Xianglong Ni (UC Berkeley)

Title:  Perfect ideals of grade three with small Betti numbers

Abstract:  Using the theory of "higher structure maps" originating from Weyman's generic ring, we give a new perspective on minimal linkage for perfect ideals of grade three. We prove that, in addition to the well-known Gorenstein and almost complete intersection cases, all such ideals with Betti numbers up to (1,7,8,2) or (1,5,8,4) are in the linkage class of a complete intersection. This is the sharpest possible result, since non-examples are known for all other Betti numbers. Moreover, for these particular Betti numbers, we also obtain analogues of the Buchsbaum-Eisenbud structure theorem for Gorenstein ideals. These "small" cases come from a surprising connection to the ADE classification, which lies at the heart of the whole theory. This is ongoing joint work with Lorenzo Guerrieri and Jerzy Weyman.