UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

August 31, 2010

939 Evans Hall

3:45PM: Resolutions over Complete Intersections; can you generalize matrix factorizations?

David Eisenbud

Free resolutions of modules over a graded ring R are interesting and mysterious when R is a polynomial ring over a field (or a regular local ring), and even more so when R is not. One of the few non-regular cases where we know a lot is the case where R is a hypersurface ring; that is R=S/(f) where S is a polynomial ring and f is a homogeneous form. Then the "infinite part" of the resolution corresponds to a matrix factorization of f; that is, a way of writing f times an identity matrix as the product of two matrices, each with entries of stricly positive degree. In some ways resolutions over all complete intersections follow similar patterns, so it seems reasonable to hope for some sort of generalization of the picture for hypersurfaces to complete intersections of higher codimension. In the codimension 2 case progress was made by Avramov and Buchweitz, and Ed Carter's recently completed Master's thesis carries this a little further. Irena Peeva and I have been working on a different approach. I'll explain some of what's known, and some of the newer ideas and open questions that have appeared.

5:00PM: 3x3 Minors of Catalecticants

Claudiu Raicu

Catalecticant matrices are a common generalization of Hankel matrices and generic symmetric matrices. Their ideals of minors give (some) equations for the secant varieties to Veronese embeddings of projective spaces. There is a connection between catalecticant matrices and Artinian Gorenstein rings which yields certain up-to-radical inclusions between ideals of minors of catalecticant matrices, and Geramita asked whether these are actual inclusions. In particular, he asked whether the 3x3 minors of the ``middle'' catalecticants are all equal, and conjectured that they should generate the ideals of the secant line varieties to the Veronese varieties. I will introduce a polarization technique and explain how to use it in order to prove the equality of 3x3 minors and Geramita's conjecture.

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