Residues and Gorenstein Contractions of Genus One Curves

TL;DR

This paper introduces a new residue theory for genus one nodal curves over local artinian rings and uses it to construct contractions that produce Gorenstein genus one singularities.

Contribution

It defines residues for curves over local artinian rings and employs them to create contractions collapsing subcurves to Gorenstein singularities, advancing the understanding of curve degenerations.

Findings

Residues for genus one curves over local artinian rings are established.

A method to contract subcurves to Gorenstein singularities is developed.

The approach links tropical data with algebraic curve contractions.

Abstract

Let $C$ be a genus one nodal curve over a local artinian base and let $E$ be a proper subcurve of genus one. We define residues for curves over local artinian rings, then define generalized residues with values in line bundles over the local artinian ring that arise from tropical data on the curve. We then use these residues to construct a contraction of $C$ that collapses $E$ to a Gorenstein genus one singularity.