Real theta characteristics and automorphisms of a real curve

Abstract

Let X be a geometrically irreductble smooth projective cruve defined over R. of genus at least 2. that admits a nontrivial automorphism, σ. Assume that X does not have any real points. Let τ be the antiholomorphic involution of the complexification λ(C) of X. We show that if the action of σ on the set S(X) of all real theta characteristics of X is trivial. then the order of sigma is even, say 2k and the automorphism τ o (σ) over cap (λ) of X-C has a fixed point, where (σ) over cap is the automorphism of X x C-R defined by sigma We then show that there exists X with a real point and admitting a nontrivial automorphism sigma, such that the action of σ on S(X) is trivial, while X/σ ≠ P-R(1) We also give an example of X with no real points and admitting a nontrivial automorphisim sigma such that the automorphism τ o (σ) over cap (λ) has a fixed point, the action of sigma on S(X) is trivial, and X/σ ≠ P-R(1).