A Betti stratification of apolar ideals to quaternary quartics, ll

Abstract

In the second of two lectures,l am going to draw people's attention to the`geometry of syzygies". We begin with a research problem from geometry: Findingall Gorenstein Calabi-Yau varieties. In dimension 3 cases, their coordinate rings have regularity 4.There are 16 Betti diagrams for such graded Gorenstein algebras with projective dimension 4. Byapolarity, they correspond to degree 4 forms in 4 variables, i.e. quaternary quartics. Note that theBetti diagram encodes how the algebraic set is embedded in the ambient space. In fact, each of the16 Betti diagrams corresponds to the geometric description of the apolar set, which is a set ofpoints in a projective space. With this description, we find a stratification of the parameter space ofquaternary quartics.

Speaker Intro

Beihui Yuan gained her Ph.D. degree from Cornell University in 2021. She has joined BIMSA in2023. Her current research interests include application of commutative algebra in pure and appliedmathematics problems.