The parity of theta characteristics is preserved by infinitesimal deformations

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.