The class of the d-elliptic locus in genus 2

2019/07/21 by Carl Lian, Lian, Carl
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1907.08991

openalex publication_date 2019/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the hyperelliptic map and varying the sixth.

Related

• d-elliptic loci in genus 2 and 3
• Families of genus two curves with many elliptic subcovers
• Curves of genus two with maps of every degree to a fixed elliptic curve
• Simply branched covers of an elliptic curve and the moduli space of curves
• The CM class number one problem for curves of genus 2