Infinitesimal deformations of nodal stable curves

TL;DR

This paper provides an analytic framework for understanding the infinitesimal deformations of nodal stable curves, generalizing classical results and offering explicit formulas for moduli space structures.

Contribution

It introduces a new analytic approach to describe the moduli cotangent sheaf for stable curves with nodes, extending the role of quadratic differentials to singular settings.

Findings

The direct image sheaf is locally free and exact for suitable stable curve families.

Residue maps at nodes lead to an exact sequence involving the vanishing residue subsheaf.

Formulas are developed for pairing infinitesimal node openings with sections of the direct image sheaf.

Abstract

An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. For suitable stable curve families, the direct image sheaf on the base is locally free and the sequence of direct images is exact. Recent work of Hubbard-Koch and a formal argument provide that the direct image sheaf is naturally identified with the moduli cotangent sheaf. The result generalizes the role of holomorphic quadratic differentials as cotangents for smooth curve families. Formulas are developed for the pairing of an infinitesimal opening of a node and a section of the direct image sheaf. Applications include an analytic description of the conormal sheaf for the locus of noded stable curves and a formula comparing infinitesimal openings of a node. The moduli action of the automorphism group of a stable curve is described. An example of plumbing an Abelian differential and the corresponding period variation is presented.