Covers of Elliptic Curves and the Lower Bound for Slopes of Effective Divisors on \mathcal M_g

2007/04/30 by Dawei Chen, Chen, Dawei
Mathematics · #05A15 #05E15 #14H10 #14H30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.0704.3994

openalex publication_date 2007/04/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Consider genus g curves that admit degree d covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family Y that naturally maps into the moduli space of stable genus g curves \mathcal M_g. We study the geometry of Y, and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold points. As a by-product of our approach, we find some equalities from classical number theory. Moreover, a correspondence between our method and the viewpoint of square-tiled surfaces is established. We also use our results to study the lower bound for slopes of effective divisors on \mathcal M_g.

Related

• Simply branched covers of an elliptic curve and the moduli space of curves
• Effective divisors on M_g and a counterexample to the Slope Conjecture
• Effective divisors on M_g, curves on K3 surfaces and the Slope Conjecture
• Complete Families of Generic Covers of Elliptic Curves
• Descendent bounds for effective divisors on the moduli space of curves