The Eisenbud-Goto Conjecture

Abstract: Let S be a polynomial ring over an algebraically closed field K. There has been considerable research into effective upper bounds for the Castelnuovo-Mumford regularity of graded ideals of S. Through the work of Bertram, Ein, Gruson, Kwak, Lazarsfeld, Peskine, and others, there are several good bounds for the defining ideals of smooth projective varieties in characteristic zero. However, for arbitrary ideals, the best upper bound is doubly exponential (in terms of the number of variables and degrees of generators), and this bound is asymptotically close to optimal due to examples derived from the Mayr-Meyer construction. In 1984, Eisenbud and Goto conjectured that the regularity of a nondegenerate prime ideal P was at most deg(P) – codim(P) + 1, and proved this when S/P was Cohen-Macaulay (even if P is not prime). In this talk, I will explain the construction of counterexamples to the Eisenbud-Goto Conjecture, joint work with Irena Peeva, through the construction of Rees-Like algebras and a special homogenization. While we show that there is no linear bound on regularity in terms of the degree (or multiplicity) of P, we later showed that some such bound exists. The latter part of this talk is joint work with Giulio Caviglia, Marc Chardin, Irena Peeva, and Matteo Varbaro.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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