Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

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 of the known results and the open problems on this fascinating subject.

Geometry of Higher-Order Markov Chains

We determine an explicit Grobner basis, consisting of linear forms and determinantalquadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains.When the states are binary, the corresponding projective variety is a linear space, the model itselfconsists of two simplices in a cross-polytope, and the likelihood function typically has two localmaxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.