Alternating Catalan numbers and curves with triple ramification

Abstract

We determine the number of minimal degree covers of odd ramification for a general curve.Moschetti is member of GNSAGA (INDAM) and is partially supported by MIUR: Dipartimenti di Eccellenza Program (2018-2022)-Dept. of Math. Univ. of Pavia; Naranjo was partially supported by the Proyecto de Investigación MTM2015-65361-P; Pirola is member of GNSAGA (INDAM) and is partially supported by PRIN Project Moduli spaces and Lie theory (2017) and by MIUR: Dipartimenti di Eccellenza Program (2018-2022) -Dept. of Math. Univ. of Pavia. Farkas was supported by the DFG Grant Syzygien und Moduli. PRELIMINARIESWe collect a few things that will be used throughout the paper.2.1. Monodromy of coverings and Hurwitz spaces of odd covers. Let f : C → 1 be a finite cover of degree d and denote by B := {P 1 , . . ., P n } its branch locus. For a point Q ∈ 1 \ B , letbe its monodromy representation. We denote by M f := Im(ρ f ) the monodromy group of f . The local monodromy of f around a branch point P i ∈ B is given by τ, where γ i is a simple loop around P i based at Q . The cover f is said to be alternating if M f ⊆ A d . We shall often consider alternating covers f : C → 1 , such that each local monodromy τ i is given by an odd cycle. We refer to such an f as being an odd cover.We denote by odd g the Hurwitz space parametrizing odd covers f : C → 1 of degree 2g + 1 branched at 3g points. We require that the local monodromy around each branch point of f be given by a 3-cycle. Such a cover is endowed with a theta characteristics ϑ := C (D ) ⊗ f * ( 1 (−1)),

A Hurwitz divisor on the moduli of Prym curves

“…). For each disk, the multiplicity of the point over [π : X → ] can be computed using curves in the universal deformation as exemplified in [8,13,18]. The multiplicity of each of the k points in…”

Hyperelliptic odd coverings

“…However, by using degeneration on Hurwitz spaces, the existence of coverings with degree d ≥ 2g −1 and alternating monodromy group has been proved in [15] for general curves in the sense of moduli. The degeneration is a powerful method and allowed in [5] to compute the number of odd ramification coverings, called the alternating Catalan numbers, in the minimal degree case d = 2g − 1. Many other cases have been covered in [13] with other techniques.…”

“…A completely different approach, also used in [5] for an alternative proof of the first step of the induction, appeared firstly in [1]. It consists in the interpretation of an odd ramification covering C → P 1 as a solution of a certain differential equation, see [5,Section 4]. A covering corresponds to a solution involving the meromorphic differential associated with a De-Rham problem.…”

“…In the same spirit of [5] we compute the number of such solutions. This is possible thanks again to Proposition 3.4, which guarantees that the forgetful map H HOC g → H g is dominant.…”

“…In the last section, we consider the case of elliptic curves with an odd spin in degree four. This was already established in [5] with different techniques. We have to add that this was our starting case and that in this case, the 4 solutions are almost explicit.…”