Mixture decompositions of exponential families using a decomposition of their sample spaces

We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m=q^{N-1}$ is the smallest number for which any distribution of $N$ $q$-ary variables can be written as mixture of $m$ independent $q$-ary variables. Furthermore, we show that any distribution of $N$ binary variables is a mixture of $m = 2^{N-(k+1)}(1+ 1/(2^k-1))$ elements of the $k$-interaction exponential family.Comment: 17 pages, 2 figure

On curvilinear subschemes of P2

AbstractLet Z be a curvilinear subscheme of P2, i.e. a zero-dimensional scheme whose embedding dimension at every point of their support is ≤1. We find bounds for the minimum degree of the plane curves on which Z imposes independent conditions and we show that the Hilbert function of Z is maximal for a “generic choice of Z”

Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of the 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

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 Segre-Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat

Regina Lectures on Fat Points

These notes are a record of lectures given in the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29--June 1, 2012. The lectures were meant as an introduction to current research problems related to fat points for an audience that was not expected to have much background in commutative algebra or algebraic geometry (although sections 8 and 9 of these notes demand somewhat more background than earlier sections).Comment: 32 pages, 3 figure

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

3-dimensional sundials

R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on P^n, n>2, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines)

Linear systems of plane curves through fixed “fat” points of P2

AbstractGiven any s points P1,…, Ps in P2 and s positive integers m1,…, ms, let Sn be the linear system of plane curves of degree n through Pi with multiplicity at least mi (1 ⩽ i ⩽ s). We give numerical bounds for the regularity of Sn in the following cases (a) the points Pi are non-singular points of an integral curve of degree d; (b) the Pi's are in general position; (c) the Pi's are in uniform position; (d) the Pi's are generic points of P2. We also study the sharpness of such bounds

Higher Secant Varieties of the Segre Varieties P^1 x ... x P^1

Let Vt=P1x \u2026 x P1 be the product of t copies of the 1-dimensional projective space P1, embedded in the N-dimensional projective space PN via the Segre embedding. Let (Vt)^s be the s-secant varieties of Vt, that is, the subvariety of PN which is the closure of the union of all the (s-1)-dimensional projective space s-secant to Vt. The expected dimension of (Vt)^s is min { st + (s-1), N }. This is not the case for (V4)^3, which we conjecture is the only defective example in this infinite family. We show that all the higher secant varieties (Vt)^s have the expected dimension\u2014except, possibly, for one higher secant variety for each such t. Moreover, whenever t +1 is a power of 2, (Vt)^s has the expected dimension for every s