Real theta characteristics and automorphisms of a real curve

Let X be a geometrically irreducible smooth projective curve defined over xs211D, 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 xxs2102 of X. We show that if the action of σ on the set xs1D4AE(X) of all real theta characteristics of X is trivial, then the order of σ is even, say 2k, and the automorphism $\tau \circ \widehat {\sigma }^k$ of Xxs2102 has a fixed point, where $\widehat {\sigma }^k$ is the automorphism of X×xs211Dxs2102 defined by σ. We then show that there exists X with a real point and admitting a nontrivial automorphism σ, such that the action of σ on xs1D4AE(X) is trivial, while X/xs2329σxs232A≠xs21191xs211D. We also give an example of X with no real points and admitting a nontrivial automorphism σ, such that the automorphism $\tau \circ \widehat {\sigma }^k$ has a fixed point, the action of σ on xs1D4AE(X) is trivial, and X/xs2329σxs232A≠xs21191xs211D. doi:10.1017/S144678870900030