On singular varieties having an extremal secant line

On secant defective varieties, in particular of dimension 4

In this paper we prove some general results on secant defective varieties. Then we focus on the 4--dimensional case and we give the full classification of secant defective 4--folds. This paper has been inspired by classical work by G. Scorza,

On the dimensions of secant varieties of Segre-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…

不变割线簇的非缺陷性

Nondefectivity of invariant secant varieties

本文证明一大类不变割线簇的非缺陷性，解决大维数 Grassmannian 割线猜想的多数情形，并改进 Chow、Segre-Veronese、Gaussian moment 等簇的非缺陷性、可辨识性与一般秩界。

An asymptotic bound for secant varieties of Segre varieties

This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varie…

Osculating spaces to secant varieties

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.

Secant varieties of Segre-Veronese embeddings of (P^1)^r

We use a double degeneration technique to calculate the dimension of the secant variety of any Segre-Veronese embedding of (P^1)^r

On the Dimension of Secant Varieties

We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.

On the regularity of varieties having an extremal secant line

The Castelnuovo-Mumford regularity of varieties of degree r and dimension n in the r-dimensional projective space that have an extremal secant line, is at least d-r+n+1. We classify these varieties and show that their regularity is exactly d-r+n+1, as predicted by the regularity conjecture.

Geometric syzygies of elliptic normal curves and their secant varieties

We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.

Secant Varieties of Grasmann Varieties

In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space to translate the problem into that of finding the dimension of a graded piece of a " fat" ideal in the exterior algebra. Among other things, our methods prove that (apart from the chordal varieties to the Grassmannians of lines) all the chordal varieties of Grassmannians have the expected dimension. We also discuss some Grassmannians with deficient secant va…

Secant Varieties and Birational Geometry

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures o…

某些非奇异射影曲线的secant簇的局部和全局性质

Some local and global properties of secant varieties of nonsingular projective curves

本文研究了代数曲线secant簇的局部和全局性质，补充了先前工作，解决了相关问题并给出了递推公式。

A characterization of secant varieties of Severi varieties among cubic hypersurfaces

It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

On the normality of secant varieties

In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal.

Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

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.

Equations for secant varieties of Veronese and other varieties

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…

Singularities of secant varieties

We study the singularities of the secant variety $Σ(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $Σ(X,L)$ is always Du Bois singular. Examples of secant varieties with worse singularities when $L$ has weak positivity are provided. We also give a necessary and sufficient condition for $Σ(X, L)$ to have rational singularities.

割线簇的有效合冲

Effective syzygies of secant varieties

该研究针对代数几何中割线簇的性质问题，给出了仅依赖于空间维度X和参数p的伴随线丛有效界，推进了割线簇合冲理论的研究。

光滑射影簇的割线簇的奇点与合冲

Singularities and syzygies of secant varieties of smooth projective varieties

本文研究光滑射影簇的高阶割线簇，证明曲面情形在正线丛下具有Du Bois奇点及线性合冲，并揭示其奇点类型由曲面结构层上同调决定，同时给出高维低阶情形的类似结果及高阶情形的否定性结论。

Grassmannian割线簇的不可识别性与奇异轨迹

Identifiability and singular locus of secant varieties to Grassmannians

本研究利用SL(V)-簇结构确定了Grassmannian线割线簇的奇异轨迹，并解决了其中点的不可识别性与切向不可识别性问题，进而确定了第二Terracini轨迹。

Higher secants of spinor varieties

Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $σ_k(S_h) $ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $σ_k(S_h) $. Then, by using an inductive argument, we get our main result: $σ_3(S_h) $ has the expected dimension except when $h\in \{7,8\} $. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.

Higher secant varieties of 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.

Distinguishing secant from cactus varieties

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…

格拉斯曼流形的受限割线簇

Restricted Secant Varieties of Grassmannians

本文研究格拉斯曼流形中带交集维数限制的受限割线簇，通过关联簇构造建立维数公式，并给出计算示例及编码理论应用。

Ranks on the boundaries of secant varieties

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point of the join of $b-2$ copies of $X$ and the tangential variety of $X$ is $>b$.

曲线割线簇奇点的不变量

Invariants of the singularities of secant varieties of curves

本文研究光滑射影曲线的 $k$-平面割线簇的奇点不变量，计算了任意曲线的交点上同调，并证明有理正规曲线的所有割线簇均为满足 Poincaré 对偶的有理同调流形，同时计算了最大非平凡割线簇的邻近圈与消没圈。

Refined de Jonquières divisors and secant varieties on algebraic curves

The aim of this paper is to provide another perspective on secant varieties on algebraic curves by reformulating the problem in terms of refined de Jonquières divisors, that is divisors on the curve with prescribed multiplicities and dimensions of their spaces of global sections. We are able to both recover some already known results and to obtain some new statements concerning the dimension theory of secant varieties. We do this via the study of the dimension theory of refined de Jonquières divisors in some relevant cases and degeneration arguments.

A probabilistic algorithm for the secant defect of Grassmann varieties

In this paper we study the higher secant varieties of Grassmann varieties in relation to Waring's problem for alternating tensors and to Alexander-Hirschowitz theorem. We show how to identify defective higher secant varieties of Grassmannians using a probabilistic method involving Terracini's Lemma, and we describe an algorithm which can compute, by numerical methods, dim(G(k,n)^{s}) for n<=14. Our main result is that, except for Grassmannians of lines, if n<=14 and k<=(n-1)/2 (if n=14 we have studied the case k<=5) there are only the four known defective cases: G(2,6)^{3}, G(3,7)^{3}, G(3,7)^…

Secant varieties of P^2 x P^n embedded by O(1,2)

We describe the defining ideal of the rth secant variety of P^2 x P^n embedded by O(1,2), for arbitrary n and r at most 5. We also present the Schur module decomposition of the space of generators of each such ideal. Our main results are based on a more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces. This extends previous work of Strassen and Ottaviani.

Secant indices of projective varieties

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…

Grassmannians of secant varieties

For an irreducible projective variety X, we study the family of h-planes contained in the secant variety Sec_k(X), for 0<h<k. These families have an expected dimension and we study varieties for which the expected dimension is not attained; for these varieties, making general consecutive projections to lower dimensional spaces, we do not get the expected singularities. In particular, we examine the family G of lines sitting in 3-secant planes to a surface S. We show that the actual dimension of G is equal to the expected dimension unless S is a cone or a rational normal scroll of degree 4 in P…

Higher secant varieties of $\mathbb{P}^n \times \mathbb{P}^m$ embedded in bi-degree $(1,d)$

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 $s$.

Singularities and syzygies of secant varieties of nonsingular projective curves

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfie…

Secant varieties and the complexity of matrix multiplication

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical computer science, and on the other hand touches on many beautiful topics in algebraic geometry such as classical and recent results on equations for secant varieties (e.g., via vector bundle and representation-theoretic methods) and the geometry and deformation theory of zero dimensional schemes.

从Hodge理论视角看割线簇的奇点

Singularities of secant varieties from a Hodge theoretic perspective

本文从Hodge理论视角，利用双有理几何方法研究光滑射影簇割线簇的奇点，给出了p-Du Bois奇点的充要条件，并证明其奇点非更高有理，进而推导出Kodaira-Akizuki-Nakano型消失定理。

Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes

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…

Nondefective secant varieties of varieties of completely decomposable forms

A variation of Waring's problem from classical number theory is the question, ``What is the smallest number $s$ such that any generic homogeneous polynomial of degree $d$ in $n+1$ variables may be written as the sum of at most $s$ products of linear forms?'' This question may be answered geometrically by determining the smallest $s$ such that the $s$\nth secant variety of the variety of completely decomposable forms fills the ambient space. If this secant variety has the expected dimension, it is called nondefective, and $s=\left\lceil\binom{n+d}{d}/(dn+1)\right\rceil$. It is conjectured that …

Finite schemes and secant varieties over arbitrary characteristic

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 …

第一和第二割簇的希尔伯特多项式

Hilbert polynomials of the first and second secant varieties

本文通过描述光滑射影复簇上2或3点的Hilbert方案上充分正线丛的对称幂的上同调群，将其维数用第一和第二割簇的希尔伯特多项式表示，并完全计算了这些多项式。

Tangential projections and secant defective varieties

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 $N\geq\binom{n+2}{2}-2$.

曲线高割线簇的Ulrich层与行列式表示

Ulrich sheaves and determinantal representations for higher secant varieties of curves

研究光滑射影曲线高割线簇的行列式表示，利用高阶Szegő核等工具，在嵌入充裕时给出对称表示，对实曲线割线簇给出正定条件，对有理正规曲线割线簇给出明确表示，还能在特定条件下将曲线凸包表示为谱面体。

关于割线簇的四讲

Four lectures on secant varieties

本文基于2012年讲座，介绍高阶割线簇理论及其应用，并附参考文献与习题解答。

Secant varieties and Hirschowitz bound on vector bundles over a curve

For a vector bundle V over a curve X of rank n and for each integer r in the range 1 \le r \le n-1, the Segre invariant s_r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan's results on rank 2 bundles which related the invariant s_1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s_r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the…

Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

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…

Duality, Bridgeland wall-crossing and flips of secant varieties

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 $\mathbb{P}^2$.

Hyperbolic Secant Varieties of M-Curves

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. Fo…

Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited

Let $X={\bf P}^2\times{\bf P}^{n-1}$ embedded with $Ø(1,2)$. We prove that its $(n+1)$-secant variety $σ_{n+1}(X)$ is a hypersurface, while it is expected that it fills the ambient space. The equation of $σ_{n+1}(X)$ is the symmetric analog of the Strassen equation. When $n=4$ the determinantal map takes $σ_5(X)$ to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hint allows to obtain some results on the jumping lines and the Brill-Noether loci of symplectic bundles on ${\bf P}^2$ by using the higher secant varieties of $X$.

Geometry of intersections of some secant varieties to algebraic curves

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

On secant dimensions and identifiability of Flag varieties

We investigate the secant dimensions and the identifiablity of flag varieties parametrizing flag of sub vector spaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.

部分对称张量与含射影直线作为因子的乘积的割线维数的非亏性

Partially symmetric tensors and the non-defectivity of secant varieties of products with a projective line as a factor

该研究在多重次数轻微限制下，证明含k-2个射影直线因子的k>2因子Segre-Veronese簇的所有割线簇具预期维数，等价于对应部分对称张量集的维数计算，结果基于k=2情形及新证的一般射影簇定理。

Secant cumulants and toric geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain ana…

All secant varieties of the Chow variety are nondefective for cubics and quaternary forms

The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We determine the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties. The main new ingredient in our proof is the generalization of a technique by [Brambilla and Ottaviani, On the Alexander--Hirschowitz theorem, J. Pure Appl. Algebra, 2008] that consists of employing Terracini's lemma and Newton's backward d…

On the ideals of secant varieties of Segre varieties

We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

A generalization of the Clifford index and determinantal equations for curves and their secant varieties

This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…