UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 09, 2010

939 Evans Hall

3:45PM: Mustafin Varieties

Bernd Sturmfels

A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of $ PGL(2)$ and in the two-dimensional building of $PGL(3)$. The latter yields the classification of Mustafin triangles into $38$ combinatorial types.

5:00PM: Finiteness conjectures about the projective coordinate rings of X^n/G as n varies

Andrew Snowden

Let G be a reductive group (such as GL(n) or a finite group) acting on a finitely generated ring R. The ring of invariants R^G is then well-behaved (in the sense that it is also finitely generated). What happens when we look at the sequence of invariant rings R^G, (R tensor R)^G, (R tensor R tensor R)^G, etc? Are the terms well-behaved, in a uniform sense? (The motivation for considering this sequence comes from geometry, where it corresponds to looking at the diagonal action on products of a given space.) I conjecture that these rings are well-behaved, in a uniform sense. I will explain the precise sense in which I mean this. I will also explain two features of the conjecture: first, that for any G and R it can be thought of as a very explicit problem in graph theory (I will give several examples of G's and R's and the corresponding graph theory problems); and second, that it can be rephrased into a more abstract and structural statement about certain algebraic objects, which may indicate that it fits into a broader theory.

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