UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 10, 2015

939 Evans Hall

3:45PM: Higher CI operators

David Eisenbud

The "CI operators" defined on free resolutions over a complete intersection R give a module structure on Ext_R(M,k) that is the main tool for analyzing such resolutions. Burke and Schreyer and I have discovered a series of higher operators extending this idea. I will explain the classical theory and some of the contexts in which these higher operators appear.

5:00PM: Counting lines on toric surfaces

Nathan Ilten

In 1849, Cayley and Salmon proved that any smooth cubic surface contains exactly 27 lines, marking the start of enumerative algebraic geometry. In this talk, I show how to count (with multiplicity) the number of lines on any projective toric surface. More precisely, for any such surface I give a complete description of its Fano scheme, which is the fine moduli space parametrizing lines contained in the surface. Using degeneration arguments, one can obtain bounds on the number of lines on certain non-toric surfaces,. In particular, I recover Cayley and Salmons classic count of 27 lines on a smooth cubic surface.

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