fact often in this section.

To prove each case, we simply show the scheme defined by the flattenings in the hypotheses coincides with the scheme of some $Gor(\vec{r})$ with $\vec{r}$ admissible and then Theorem 3.1.4 combined with Theorem 3.1.9 implies the result. The first step is to determine which $\vec{r}$ are admissible.

The only admissible sequences $\vec{r}$ with $r_{1}=3$ and $r_{i}\leq 5$ are

We may assume that ${\rm rank}(\phi_{1,d-1})=3$, otherwise we are in the case of two variables. By inheritance, it is sufficient to prove the result for $n=2$. In this case, if ${\rm rank}(\phi_{a,d-a})\leq 3$ for $a=\lfloor\frac{d}{2}\rfloor$, then ${\rm rank}(\phi_{a,d-a})\leq 3$ for all $a$ such that $2\leq a\leq d-2$, and $\vec{r}=(3,\ldots,3)$ is admissible . ∎

Otherwise ${\rm rank}(\phi_{2,d-2})\leq 5$. In this case, by an extension of Lemma 3.2.3 , there are just two other possibilities for $\vec{r}$, namely $\vec{r}_{1}=(1,3,5,6\ldots,6,5,3,1)$ and $\vec{r}_{2}=(1,3,4,5,6\ldots,6,5,4,3,1)$ for $d\geq 8$.

Let $\operatorname{dim}W=3$. We will need the following facts to prove Lemma 3.2.3:

For $\phi\in S^{d}W$, consider the ideal $\phi{}^{\perp}$ generated in degrees $\leq d$, by $\operatorname{ker}(\phi_{a,d-a})\subset S^{a}W^{*}$, $1\leq a\leq d$, and the ring $A_{\phi}:=Sym(W^{*})/\phi{}^{\perp}$. Note that the values of the Hilbert function of $A_{\phi}$, $H_{A_{\phi}}(j)$, are $T(\vec{r}):=(1,r_{1},\ldots,r_{\llcorner\frac{d}{2}\lrcorner},r_{\llcorner\frac{d}{2}\lrcorner},\ldots,r_{1},1,0,\ldots,0)$, where recall that $r_{j}={\rm rank}(\phi_{j,d-j})$.

[5, Thm. 2.1] The ring $A_{\phi}$ has a minimal free resolution of the form

where the $q_{j}$ are non-decreasing. Here $Sym(W^{*})(j)$ denotes $Sym(W^{*})$ with the labeling of degrees shifted by $j$, so $\operatorname{dim}(Sym(W^{*})(j))_{d}=\binom{d+j+2}{2}$.

Moreover [12, Thm 1.1] there is a unique resolution with the properties $q_{1}\leq\frac{d+3}{2}$, $q_{i}+q_{u-i+2}=d+2$, $2\leq i\leq\frac{u+1}{2}$ having $T(\vec{r})$ as the values of its Hilbert function.

Recall that $H_{A_{\phi}}(j)$ is also the alternating sum of the dimensions of the degree $j$ term in (5) (forgetting the last term). Thus the $q_{j}$ determine the $r_{i}$.

[12, Thm. 3.3] Letting $j_{0}$ be the smallest $j$ such that $\phi_{j,d-j}$ is not injective, then $u=2j_{0}+1$ in the resolution above.

If there are three generators in degree $2$ then $q_{i}=(2,2,2,d,d)$, and computing the Hilbert function via the Euler characteristic, we are in case (1) of the Lemma.

If there are two generators in degree $2$ then there are the two possibilities $q_{i}=(2,2,3,n-1,n)$ (case(2)) or $q_{i}=(2,2,4,n-2,n)$ (case (4)). Note that $q_{i}=(2,2,5,n-3,n)$ for $n\geq 8$ is impossible because otherwise $\vec{r}=(1,3,4,5,6,\ldots)$ contrary to assumption. By similar arguments one shows that there are no other possibilities.

If there is one generator in degree $2$ then $q_{i}=(2,3,3,n-1,n-1)$ (case (3)). ∎

The result follows by solving the inequality. ∎

For the case $(p,r)=(2,4)$ see Theorem 3.2.1(2) and for the case $(p,r)=(3,9)$ see Theorem 4.2.9.

The values of the right hand side of the inequality for $p=1,\ldots 4$ are respectively $4$, $112$, $28314$, $81662152$.

Equality holds in Corollary 3.2.6 for the cases $p=1,2,3$ by Proposition 2.4.2. The case $p=1$ corresponds to the quadratic Veronese surface and the cases $p=2,3$ will be considered respectively in Thm.3.2.1 (1) and Theorem 4.2.8. For $p\geq 4$ these numbers are out of the range of the results in (the points are too few to be fixed points of a torus action) and we do not know if equality holds.

Young flattenings for Veronese varieties

In what follows we fix $n+1=\operatorname{dim}V$ and endow $V$ with a volume form and thus identify (as $SL(V)$-modules) $S_{(p_{1},\ldots,p_{n+1})}V$ with $S_{(p_{1}-p_{n+1},p_{2}-p_{n+1},\ldots,p_{n}-p_{n+1},0)}V$. We will say $(p_{1}-p_{n},p_{2}-p_{n},\ldots,p_{n-1}-p_{n},0)$ is the reduced partition associated to $(p_{1},\ldots,p_{n})$.

The Pieri formula states that $S_{\pi}V^{*}\subset S_{\nu}V^{*}{\mathord{\otimes}}S^{d}V^{*}$ iff the Young diagram of $\pi$ is obtained by adding $d$ boxes to the Young diagram of $\nu$, with no two boxes added to the same column. Moreover, if this occurs, the multiplicity of $S_{\pi}V^{*}$ in $S_{\nu}V^{*}{\mathord{\otimes}}S^{d}V^{*}$ is one.

Say $S_{\pi}V^{*}\subset S_{\nu}V{\mathord{\otimes}}S^{d}V^{*}$ and consider the map $S^{d}V\rightarrow S_{\pi}V{\mathord{\otimes}}S_{\nu}V^{*}$. Let $S_{\mu}V=S_{\nu}V^{*}$ where $\mu$ is the reduced partition with this property. We obtain an inclusion $S^{d}V\rightarrow S_{\pi}V{\mathord{\otimes}}S_{\mu}V$.

Given $\phi\in S^{d}V$, let $\phi_{\pi,\mu}\in S_{\pi}V{\mathord{\otimes}}S_{\mu}V$ denote the corresponding element. If $S_{\mu}V=S_{\nu}V^{*}$ as an $SL(V)$-module, we will also write $\phi_{\pi,\nu^{*}}=\phi_{\pi,\mu}$ when we consider it as a linear map $S_{\nu}V\rightarrow S_{\pi}V$.

The following proposition is an immediate consequence of the subadditivity for ranks of linear maps.

So far we have not obtained any new equations using three way symmetric flattenings and Young flattenings. We mention three-way symmetric flattenings because they may be useful in future investigations, especially when further modules of equations for secant varieties of triple Segre products are found.

Recall $YF_{d,n}^{r}$ from §1.2 whose description had redundancies. We can now give an irredundant description of its defining equations:

This is a consequence of Schur’s Lemma, because the module $S_{(\delta+1,1^{a})}V$ is the only one appearing in both sides of $S^{d}V\otimes S^{\delta}V^{*}\otimes\wedge^{a}V\to S^{\delta}V\otimes\wedge^{a+1}V$

Note that if $S_{\pi}V\simeq S_{\mu}V$ as $SL(V)$-modules and the map is symmetric, then

In this subsection fix $\operatorname{dim}V=3$ and a volume form $\Omega$ on $V$. From the general formula for $\operatorname{dim}S_{\pi}V$ (see, e.g., [17, p78]), we record the special case:

Let $a\geq b$. Write $d=\alpha+\beta+\gamma$ with $\alpha\leq b$, $\beta\leq a-b$ so $S_{(a+\gamma-\alpha,b+\beta-\alpha)}V\subset S_{a,b}V{\mathord{\otimes}}S^{d}V$. For $\phi\in S^{d}V$, consider the induced map

In the following picture we label the first row containing $a$ boxes with $a$ and so on.

Assume we have chosen a weight basis $x_{1},x_{2},x_{3}$ of $V$ and $x=x_{3}$ is a vector of lowest weight. Consider the image of a weight basis of $S_{a,b}V$ under $(x^{3}_{3})_{(a,b),(a+\gamma-\alpha,b+\beta-\alpha)}$. Namely consider all semi-standard fillings of the Young diagram corresponding to $(a,b)$, and count how many do not map to zero. By construction, the images of all the vectors that do not map to zero are linearly independent, so this count indeed gives the dimension of the image.

In order to have a vector not in the kernel, the first $\alpha$ boxes of the first row must be filled with $1$’s and the first $\alpha$ boxes of the second row must be filled with $2$’s.

We are particularly interested in cases where $(a,b)=(a+\gamma-\alpha,b+\beta-\alpha)$. In this case

Plugging into the conclusion of Lemma 4.2.1, the rank of the image of a $d$-th power in this situation is

To keep this small, it is convenient to take $d=a+b$ so the rank is $a-b+1$. One can then fix this number and let $a,b$ grow to study series of cases.

If (10) has rank one when $d=2p$, we just recover the usual symmetric flattenings as $S_{(p,p)}V=S_{p}V^{*}$. We consider the next two cases in the theorems below, $(a,b)=(p+1,p)$ when $d=2p+1$ and $(a,b)=(p+2,p)$ when $d=2p+2$. Recall that $(p+q,p)^{*}=(p+q,q)$ in the notation of §4.1.1.

Let $d=2p+1$. The skew analog of Proposition 3.2.5 is the following proposition, which shows that the bound in the assumption of Theorem 1.2.3 is sharp.

Here is a pictorial description when $p=2$ of $\phi_{31,31}\colon S_{32}V\to S_{31}V$ in terms of Young diagrams:

The values of the right hand side for $p=1,\ldots 4$ are respectively $4$, $140$, $65780$, $563178924$.

In Corollary 4.2.4 equality holds in the cases $p=1,2$ by Proposition 2.4.2. The case $p=1$ is just the Aronhold case and the case $p=2$ will be considered in Theorem 4.2.7 (2). For $p\geq 3$ these numbers are out of the range of and we do not know if equality holds.

Now let $d=2p+2$ be even, requiring $\pi=\mu$, the smallest possible ${\rm rank}((x^{d})_{\pi,\mu})$ is three, which we obtain with $\phi_{(p+2,p),(p+2,p)}$.

A pictorial description when $p=2$ is as follows:

and define $M_{\phi}$ for arbitrary $\phi\in S^{2p+2}V$ by linearity and polarization. If we take bases of $S^{2}V{\mathord{\otimes}}S^{2}(\wedge^{2}V)$ as above, with indices $((i_{1},\ldots,i_{p}),(kl),(k^{\prime}l^{\prime}))$, most of the matrix of $M_{e_{1}^{2p+2}}$ is zero. The upper-right hand $6\times 6$ block, where $(i_{1},\ldots,i_{p})=(1,\ldots,1)$ in both rows and columns and the order on the other indices

where all the terms in $stuff$ have partitions with at least three parts. On the other hand, from the nature of the image we conclude it is just the first factor and $M_{\phi}\in S^{2}(S_{p+2,2}V)$. ∎

is skew-symmetric if $p$ is even and symmetric if $p$ is odd.

Consider $M_{\phi}:S^{p-1}V^{*}{\mathord{\otimes}}S^{q}(\wedge^{2}V^{*})\rightarrow S^{p}V{\mathord{\otimes}}S^{q}(\wedge^{2}V)$ given for $\phi=x^{p+4q-1}$ by

The following is a special case of the Thm. 3.2.1(4), we include this second proof because it is very short.

$\det(\phi_{42,42})$ is a polynomial of degree 27, which is not the power of a lower degree polynomial. This can be proved by cutting with a random projective line, and using Macaulay2. The two variable polynomial obtained is not the power of a lower degree polynomial.

To prove (2), we picked a polynomial $\phi$ which is the sum of $8$ random fifth powers of linear forms, and a submatrix of $\phi_{42,42}$ of order $24$ which is invertible. The matrix representing $\phi_{42,42}$ can be constructed explicitly by the package PieriMaps of Macaulay2 .

(1) can be proved in the same way. (3) is well known. ∎

(1) follows from Theorem 1.2.3. (2) is obvious. ∎

Construction of equations from vector bundles

Write $e:={\rm rank}(E)$ and consider the determinantal varieties or rank varieties defined by the minors of $A^{E}_{v}:H^{0}(E)\to H^{0}(E^{*}\otimes L)^{*}$ (defined by equation (3)),

i.e., the size $(re+1)$ minors of $A_{v}^{E}$ give equations for $\sigma_{r}(X)$.

When $E$ is understood, we will write $A_{v}$ for $A_{v}^{E}$.

By (3), if $x=[v]\in X$, then $H^{0}(I_{x}\otimes E)\subseteq\ker A_{v}$. The subspace $H^{0}(I_{x}\otimes E)\subseteq H^{0}(E)$ has codimension at most $e$ in $H^{0}(E)$, hence the same is true for the subspace $\ker A_{x}$ and it follows ${\rm rank}(A_{x})\leq e$. If $v\in\hat{\sigma}_{r}(X)$ is a general point, then it may be expressed as $v=\sum_{i=1}^{r}x_{i}$, with $x_{i}\in\hat{X}$. Hence

as claimed. Since the inequality is a closed condition, it holds for all $v\in\hat{\sigma}_{r}(X)$. ∎

The generalization of symmetric flattenings to Young flattenings for Veronese varieties is a representation-theoretic version of the generalization from line bundles to higher rank vector bundles.

A general source of examples is given by curves obtained as determinantal loci. This topic is studied in detail in . In , A. Ginensky considers the secant varieties $\sigma_{k}(C)$ to smooth curves $C$ in their bicanonical embedding. With our notations this corresponds to the symmetric pair $(E,L)=(K_{C},K_{C}^{2})$. He proves [21, Thm. 2.1] that $\sigma_{k}(C)=Rank_{k}(K_{C})$ if $k<\textrm{Cliff}(C)$ and $\sigma_{k}(C)\subsetneq Rank_{k}(K_{C})$ for larger $k$. Here $\textrm{Cliff}(C)$ denotes the Clifford index of $C$.

2. The construction in the symmetric and skew-symmetric cases

We say that $(E,L)$ is a symmetric pair if the isomorphism $E\smash{\mathop{\longrightarrow}\limits^{\alpha}}E^{*}\otimes L$ is symmetric, that is the transpose isomorphism $E\otimes L^{*}\smash{\mathop{\longrightarrow}\limits^{\alpha^{t}}}E^{*}$, after tensoring by $L$ and multiplying the map by $1_{L}$ equals $\alpha$, i.e., $\alpha=\alpha^{t}\otimes 1_{L}$. In this case $S^{2}E$ contains $L$ as a direct summand, the morphism $A_{v}$ is symmetric, and $Rank_{k}(E)$ is defined by the $(ke+1)$-minors of $A_{v}$.

Similarly, $(E,L)$ is a skew-symmetric pair if $\alpha=-\alpha^{t}\otimes 1_{L}$. In this case $e$ is even, $\wedge^{2}E$ contains $L$ as a direct summand, the morphism $A_{v}$ is skew-symmetric, and $Rank_{k}(E)$ is defined by the size $(ke+2)$ subpfaffians of $A_{v}$, which are equations of degree $\frac{ke}{2}+1$.

3. The conormal space

The results reviewed in §2.5, restated in the language of vector bundles, say the affine conormal space of $Rank_{k}(E)$ at $[v]\in Rank_{k}(E)_{smooth}$ is the image of the map

If $(E,L)$ is a symmetric (resp. skew symmetric) pair, there is a symmetric (resp. skew symmetric) isomorphism $\ker A_{v}\simeq\textrm{Im\ }A_{v}^{\perp}$ and the conormal space of $Rank_{k}(E)$ at $v$ is given by the image of the map $S^{2}\left(\ker A_{v}\right)\to H^{0}(L)$, (resp. $\wedge^{2}\left(\ker A_{v}\right)\to H^{0}(L)$).

Let $v\in\hat{X}$, in the proof of Proposition 5.1.1, we saw $H^{0}(I_{v}\otimes E)\subseteq\ker A_{v}$. In the same way, $H^{0}(I_{v}\otimes E^{*}\otimes L)\subseteq\textrm{Im\ }A_{v}^{\perp}$, by taking transpose. Equality holds if $E$ is spanned at $x=[v]$. This is generalized by the following Proposition.

Let $v=\sum_{i=1}^{k}x_{i}\in V$, with $[x_{i}]\in X$, and let $Z=\{[x_{1}],\ldots,[x_{k}]\}$. Then

The first inclusion is an equality if $H^{0}(E^{*}\otimes L)\to H^{0}(E^{*}\otimes L_{|Z})$ is surjective. The second inclusion is an equality if $H^{0}(E)\to H^{0}(E_{|Z})$ is surjective.

the inclusion for the kernel follows. To see the equality assertion, if $H^{0}(E^{*}\otimes L)\to H^{0}(E^{*}\otimes L_{|Z})$ is surjective, for every $j=1,\ldots,k$ we can choose $t_{h,j}\in H^{0}(E^{*}\otimes L)$, for $h=1,\ldots,e$, such that $t_{h,j}(x_{i})=0$ for $i\neq j$, $\forall h$ and $t_{h,j}$ span the fiber of $E^{*}\otimes L$ at $x_{j}$. It follows that if $s\in\ker(A_{v})$ then $t_{h,j}(x_{j})\cdot s(x_{j})=0$ for every $h,j$ which implies $s(x_{j})=0$, that is $s\in H^{0}(I_{Z}\otimes E)$. The dual statement is similar. ∎

The following theorem gives a useful criterion to find local equations of secant varieties.

Let $v=\sum_{i=1}^{r}x_{i}\in V$ and let $Z=\{[x_{1}],\ldots,[x_{r}]\}$, where $[x_{j}]\in X$. If

is surjective, then $\sigma_{r}(X)$ is an irreducible component of $Rank_{r}(E)$.

If $(E,L)$ is a symmetric, resp. skew-symmetric pair, and

is surjective, then $\sigma_{r}(X)$ is an irreducible component of $Rank_{r}(E)$.

The surjectivity of the map in the first row implies that the rank of map in the second row is at least ${\rm dim}\;H^{0}(I_{Z^{2}}\otimes L)$. By Proposition 5.4.1, $\ker A_{v}\otimes\textrm{Im\ }A_{v}^{\perp}\to H^{0}(I_{Z^{2}}\otimes L)$ is surjective, so that the conormal spaces of $\sigma_{k}(X)$ and of $Rank_{k}(E)$ coincide at $v$, proving the general case. The symmetric and skew-symmetric cases are analogous. ∎

The induction Lemma

The notion of weak defectivity, which dates back to Terracini, was applied by Ciliberto and Chiantini in to show many cases of tensors and symmetric tensors admitted a unique decomposition as a sum of rank one tensors. We review here it as it is used to prove the promised induction Lemma 6.2.1.

Notational warning: what we call $k$-weakly defective is called $(k-1)$-weakly defective in . We shifted the index in order to uniformize to our notion of $k$-defectivity: by Terracini’s lemma, $k$-defective varieties (i.e., those where $\operatorname{dim}\sigma_{k}(X)$ is less than the expected dimension) are also $k$-weakly defective.

The Veronese varieties which are weakly defective have been classified.

(i) the $k$-defective varieties, namely $(k,2,n)$, $k=2,\ldots,{{n+2}\choose 2}$, $(5,4,2)$, $(9,4,3)$, $(14,4,4)$, $(7,3,4)$,

If $X$ is not $k$-weakly defective, then it is not $k^{\prime}$-weakly defective for any $k^{\prime}\leq k$. Hence we may assume $k^{\prime}=k-1$. Consider the projection $\pi$ of $X$ centered at the span of $k^{\prime}$ general tangent spaces at $X$. The image $\pi(X)$ is not $1$-weakly defective, [10, Prop 3.6] which means that the Gauss map of $\pi(X)$ is nondegenerate [10, Rem. 3.1 (ii)]. Consider the dual variety of $\pi(X)$, which is contained in the discriminant, see [34, p. 810]. If the dual variety of $\pi(X)$ were contained in a hyperplane, then $\pi(X)$ would be a cone (see, e.g. [13, Prop. 1.1]), and thus be $1$-weakly defective, which is a contradiction. Hence the linear span of the discriminant variety is the ambient space. ∎

is surjective for general $Z^{\prime}$ of length $k^{\prime}\leq k$.

The analogous results hold in the symmetric and skew-symmetric cases.

It is enough to prove the case when $k^{\prime}=k-1$. Let $s\in H^{0}(I_{Z^{\prime}}^{2}\otimes L)$. By the assumption and Proposition 6.1.3 there are sections $s_{1},\ldots,s_{t}$, with $t$ at most $\operatorname{dim}H^{0}(I_{Z^{\prime}}^{2}\otimes L)$, with singular points respectively $p_{1},\ldots,p_{t}$ outside $Z^{\prime}$, such that $s=\sum_{i=1}^{t}s_{i}$. Let $Z_{i}=Z^{\prime}\cup\{p_{i}\}$ for $i=1,\ldots,t$. We may assume that the $p_{i}$ are in general linear position. By assumption $s_{i}$ is in the image of $H^{0}(I_{Z_{i}}\otimes E)\otimes H^{0}(I_{Z_{i}}\otimes E^{*}\otimes L)\smash{\mathop{\longrightarrow}\limits^{\ }}H^{0}(I_{Z_{i}^{2}}\otimes L)$. So all $s_{i}$ come from $H^{0}(I_{Z^{\prime}}\otimes E)\otimes H^{0}(I_{Z^{\prime}}\otimes E^{*}\otimes L)$. The symmetric and skew-symmetric cases are analogous.∎

Proof of Theorem 1.2.3

By Theorem 5.4.3 it is sufficient to prove the map

is surjective. In the case $n=2a$, $a$ odd we have to prove that the map

is surjective. The arguments are similar.

We need the following lemma, whose proof is given below:

Let $h_{i}\in V^{*}$ be an equation for $H_{i}$. A basis of the space of polynomials of degree $2{\delta}+1$ which are singular on $Z$ is given by $P_{I,J}:=\left(\prod_{i\in I}h_{i}\right)\cdot\left(\prod_{j\in J}h_{j}\right)$ where $I,J\subseteq\{1,\ldots{\delta}+n\}$ are multi-indices (without repetitions) satisfying $|I|={\delta+1}$, $|J|={\delta}$, and $|I\cap J|\leq{\delta}-1$ (that is $J$ cannot be contained in $I$).

We prove that the map (15) is surjective for $t={{\delta+n}\choose n}$ by degeneration to the case when the ${{\delta+n}\choose n}$ points are the vertices of a configuration given by the union of $\delta+n$ general hyperplanes given by linear forms $h_{1},\ldots,h_{\delta+n}\in V^{*}$.

First consider the case $n$ is even. For any multi-index $I\subset\{1,\ldots,\delta+n\}$ with $|I|=\delta$ write $q_{I}=\prod_{k\in I}h_{k}\in H^{0}({\mathcal{O}}(\delta))$ and for any multi-index $H$, of length $a+1$, let $s_{H}$ be the section of $\wedge^{a}Q$ represented by the linear subspace of dimension $a-1$ given by $h_{H}=\{h_{i}=0,i\in H\}$. Indeed the linear subspace is a decomposable element of $\wedge^{a}V=H^{0}(\wedge^{a}Q)$. The section $s_{H}$ is represented by the Plücker coordinates of $h_{H}$.

The section $q_{I}s_{H}\in H^{0}(\wedge^{a}Q(\delta))$ vanishes on the reducible variety consisting of the linear subspace $h_{H}$ of dimension $a-1$ and the degree $\delta$ hypersurface $q_{I}=0$. In particular, if $I\cap H=\emptyset$, then $q_{I}s_{H}\in H^{0}(I_{Z}\otimes\wedge^{a}Q(\delta))$.

Let $I=I_{0}\cup\{u\}$ with $|I_{0}|=\delta$, consider $|J|=\delta$ such that $u\notin J$ and $|I\cap J|\leq\delta-1$. It is possible to choose $K_{1},K_{2}$ such that $|K_{1}|=|K_{2}|=a$ and such that $K_{1}\cap K_{2}=K_{1}\cap I=K_{2}\cap J=\emptyset$.

We get $q_{I_{0}}s_{\{u\}\cup K_{1}}\wedge q_{J}s_{\{u\}\cup K_{2}}$ is the degree $2\delta+1$ hypersurface $q_{I_{0}}q_{J}h_{u}=q_{I}q_{J}$. By Lemma 7.0.1 these hypersurfaces generate $H^{0}(I_{Z}^{2}(2\delta+1))$. Hence $W_{Z}$ is surjective and the result is proved in the case $n$ is even.

When $n$ is odd, the morphism $A_{\phi}$ is represented by a rectangular matrix, and it induces a map $B\colon H^{0}(\wedge^{a}Q(\delta))\otimes H^{0}(\wedge^{n-a}Q(\delta))\to H^{0}({\mathcal{O}}(2\delta+1))$.

For any multi-index $H$, of length $n-a+1$, let $s_{H}$ be the section of $\wedge^{a}Q$ represented by the linear subspace of dimension $a-1$ given by $h_{H}=\{h_{i}=0,i\in H\}$. Indeed the linear subspace is a decomposable element of $\wedge^{a}V=H^{0}(\wedge^{a}Q)$. The section $s_{H}$ is represented by the Plücker coordinates of $h_{H}$.

The section $q_{I}s_{H}\in H^{0}(\wedge^{a}Q(\delta))$ vanishes on the reducible variety consisting of the linear subspace $h_{H}$ of dimension $a-1$ and the degree $\delta$ hypersurface $q_{I}$. In particular, if $I\cap H=\emptyset$, then $q_{I}s_{H}\in H^{0}(I_{Z}\otimes\wedge^{a}Q(\delta))$.

Let $I=I_{0}\cup\{u\}$ with $|I_{0}|=\delta$, consider $|J|=\delta$ such that $u\notin J$ and $|I\cap J|\leq\delta-1$. The modification to the above proof is that it is possible to choose $K_{1},K_{2}$ such that $|K_{1}|=n-a=a+1$, $|K_{2}|=a$ and such that $K_{1}\cap K_{2}=K_{1}\cap I=K_{2}\cap J=\emptyset$.

Then $q_{I_{0}}s_{\{u\}\cup K_{1}}\in H^{0}(\wedge^{a+1}Q(\delta))$, $q_{J}s_{\{u\}\cup K_{2}}\in H^{0}(\wedge^{a}Q(\delta))$,

and $q_{I_{0}}s_{\{u\}\cup K_{1}}\wedge q_{J}s_{\{u\}\cup K_{2}}$ is the degree $2\delta+1$ hypersurface $q_{I_{0}}q_{J}h_{u}=q_{I}q_{J}$. By Lemma 7.0.1 these hypersurfaces generate $H^{0}(I_{Z}^{2}(2\delta+1))$. Hence the map (15) is surjective and the result is proved.

We begin by discussing properties of the hypersurfaces through $Z$, which may be of independent interest.

(i) For $d\geq{\delta}$, the products $\prod_{i\in I}h_{i}$ for every $I\subseteq\{1,\ldots,n+{\delta}\}$ such that $|I|=d$ are independent in $S^{d}V^{*}$.

(ii) For $d\leq{\delta}$, the products $\prod_{i\in I}h_{i}$ for every $I\subseteq\{1,\ldots,n+{\delta}\}$ such that $|I|=d$ span $S^{d}V^{*}$.

Write $h_{I}=\prod_{\in I}h_{i}$. Consider a linear combination $\sum_{I}a_{I}h_{I}=0$. Let $d\geq{\delta}$. If $d\geq n+\delta$ the statement is vacuous, so assume $d<n+\delta$. For each $I_{0}$ such that $|I_{0}|=d$ there is a point $P_{0}\in V$ such that $h_{i}(P_{0})=0$ for $i\notin I_{0}$ and $h_{i}(P_{0})\neq 0$ for $i\in I_{0}$. The equality $\sum_{I}a_{I}h_{I}(P_{0})=0$ implies $a_{I_{0}}=0$, proving (i). For $d={\delta}$, (ii) follows as well by counting dimensions, as $\operatorname{dim}S^{\delta}V^{*}={{{\delta}+n}\choose n}$. For $d\leq{\delta}-1$ pick $f\in S^{d}V^{*}$. By the case already proved, there exist coefficients $a_{I}$, $b_{J}$ such that

where the first sum is over all $I$ with $|I|={\delta}$ such that $\{1,\ldots,{\delta}-d\}\subseteq I$ and the second sum over all $J$ with $|J|={\delta}$ such that $\{1,\ldots,{\delta}-d\}\not\subseteq J$.

For every $J_{0}$ such that $|J_{0}|={\delta}$ and $\{1,\ldots,{\delta}-d\}\not\subseteq J_{0}$ there is a (unique) point $[P_{0}]$ such that $h_{i}(P_{0})=0$ for $i\notin J_{0}$ and $h_{i}(P_{0})\neq 0$ for $i\in J_{0}$. In particular $\prod_{i=1}^{{\delta}-d}h_{i}(P_{0})=0$ and substituting $P_{0}$ into (17) shows $b_{J_{0}}=0$, for each $J_{0}$. Thus

and dividing both sides by $\prod_{i=1}^{{\delta}-d}h_{i}$ proves (ii). ∎

Recall that the subspace of $S^{d}V^{*}$ of polynomials which pass through $p$ points and are singular through $q$ points has codimension $\leq\min\left({{n+d}\choose n},p+q(n+1)\right)$. When equality holds one says that the conditions imposed by the points are independent and that the subspace has the expected codimension.

(i) $H^{0}(I_{Z}(d))=0$ if $d\leq{\delta}$.

(ii) $H^{0}(I_{Z}({\delta+1}))$ has dimension ${{{\delta}+n}\choose{\delta+1}}$ and it is generated by the products $h_{I}$ with $|I|={\delta}+1$.

Since $Z$ is finite, $\operatorname{dim}H^{0}({\mathcal{O}}_{Z}(d))={{{\delta}+n}\choose n}=\deg Z$ for every $d$.

In order to prove (ii), consider the products $h_{I}$ with $|I|={\delta}+1$, which are independent by Proposition 7.0.2.i, so they span a ${{{\delta}+n}\choose{\delta+1}}$-dimensional subspace of $H^{0}(I_{Z}({\delta}+1))$.