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Families of cyclic curve coverings with maximal monodromy

Irene Spelta, Carolina Tamborini

Abstract

We study the algebraic monodromy of families of cyclic Galois coverings of curves. Under a condition on the $G$-decomposition of the associated variation of Hodge structures, we prove a criterion for the maximality of the monodromy. The proof combines the genus-zero case with a degeneration argument involving Prym varieties of certain admissible coverings. As a consequence of our criterion, we show that for $g\geq 8$ there exists no special family of Galois covers of the type we consider, providing new evidence towards the Coleman-Oort conjecture. Finally, we determine when the loci of double and triple Galois covers are totally geodesic.

Families of cyclic curve coverings with maximal monodromy

Abstract

We study the algebraic monodromy of families of cyclic Galois coverings of curves. Under a condition on the -decomposition of the associated variation of Hodge structures, we prove a criterion for the maximality of the monodromy. The proof combines the genus-zero case with a degeneration argument involving Prym varieties of certain admissible coverings. As a consequence of our criterion, we show that for there exists no special family of Galois covers of the type we consider, providing new evidence towards the Coleman-Oort conjecture. Finally, we determine when the loci of double and triple Galois covers are totally geodesic.
Paper Structure (6 sections, 13 theorems, 55 equations)

This paper contains 6 sections, 13 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on families of $G$-curves
  3. An admissible covering and its Prym variety
  4. Maximal monodromy
  5. The Coleman-Oort conjecture
  6. Application to double and triple coverings

Key Result

Proposition 1

All known examples of special families of cyclic coverings (namely, the ones appearing in fgpfpp) have no repeating factors in the monodromy.

Theorems & Definitions (27)

  • Proposition
  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition 4.3
  • Theorem 4.4
  • Remark 4.5
  • ...and 17 more