UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 13, 2010

939 Evans Hall


3:45PM: Push-outs in algebraic geometry

David Rydh

Push-outs are frequently used in topology and simple to describe. In algebraic geometry on the other hand, push-outs are more unwieldy and far less utilized. In this talk I'll motivate why push-outs nevertheless constitute a powerful technique and explain the ins and outs of two fundamental classes of push-outs.

5:20PM: Linear spaces of multigraded Betti diagrams

Thanh Vu

Let S = k[x_1, ..., x_n] be the polynomial ring. For each strictly increasing sequence of integers d = (d_0 < ... < d_n), denote the difference sequence by e = (d_1 - d_0, ..., d_n - d_n-1). For each such d, there exists a pure free resolution of the form S(-d_0)^b_0 <- ... <- S(-d_n)^b_n which is also GL(n)- equivariant, therefore has a natural multi-graded structure with multi-graded Betti diagram B. It turns out that all multigraded Betti diagrams of possible multigraded structures of pure free resolutions with difference sequences e are linear combination of the twists of B as announced by Floeystad. I'll explain this result and the description of the positive cone of these multigraded Betti diagrams.

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