MIT Combinatorics Seminar
Combinatorial Secant Varieties
Seth Sullivant, (Harvard University)
Wednesday, December 7, 2005 4:30 pm Room 2-142
Given two projective varieties X and Y, their join X*Y is obtained by
taking the Zariski closure of the union of
all lines spanned by a point in X and a
point in Y. The join of a variety X with
itself is called the secant variety of X.
In this talk, I will describe the
construction of joins and secants in the
combinatorial context of monomial ideals.
For ideals generated by squarefree
quadratic monomials, the generators of the
secant ideals are obstructions to graph
colorings and this leads to a commutative
algebra version of the Strong Perfect Graph
Theorem. Questions about secant varieties
of combinatorially defined varieties
(e.g. Grassmannians, determinantal
varieties, toric varieties) can often be
reduced to the monomial case. I will try
to emphasize the combinatorial aspects of
all of this, including the connections to
graph theory, regular triangulations, and
partially ordered sets.
This is joint work with Bernd Sturmfels.