Monodromy of the Prym map and semicanonical pencils in genus 6

TL;DR

This work analyzes the genus‑6 Prym map $\mathcal{P}_6:\mathcal{R}_6\to\mathcal{A}_5$, focusing on the divisor $\mathcal{T}^o_6$ of odd semicanonical pencils. It proves that $\mathcal{P}_6|_{\mathcal{T}^o_6}$ is birational onto its image and that its monodromy group is the Weyl group $WD_5$, revealing two additional Prym divisors of degrees $10$ and $16$ in $\mathcal{R}_6$ and a detailed study of the degree‑$10$ locus. The authors deploy trigonal/tetragonal constructions, Prym–Brill–Noether theory, and compactifications to connect genus‑6 trigonal curves with plane sextics having a tritangent line, and they compute the monodromy by analyzing degenerations to degree‑4 del Pezzo surfaces and their line configurations. Consequently, the preimage $\mathcal{P}_6^{-1}(\mathcal{Z})$ (with $\mathcal{Z}=\overline{\mathcal{P}_6(\mathcal{T}^o_6)}$) splits into three irreducible components of degrees $1$, $10$, and $16$, and the Jacobian locus lies inside $\mathcal{Z}$, offering a refined structural picture of Prym maps in genus $6$ and their geometric significance for moduli spaces.

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.

Theorems & Definitions (33)

• Theorem A
• Lemma 1
• Theorem 2.1
• Definition 1
• Lemma 2
• proof
• Proposition 1
• proof
• Claim
• Remark 1