UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 18, 2014

939 Evans Hall


3:45PM: Boij-S\"oderberg theory of coherent sheaves on projective spaces and squarefree modules over polynomial rings

Gunnar Floystad

We consider i) the positive cone of cohomology diagrams of complexes of coherent sheaves on projective spaces, with specified regularity conditions on the cohomology sheaves, and ii) the positive cone of homological invariants of complexes of squarefree modules over polynomial rings. We show that these cones are closely related and conjecturally isomorphic.

5:00PM: Linear Spaces of Singular Matrices

Nathan Ilten

Simple questions in linear algebra often give rise to interesting objects in algebraic geometry. For example, the space of singular $n \times n$ matrices is the classical hypersurface given by the vanishing of the $n\times n$ determinant. This hypersurface has interesting geometric properties. In particular, it contains many linear subspaces, whose study is related to (among other things) geometric complexity theory. In this talk, I will introduce a geometric object, called a Fano scheme, parameterizing these linear subspaces. We’ll look at some basic examples of Fano schemes, and then see a characterization of exactly when our Fano schemes are connected. This is joint work with Melody Chan.

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