28 research outputs found
Mixture decompositions of exponential families using a decomposition of their sample spaces
We study the problem of finding the smallest such that every element of
an exponential family can be written as a mixture of 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 is the smallest number for which any
distribution of -ary variables can be written as mixture of
independent -ary variables. Furthermore, we show that any distribution of
binary variables is a mixture of elements
of the -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 â€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 âgeneric choice of Zâ
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 âfatâ points of P2
AbstractGiven any s points P1,âŠ, Ps in P2 and s positive integers m1,âŠ, ms, let Sn be the linear system of plane curves of degree n through Pi with multiplicity at least mi (1 â©œ i â©œ 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