UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 20, 2011

939 Evans Hall

3:45PM: Free Resolutions of Determinantal Ideals Modulo Hyperplanes

Adam Boocher

In 1982, Giusti and Merle studied matrices whose entries are distinct variables and zeros. In particular, they studied the ideal of maximal minors, and showed that many invariants depended only on the size of the biggest rectangle of zeros. In this talk, I'll present a technique for computing the minimal free resolution of these ideals. As a consequence we can show that the ideals studied by Merle and Giusti all have a linear resolution, and that the projective dimension depends only on the number of columns of the matrix which are identically zero. Along the way, we'll discuss related questions Gro\''bner bases, initial ideals, and their free resolutions.

3:45PM: Algebraic boundaries of Hilbert's SOS cones

Bernd Sturmfels

I will present joint work with Greg Blekherman, Jon Hauenstein, John Ottem and Kristian Ranestad on the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. A computation of their degrees using numerical algebraic geometry verifies a formula of Maulik and Pandharipande. The non-sos extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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