Monodromy of the Prym map and semicanonical pencils in genus 6

TL;DR

This paper investigates the monodromy and degree of the Prym map in genus 6, revealing its birationality on a specific divisor and identifying the monodromy group as the Weyl group WD_5, with detailed geometric analysis.

Contribution

It proves the Prym map in genus 6 is birational on the divisor of curves with an odd semicanonical pencil and determines the monodromy group as WD_5, providing new insights into its geometric structure.

Findings

Prym map in genus 6 is birational on the divisor of curves with an odd semicanonical pencil.

Monodromy group over the image of this divisor is the Weyl group WD_5.

Degree of the Prym map restricted to other divisors is 10 and 16.

Abstract

The Prym map $\mathcal{P}_6$ in genus 6 is dominant and generically finite of degree 27. When restricted to the divisor of curves with an odd semicanonical pencil $\mathcal{T}_6^o$, it is still generically finite, but of degree strictly smaller. In this paper, we prove that $\mathcal{P}_6$ restricted to $\mathcal{T}_6^o$ is birational and that the monodromy group over the image of $\mathcal{T}_6^o$ is the Weyl group $WD_5$. Thus, there are two other irreducible divisors in the moduli space of Prym curves $\mathcal{R}_6$ and the degree of $\mathcal{P}_6$ restricted to them is 10 and 16. Moreover, we study the geometry of the divisor where $\mathcal{P}_6$ has degree 10.