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Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

Published online by Cambridge University Press:  04 December 2007

J. HUISMAN
J. HUISMAN
Affiliation:
Institut Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; e-mail: [email protected]
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Abstract

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The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M$_g/R$ of nonsingular real algebraic curves of genus g (g[les ]2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M$_g/R$. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M$_g/R$is real analytic. We conclude that M$_g/R$ is not a real analytic variety.

Keywords

moduli spacesreal algebraic curvesreal quotient singularitiesreal analytic manifoldsproperly discontinuous group actionssemianalytic varieties.
Type
Research Article
Information
Compositio Mathematica , Volume 118 , Issue 1 , August 1999 , pp. 43 - 60
Copyright
© 1999 Kluwer Academic Publishers