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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, Teichmüller theory (complex-analytic aspects in several variables) |

### Keywords:

deformations of curve; cyclic cover of projective line; period space for Riemann surfaces; holomorphic differentials; group of automorphisms### 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. |

[3] | G.A. Bliss : Algebraic Functions . Dover Publications, Inc. New York, 1966. · Zbl 0141.05002 |

[4] | H. Clemens : Double Solids. Preprint , University of Utah, May 1979. |

[5] | P. Griffiths and J. Harris : Principles of Algebraic Geometry . John Wiley & Sons Interscience Publication, 1978. · Zbl 0408.14001 |

[6] | P. Griffiths : On the periods of certain rational integrals. Part I . Annals Math. 90 (1969) 460-541. · Zbl 0215.08103 · doi:10.2307/1970746 |

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