Skip to main content
×
×
Home

REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE

  • INDRANIL BISWAS (a1) and SIDDHARTHA GADGIL (a2)
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE
      Available formats
      ×
Copyright
Corresponding author
For correspondence; e-mail: indranil@math.tifr.res.in
References
Hide All
[1]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of Algebraic Curves, Volume I, Grundlehren der Mathematischen Wissenschaften, 267 (Springer, New York, 1985).
[2]Atiyah, M. F., ‘Riemann surfaces and spin structures’, Ann. Sci. École Norm. Sup. 4 (1971), 4762.
[3]Biswas, I., Gadgil, S. and Sankaran, P., ‘On theta characteristics of a compact Riemann surface’, Bull. Sci. Math. 131 (2007), 493499.
[4]Gross, B. H. and Harris, J., ‘Real algebraic curves’, Ann. Sci. École. Norm. Sup. 14 (1981), 157182.
[5]Kallel, S. and Sjerve, D., ‘Invariant spin structures on Riemann surfaces’, http://arxiv.org/abs/math/0610568.
[6]Mumford, D., ‘Theta characteristics of an algebraic curve’, Ann. Sci. École. Norm. Sup. 4 (1971), 181192.
[7]Natanzon, S. M., ‘Spinors and differentials of real algebraic curves’, in: Topology of Real Algebraic Varieties and Related Topics, American Mathematical Society Translations, Series 2, 173 (American Mathematical Society, Providence, RI, 1996), pp. 179186.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed