Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: The extremal Betti number of a canonical curve

Seminar | October 17 | 3:45-4:45 p.m. | 939 Evans Hall

 Michael Kemeny, Stanford University

 Department of Mathematics

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.