Deformations of half-canonical Gorenstein curves in codimension four

TL;DR

This paper explores how singular Gorenstein codimension four varieties can be deformed into nonsingular ones within the same Hilbert scheme, expanding understanding of their geometric relationships.

Contribution

It demonstrates flat deformations connecting singular and nonsingular Gorenstein codimension four varieties, illustrating their place within the same Hilbert scheme.

Findings

Many singular Gorenstein varieties are specialisations of nonsingular varieties.

Deformations are realized within the same Hilbert scheme.

The work links different Betti table constructions.

Abstract

Recent work of Ablett arXiv:2112.03400 and Kapustka, Kapustka, Ranestad, Schenck, Stillman and Yuan arXiv:2111.05817 outlines a number of constructions for singular Gorenstein codimension four varieties. Earlier work of Coughlan, Go{\l}\c{e}biowski, Kapustka and Kapustka arXiv:1609.01195 details a series of nonsingular Gorenstein codimension four constructions with different Betti tables. In this paper we exhibit a number of flat deformations between Gorenstein codimension four varieties in the same Hilbert scheme, realising many of the singular varieties as specialisations of the earlier nonsingular varieties.