Halfcanonical Gorenstein curves of codimension four

TL;DR

This paper explores the classification of Gorenstein curves of codimension four, providing explicit examples and addressing deformation possibilities between different Betti table types, advancing understanding of their algebraic structure.

Contribution

It constructs explicit examples of stable Gorenstein curves filling the second half of the Betti table classification and solves a specific deformation case between two such curves.

Findings

Constructed examples of stable Gorenstein curves with specific Betti tables.

Established a deformation between a reducible curve and a del Pezzo intersection.

Contributed to the classification of Artin Gorenstein algebras with regularity 4.

Abstract

Recent work of Schenck, Stillman and Yuan arXiv:2011.10871 outlines all possible Betti tables for Artin Gorenstein algebras $A$ with regularity($A$) = 4 = codim($A$). We populate the second half of this list with examples of stable curves, and ask if there are further possible constructions. The problem of deformation between curves with the same Hilbert series but different Betti tables is ongoing work, but our work solves one case: a deformation (due to Jan Stevens) between a reducible curve corresponding to Betti table type 2.7 in arXiv:2011.10871 and the curve obtained as the intersection of a del Pezzo surface of degree 5 and a cubic hypersurface.