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The geometry of the period mapping on cyclic covers of \({\mathbb{P}}_ 1\). (English) Zbl 0542.14017

The tangent space to the period space for Riemann surfaces of genus g at a curve C is naturally isomorphic to the second symmetric product of the dual of the vector space of holomorphic differentials on C. If C is Galois, then its group of automorphisms acts on this tangent space. The object of this and future papers of the author is to study the relationship between subspaces described representation-theoretically and the geometric properties of deformations of C in directions lying in these subspaces. The present paper deals with the case where C is a cyclic cover of \({\mathbb{P}}_ 1\). The results are rather too technical to be conveniently reproduced here.
Reviewer: H.H.Martens

MSC:

14H30 Coverings of curves, fundamental group
14H10 Families, moduli of curves (algebraic)
30F30 Differentials on Riemann surfaces
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
14H05 Algebraic functions and function fields in algebraic geometry
32G15 Moduli of Riemann surfaces, Teichm├╝ller theory (complex-analytic aspects in several variables)
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References:

[1] E. Arbarello , M. Cornalba , P. Griffiths and J. Harris : Special Divisors on Algebraic Curves . Regional Algebraic Geometry Conference Athens, Georgia, May 1979. · Zbl 0559.14017
[2] E. Arbarello and M. Cornalba : Su Una Congettura di Petri Preprint Gruppi per la Matematica del C.N.R. , June 1980.
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[4] H. Clemens : Double Solids. Preprint , University of Utah, May 1979.
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