UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 22, 2016

939 Evans Hall

3:45PM: Degenerating Grassmannians into toric varieties

Fatemeh Mohammadi

A toric variety is a certain algebraic variety modeled on a convex polyhedron. Toric varieties play an important role in commutative algebra. I will talk about the method of degenerating the Grassmannian Gr$(d,n)$ into toric varieties. The Grassmannian Gr$(d,n)$ is the subvariety of the projective space $\mathbb{P}^{{n\choose d}-1}$ consisting of all $d$-dimensional subspaces of $\mathbb{K}^n$ (defined by the Pl\"ucker ideal). A point in the tropical Grassmannian arises from a classical plane defined by a rank $d$ matrix of size $d\times n$. The top dimensional faces (facets) of the tropical Grassmannian are good candidates for toric degenerations via initial ideals of the Pl\"ucker ideal. For the Grassmannians Gr$(2, n)$ each facet of the tropical Grassmannian gives a toric degeneration. But for Gr$(3, n)$ the situation is not as nice, and there are facets of the tropical Grassmannian which do not produce toric degenerations. However, each top dimensional cone of the tropical Gr$(3,n)$ determines a combinatorial tree arrangement. In an ongoing joint work with Kristin Shaw, we identify which facets of the tropical Gr$(3, n)$ produce toric degenerations in terms of the combinatorics of their associated tree arrangements.

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