UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 17, 2017

939 Evans Hall

3:45PM: The extremal Betti number of a canonical curve

Michael Kemeny

The resolution of the coordinate ring of a canonically embedded curve has been studied since the beginnings of algebraic geometry. In the 80s, Mark Green famously predicted that the length of the linear strand could be given in terms of a particular invariant of the curve (the Clifford index). A conjecture of Schreyer gives a proposed explanation for this conjecture via the Eagon-Northcott resolution of the scroll associated to a “minimal pencil”. I will explain what all this means and outline a proof of an extension of Schreyer’s conjecture, stating that all syzygies at the end of the linear resolution comes from such scrolls, provided there are only finitely many minimal pencils and up to explicit generality hypotheses. This is joint work with Gavril Farkas.

5:00PM: Introduction to Deformation Theory

Ritvik Ramkumar

In this talk I will give an introduction the notion of a deformation, give examples and explain why deformations can fail to exist. This will include deformations of k -algebras (varieties) and modules over a fixed algebra (coherent sheaves). I will end by describing how deformation theory is used to understand the global geometry of the Hilbert Scheme. In particular, motivate why the Hilbert scheme parameterizing nice objects can be badly behaved.

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