4:10PM: The Eisenbud-Goto Regularity Conjecture
Irena PeevaRegularity is a numerical invariant that measures the complexity of the structure of homogeneous ideals in a polynomial ring. Papers of Bayer-Mumford and others give examples of families of ideals attaining doubly exponential regularity. In contrast, Bertram-Ein-Lazarsfeld, Chardin-Ulrich, and Mumford have proven that there are nice bounds on the regularity of the ideals of smooth (or nearly smooth) projective varieties. As discussed in an influential paper by Bayer and Mumford (1993), the biggest missing link between the general case and the smooth case is to obtain a decent bound on the regularity of all prime ideals (the ideals that define irreducible projective varieties). The long standing Eisenbud-Goto Regularity Conjecture (1984) predicts an elegant linear bound, in terms of the degree of the variety (also called multiplicity). The conjecture was proven for curves by Gruson-Lazarsfeld-Peskine, for smooth surfaces by Lazarsfeld and Pinkham, for most smooth 3-folds by Ran, and in many other special cases. Recently, McCullough and I introduced two new techniques related to the graphs of rational maps of projective spaces, and used them to provide many counterexamples to the Eisenbud-Goto conjecture. In fact we show that the regularity of prime ideals is not bounded by any polynomial function of the degree. I will explain the ideas behind these new techniques.