Moduli of abelian covers of elliptic curves

Abstract

For any finite abelian group G, we study the moduli space of abelian Gcovers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M1,n, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric. 1 2 NICOLA PAGANIfinite map is in fact an isomorphism), the latter is naturally a codimension-(g − 1) moduli subscheme of M g .We now summarize the main results of our work. In Theorem 1, we give a geometric description of the irreducible components of the moduli spaces of abelian G-covers of elliptic curves. From this description, we are able to deduce further results for those components that parametrize totally ramified covers. In fact, in Section 3.2, we prove that each component of the moduli space parametrizing totally ramified abelian G-covers is birational to M 1,n where n is the number of marked points (in general, a superset of the set of branch points). In Theorem 3, we exploit our description to prove the vanishing of the rational Picard group of such components of the moduli space, and to find the number of independent relations among the boundary divisors of the admissible covers compactification. In the last section we make our results more explicit in the case of the rational Picard group of the moduli spaces of admissible bielliptic curves, see Corollaries 6 and 7.The Picard number of the compact moduli of hyperelliptic curves H g is g [21]. In Corollary 6, we see that, when g > 2, it is 2g for the compact moduli B g of bielliptic curves (without ordering the branch). A simple analysis of the number of boundary divisors of the compact moduli of double covers of genus-h curves leads us to conjecture that the Picard number of the latter moduli space grows is (h + 1)g + g · o(1) for g → ∞.

Picard Groups of Moduli Spaces of Curves with Symmetry

“…As far as the author can tell, the orbifold approach to this subject, in which symmetric mapping class groups play a central role, is novel. Where there is overlap, our results appear to agree with those found in, for example, [4], [34], [36].…”

Stacks of uniform cyclic covers of curves and their Picard groups

“…, (1)), has trivial rational Picard group for nd > 0, which also implies the vanishing of the rational Picard group of B h,1,n . In this paper we recover this last result by explicitly describing the integral Picard group of B h,1,n , but we can not directly deduce the result in [Pag13,Theorem 3].…”

“…In [Pag13] the author introduces moduli stacks of abelian covers of curves, which are related to our stacks B h,g,n in the cyclic, totally ramified case. Let Y g,r,n be the stack of tuples (D, C, f, σ 1 , .…”

“…By forgetting the sections we obtain a functor Y g,nd,n −→ B h,g,n , where d, g, h, n are related by the expression (1) below, which is a S ndtorsor. In [Pag13,Theorem 3] is proved that Y g,nd,n , which is denoted by M 1,nd (B(Z/nZ), (1), . .…”

Stack of $S_{3}$-covers

“…[Ber13]). In this context we recall also [Pag16,AV04,BV12,PTT15]. In [Ton13a] I introduced the moduli stack G-Cov of G-covers for a finite, flat and finitely presented group scheme G over some base S. Although G-Cov and the various moduli of covers of curves/surfaces are both stacks parametrizing covers, we want to stress that they are very different objects: the geometric points in the first case are particular finite schemes (covers of a point) while in the second case are covers of curves/surfaces.…”