The cone of Betti diagrams over a hypersurface ring of low embedding dimension

TL;DR

This paper characterizes the cone of Betti diagrams over a specific hypersurface ring and introduces an algorithm for decomposing these diagrams into pure components, extending Boij–Söderberg theory beyond polynomial rings.

Contribution

It provides the first complete description of the Betti diagram cone over a hypersurface ring of low embedding dimension and an algorithm for their decomposition.

Findings

Complete description of the Betti diagram cone over the hypersurface ring

Finite algorithm for decomposing Betti diagrams into pure diagrams

Extension of Boij–Söderberg theory to new class of rings

Abstract

We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij--Soederberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood.