The parity of theta characteristics is preserved by infinitesimal deformations

TL;DR

This paper proves that the parity of relative theta characteristics remains invariant under infinitesimal deformations in families over smooth curves, extending classical invariance results.

Contribution

It establishes the invariance of theta characteristic parity in infinitesimal deformations and describes the splitting of related sheaves.

Findings

Parity of theta characteristics is preserved in infinitesimal deformations.

The torsion subsheaf of the higher direct image sheaf splits into two isomorphic sheaves.

Abstract

In this note, given a family of relative dimension one over a smooth curve, we determine the parity of the restriction of a relative theta characteristic to an arbitrary multiple of a fiber in terms of the parity of the restriction to a general fibre. This result can be regarded as a variant of the well-known theorem on the invariance of the parity of theta characteristics in families. As a corollary, we obtain that the torsion subsheaf of the first higher direct image sheaf of a relative theta characteristic splits as a direct sum of two isomorphic sheaves.