Automorphisms of curves fixing the order two points of the Jacobian

TL;DR

This paper characterizes hyperelliptic curves by their automorphisms that fix all order two points of the Jacobian or all theta characteristics, showing such automorphisms are hyperelliptic involutions.

Contribution

It proves that automorphisms fixing all order two points or theta characteristics must be the hyperelliptic involution, providing a new characterization of hyperelliptic curves.

Findings

Automorphisms fixing all order two points are hyperelliptic involutions.

Automorphisms fixing all theta characteristics are hyperelliptic involutions.

Such automorphisms uniquely determine the hyperelliptic structure.

Abstract

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism \sigma that fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with \sigma being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms \sigma' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with \sigma' being its hyperelliptic involution.