Gorenstein contractions of multiscale differentials

TL;DR

This paper proves that multiscale differentials in one-parameter degenerations can be contracted to Gorenstein singularities, with the residue conditions governing their smoothability.

Contribution

It confirms the conjecture that multiscale differentials can be contracted to Gorenstein singularities level by level in degenerations.

Findings

Multiscale differentials can be contracted to Gorenstein singularities.

Residue conditions for smoothability are special cases of descent differential conditions.

Abstract

Multiscale differentials arise as limits of holomorphic differentials with prescribed zero orders on nodal curves. In this paper, we address the conjecture concerning Gorenstein contractions of multiscale differentials, originally proposed by Ranganathan and Wise and further developed by Battistella and Bozlee. Specifically, in the case of a one-parameter degeneration, we show that multiscale differentials can be contracted to Gorenstein singularities, level by level, from the top down. At each level, these differentials descend to generators of the dualizing bundle at the resulting singularities. Moreover, the global residue condition, which governs the smoothability of multiscale differentials, appears as a special case of the residue condition for descent differentials.