Speaker: Jason McCullough, Iowa State University, Ames, IA, USA
Date/Time: 26 March 2021, 6:30pm IST/ 1:00pm GMT / 9:00am EDT (joining
time 6:15pm IST).
Google meet link:
https://meet.google.com/qcy-jrqf-naa
Title: * 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 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.
For more information and links to previous seminars, visit the website of
VCAS:
https://sites.google.com/view/virtual-comm-algebra-seminar