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REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE

Published online by Cambridge University Press:  22 January 2010

INDRANIL BISWAS*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India (email: indranil@math.tifr.res.in)
SIDDHARTHA GADGIL
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560003, India (email: gadgil@math.iisc.ernet.in)
*
For correspondence; e-mail: indranil@math.tifr.res.in
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Abstract

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Let X be a geometrically irreducible smooth projective curve 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 x 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 σ is even, say 2k, and the automorphism of X has a fixed point, where is the automorphism of X×ℂ 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 𝒮(X) is trivial, while X/〈σ〉≠ℙ1. We also give an example of X with no real points and admitting a nontrivial automorphism σ, such that the automorphism has a fixed point, the action of σ on 𝒮(X) is trivial, and X/〈σ〉≠ℙ1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

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