A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\mathbb{P}^n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity of $σ_s(O_{k,n,d})$ to the Hilbert function of certain sets of fat points in $\mathbb{P}^n$.
We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension of their higher secant varieties. Via inverse systems (apolarity) and the study of certain spaces of forms we are able, in several cases, to determine whether those secant varieties are defective or not.
In this paper we study singularities of third secant varieties of Veronese embedding $v_d(\mathbb{P}^n)$, which corresponds to the variety of symmetric tensors of border rank at most three in $(\mathbb{C}^{n+1})^{\otimes d}$.
We define new classes of modules of equations for secant varieties of Veronese varieties using representation theory and geometry. We also revisit some old modules of equations (catalecticant minors) to determine when they are sufficient to give scheme-theoretic defining equations.
Here we present a partial generalization to higher order osculating spaces of the classical Lemma of Terracini on ordinary tangent spaces. As an application, we investigate the secant varieties to the osculating varieties to the Veronese embeddings of the projective plane.
We prove that the ideal of the variety of secant lines to a Segre--Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre--Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.
We generalize the classical Terracini's Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of the $d$-Veronese embedding of the projective 3-space.
We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $\dim V_i \geq 3$ for all $i$'s, then also all the ranks between $k$ and $2k-1$ arise.
We give an almost asymptotically sharp bound for the non secant defectiveness and identifiability of Segre-Veronese varieties. We also provide new examples of defective Segre-Veronese varieties.
We prove the existence of defective secant varieties of three-factor and four-factor Segre-Veronese varieties embedded in certain multi-degree. These defective secant varieties were previously unknown and are of importance in the classification of defective secant varieties of Segre-Veronese varieties with three or more factors.
If $X\subset \mathbb{P}^n$ is a projective non degenerate variety, the $X$-rank of a point $P\in \mathbb{P}^n$ is defined to be the minimum integer $r$ such that $P$ belongs to the span of $r$ points of $X$. We describe the complete stratification of the fourth secant variety of any Veronese variety $X$ via the $X$-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers $r$ that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of $X$-border rank…
In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in \mathbb{Z}^+$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.
This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high dimensional case to the computation of the dimensions of secant varieties in low dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective s-th S…
In this note we consider multiples aD, where D is a divisor of the blow-up of P^n along points in general position which appears in the Alexander and Hirschowitz list of Veronese embeddings having defective secant varieties. In particular we show that there is such a D with h^1(X,D) > 0 and h^1(X,2D) = 0.
Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the $s^{\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \bar{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.
We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by ${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it a {\it Segre-Veronese variety} and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for 3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the expected dimension except for a few cases. A list of examples of defective $X^s$'s is given.
We introduce a method to produce bounds for the non secant defectivity of an arbitrary irreducible projective variety, once we know how its osculating spaces behave in families and when the linear projections from them are generically finite. Then we analyze the relative dimension of osculating projections of Grassmannians, and as an application of our techniques we prove that asymptotically the Grassmannian $\mathbb{G}(r,n)$, parametrizing $r$-planes in $\mathbb{P}^n$, is not $h$-defective for $h\leq (\frac{n+1}{r+1})^{\lfloor\log_2(r)\rfloor}$. This bound improves the previous one $h\leq \fr…
We give a partial "quasi-stratification" of the secant varieties of the order $d$ Veronese variety $X_{m,d}$ of $\mathbb {P}^m$. It covers the set $σ_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of curvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. For low border rank two different "quasi-strata" are disjoint and we compute the symmetric rank of their elements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties. To get a stratification we attach to each $P\in σ_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratum containi…
New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homoge…
We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty…
We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to p…
Let $SV^{\pmb n}_{\pmb d}$ be the Segre-Veronese given as the image of the embedding induced by the line bundle $\mathcal{O}_{\mathbb{P}^{n_1}\times\dots\times\mathbb{P}^{n_r}}(d_1,\dots, d_r)$. We prove that asymptotically $SV^{\pmb n}_{\pmb d}$ is not $h$-defective for $h\leq n_1^{\lfloor \log_2(d-1)\rfloor}$.
We compute the dimensions of all the secant varieties to the tangential varieties of all Segre-Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.
We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.
A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal d…
We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space $\mathbb{P}^n$ have the expected dimension, modulo a few well-known exceptions. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $n$ of the projective space via a specializati…
Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $\mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.
Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$
We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersects in the same dimension. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possible altered vector space. Ralf Koetter and Frank R. Kschischang introduced a metric on the set af vector spaces and showed that a minimal dis…
If $\X \subset ¶^n$ is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of $\X$. In this paper we find those equations in the following cases: $\X = ¶^{n_1}\times...\times¶^{n_t}\times¶^n$ is the Segre embedding of the product and $n$ is "large" with respect to the $n_i$ (Theorem 2.4); $\X$ is a Segre-Veronese embedding of some products with 2 or three factors; $\X$ is a Del Pezzo surface.
To each subvariety $X$ in projective $n$-space of codimension $m$ we associate an integer sequence of length $m + 1$ from $1$ to the degree of $X$ recording the maximal cardinalities of finite, reduced intersections of $X$ with linear subvarieties. We call this the sequence of secant indices of $X$. Similar numbers have been studied independently with the aim of classifying subvarieties with extremal secant spaces. Our focus in this note is the study of the combinatorial properties that the secant indices satisfy collectively. We show these sequences are strictly increasing for nondegenerate s…
Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
Mathematical School of Science, Xiamen University（厦门大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the 3x3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3x3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the…
We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Plücker embedding and of Spinor varieties in both their Plücker and Spinor embeddings.
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most o…
Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the ground-breaking but difficult work of Alexander and Hirschowitz. Also, the non-defectiveness of certain Segre embeddings is proved, w…
Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $¶^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $¶^N$ that can be projected isomorphically into $¶^{2n}$ when $N\geq\binom{n+2}{2}-2$.
In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \subset ¶^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \geq 2$, $q \in ¶^r$ a closed point outside of $X$, and $X_q := π_q (X) \subset ¶^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $Σ_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on $X$ spanning secant lines passing through $q$, is either empty or a quadric in a subspace of $¶^r$.…
Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant varieties use only isolated reduced points. In particular, any secant variety is always contained in the respective cactus variety, and, except in a few initial cases, the inclusion is strict. It is known that lots of natural criteria that test membership in secant varieties are actually only tests for membership in cactus varieties. In this article, we propose the first techniques to distinguish actual secant variety from the cactus variety…
Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.
Let $v_d(\mathbb{P}^2)\subset |\mathcal{O}_{\mathbb{P}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d(\mathbb{P}^2)$ by embedding the blow-up $\mbox{bl}_{v_d(\mathbb{P}^2)}|\mathcal{O}_{\mathbb{P}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on $\mathbb{P}^2$.
We consider higher secant varieties to Veronese varieties. Most points on the r-th secant variety are represented by a finite scheme of length r contained in the Veronese variety --- in fact, for generic point, it is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note we investigate those special points for which the smooth…
The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree $d$ in $n+1$ variables on an algebraically closed field, called $\Split_{d}(\PP n)$, with the Grassmannian of $n-1$ dimensional projective subspaces of $\PP {n+d-1}$. We compute the dimension of some secant varieties to $\Split_{d}(\PP n)$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $\GG (n-1, n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Plücker space, then we are able…
Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. We prove that if the $(k+n-1)$-secant variety of $X$ has (the expected) dimension $(k+n-1)(n+1)-1<r$ and $X$ is not uniruled by lines, then $X$ is not $k$-weakly defective and hence the $k$-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique $S\subset X$ with $\sharp (S) =k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $G_{1,d}$. If $X$ is the Segre embedding of a multiprojective space we prove …
We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability …
We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.
We prove that vector bundles give equations of cactus varieties. We derive from it that equations coming from vector bundles are not enough to define secant varieties of Veronese varieties in general.
Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Let $σ_{a,b}(X)\subseteq \mathbb{P}^r$, $(a,b)\in \mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $\dim σ_{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $\mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $\dim σ_{0,b}(X)$ is the expected o…
A projective variety $X\subset\mathbb{P}^N$ is $h$-identifiable if the generic element in its $h$-secant variety uniquely determines $h$ points on $X$. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect. In this way we are able to improve all known bounds on identifiability. In particular we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.
In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3--secant planes to a variety. Precisely we prove that if $X\subseteq \PP^r$ is an irreducible, non--degenerate, projective complex variety of dimension $n$ with $r\geq 3n+2$, such that the variety of osculating planes to curves in $X$ has the expected dimension $3n$ and for every $0$--dimensional, curvilinear scheme $γ$ of length 3 contained in $X$ the family of hyperplanes sections of $X$ which are singular along $γ$ has dimension larger that $r-3(n+1)$, then $X$ is $2$--secant defective.
The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is the unique case of a Veronese image of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable. The title originates from a paper of 1981, where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. We explore the apolar ideal of a sextic $F$ and the …
