The Hurwitz space of covers of an elliptic curve $E$ and the Severi variety of curves in $E \times \mathbb{P}^1$

Abstract: We describe the hyperplane sections of the Severi variety of curves in $E \times \mathbb{P}^1$ 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 $E \times \mathbb{P}^1$, in a restricted range of degrees.

• The paper establishes algebraic irreducibility of Hurwitz spaces of primitive covers of elliptic curves using deformation theory.
• It develops recursive formulas for Severi degrees on E × P¹, detailing component structures and hyperplane intersections.
• The analysis links isogeny data with moduli space connectedness, providing insights into the geometry of algebraic surfaces.

A Detailed Summary of "The Hurwitz space of covers of an elliptic curve $E$ and the Severi variety of curves in $E \times \mathbb{P}^1$" (1409.0927)

Introduction

The paper explores the intersection of two classical parameter spaces: the Hurwitz space of simply branched covers of a genus-one curve and the Severi variety on a surface $S = E \times \mathbb{P}^1$. The authors focus on how the study of one space aids the understanding of the other, particularly in determining components and irreducibility. The key objectives are algebraic proofs of irreducibility, recursive formulae for Severi degrees, and applications to characterizing Hurwitz components and Severi variety components within specific degree ranges.

The Hurwitz Spaces

In the classical algebraic geometry setting, Hurwitz spaces ($\mathcal{H}_{d,g}(C)$) parameterize degree $d$ covers of a curve $C$ of genus $g$. For genus-one curves ($C=E$), the paper presents an algebraic proof of the irreducibility of $\mathcal{H}_{d,g}^0(E)$, the space of primitive covers, leveraging deformation theory and algebraic geometry techniques. Specifically, it describes conditions under which the map $f: X \to E$ and its ramification properties can determine the structure and components of $\mathcal{H}_{d,g}(E)$.

Severi Varieties and Deformation Theory

Severi varieties, parameterizing integral curves of fixed class on a surface, are extended using algebraic techniques to $E \times \mathbb{P}^1$. The paper develops a deformation theory approach to assess the dimension and singularity structure of these varieties. It builds on Caporaso-Harris techniques, extending them to characterize the intersection of Severi varieties with hyperplanes.

Hyperplane Sections and Irreducibility

The work explores hyperplane sections in depth, identifying necessary conditions for the variety's existence and irreducibility criteria. It establishes new results such as the specific multiplicity with which hyperplane sections appear and characterizes the tangency of curves intersecting fixed fibers of the projection $E\times \mathbb{P}^1 \to E$.

Intersection Theory

A key feature of the approach is determining components of the intersected Severi varieties. Two scenarios are described based on the multiplicity $m$ with which the fiber $E_0$ is split off in the generic point. The methodology mirrors classical intersection theory but is adapted to the context of non-classical moduli spaces.

Applications and Implications

The study provides practical results concerning the components of Hurwitz spaces and Severi varieties with implications extending to the broader study of algebraic surface theory. One significant insight is the bijection between component structure in Hurwitz spaces and Severi varieties based on isogeny data. It culminates in broader implications for understanding nontrivial isogeny structures and their role in moduli space connectedness.

Conclusion

The paper concludes with acknowledgments of the complexities encountered when integrating topological and algebraic techniques, recommending further investigation into more intricate surfaces (e.g., higher-genus fibrations) using similar approaches.

This work provides a foundation for extending Severi and Hurwitz space techniques to more general surfaces, contributing to a deeper understanding of modern algebraic geometry while addressing specific conjectures regarding component structure and reducibility within this domain.