Algebra and Combinatorics Seminar

Date: April 28, 2023

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Patricia Klein, Texas A&M University

Title: Geometric vertex decomposition and liaison

Abstract: Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we will describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras. This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gröbner bases. Time permitting, we will describe briefly, as an application of this work, a proof of a conjecture of Hamaker, Pechenik, and Weigandt on diagonal Gröbner bases of certain Schubert determinantal ideals. This talk is based on joint work with Jenna Rajchgot.