The Hurwitz space of covers of an elliptic curve E and the Severi variety of curves in E × ℙ¹

2014/09/03 by Gabriel Bujokas, Bujokas, Gabriel
Computer Science · Engineering · Mathematics · #14H10 #14H30 #14H52 #14N35 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1409.0927

openalex publication_date 2014/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We describe the hyperplane sections of the Severi variety of curves in E × ℙ¹ in a similar fashion to Caporaso-Harris' seminal work. From this description we almost get a recursive formula for the Severi degrees (we get the terms, but not the coefficients). As an application, we determine the components of the Hurwitz space of simply branched covers of a genus one curve. In return, we use this characterization to describe the components of the Severi variety of curves in E × ℙ¹, in a restricted range of degrees.

Citations

• A note on Hurwitz schemes of covers of a positive genus curve
• Irreducibility of Hurwitz spaces

Related

• Short equations for the genus 2 covers of degree 3 of an elliptic curve
• On Hurwitz--Severi numbers
• Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties
• The Caporaso-Harris-Ran degeneration principle: proof and applications
• Compactifications of Hurwitz spaces