Some Results on Secant Varieties Leading to a Geometric Flip Construction

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, a flip first described in the case of curves by M. Thaddeus via Geometric Invariant Theory.     

On the Ideals of Secant Varieties of Segre Varieties

We establish basic techniques for determining 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.     

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 homogeneous examples of this method.     

Secant varieties of Grassmann varieties

We consider the dimensions of the higher secant varieties of the Grassmann varieties. We give new instances where these secant varieties have the expected dimension and also a new example where a higher secant variety does not.

     

On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces 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.     

Combinatorics of the Toric Hilbert Scheme

The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P ⁴ whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Received May 15, 2000, and in revised form March 8, 2001. Online publication January 7, 2002.     

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 sth Segre–Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre–Veronese varieties.     

Products of Young symmetrizers and ideals in the generic tensor algebra

We describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau, generalizing the classical quasi-idempotence of Young symmetrizers. We derive some consequences to the structure of ideals in the generic tensor algebra and its partial symmetrizations. Instances of these generic algebras appear in the work of Sam and Snowden on twisted commutative algebras, as well as in the work of the author on the defining ideals of secant varieties of Segre–Veronese varieties, and in joint work of Oeding and the author on the defining ideals of tangential varieties of Segre–Veronese varieties.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Distinguishing secant from cactus varieties

Foundations of Computational Mathematics - Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant...     

On the secant varieties to the tangential varieties of a Veronesean

We study the dimensions of the higher secant varieties to the tangent varieties of Veronese varieties. Our approach, generalizing that of Terracini, concerns 0-dimensional schemes which are the union of second infinitesimal neighbourhoods of generic points, each intersected with a generic double line.

We find the deficient secant line varieties for all the Veroneseans and all the deficient higher secant varieties for the quadratic Veroneseans. We conjecture that these are the only deficient secant varieties in this family and prove this up to secant projective 4-spaces.

     

Three notions of tropical rank for symmetric matrices

We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.     

A tropical approach to secant dimensions

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. 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; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included.     

Binary Cumulant Varieties

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2×2×···×2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary cumulants. Our primary focus lies on hidden subset models. Parametrizations and implicit equations in cumulants are derived for hyperdeterminants, for secant and tangential varieties of Segre varieties, and for certain context-specific independence models. Extending work of Rota and collaborators, we explore the polynomial inequalities satisfied by cumulants.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Different Definitions of Algebraically Closed Skew Fields

We consider an algebraically closed (in the sense of solvability of arbitrary polynomial equations) skew field constructed by Makar-Limanov. It is shown that every generalized polynomial equation with more than one homogeneous component has a non-zero solution. We also look into P. Cohn's approach to defining algebraically closed non-commutative skew fields and treat some related problems.     

Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points.     

Error-Correcting Codes from Higher-Dimensional Varieties

In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric error-correcting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from Deligne–Lusztig varieties, complete intersections of Hermitian hyper-surfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.     

Singularities of the secant variety

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.