Explicit analysis of positive dimensional fibres of $ \mathcal{P}_{g,r} $ and Xiao conjecture

TL;DR

This paper investigates the positive dimensional fibers of the Prym map using Galois coverings, providing a systematic method to find counterexamples to Xiao's conjecture on relative irregularity.

Contribution

It introduces a direct procedure to analyze positive dimensional fibers of the Prym map via Galois coverings, extending to higher degrees and identifying all known counterexamples to Xiao's conjecture.

Findings

Developed a method to find infinitely many positive dimensional fibers.

Generalized the procedure to higher degree Galois coverings.

Identified all known counterexamples to Xiao's conjecture.

Abstract

We focus on the positive dimensional fibres of the Prym map $\mathcal{P}_{g,r}$. We present a direct procedure to investigate infinitely many examples of positive dimensional fibres. Such procedure uses families of Galois coverings of the line admitting a 2-sheeted Galois intermediate quotient. Then we generalize to families of Galois coverings of the line admitting a Galois intermediate quotient of higher degree and we show that the higher degree analogue of the aforementioned procedure gives all the known counterexamples to a conjecture by Xiao on the relative irregularity of a fibration.