MIT Combinatorics Seminar
Combinatorial Secant Varieties
Seth Sullivant (Harvard University)
Email: seths@math.harvard.edu
Wednesday, February 22, 2006 4:30 pm Room 2105
ABSTRACT

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.


