On the minimal number of singular fibers with non-compact Jacobians for families of curves over $\mathbb P^1$

TL;DR

This paper establishes lower bounds on the number of singular fibers with non-compact Jacobians in families of semi-stable curves over the projective line, depending on the field characteristic and genus.

Contribution

It proves new minimal bounds for the count of such singular fibers in non-isotrivial families, extending understanding in algebraic geometry.

Findings

At least 5 non-compact Jacobian singular fibers over complex numbers for genus ≥ 5.

At least 4 such fibers in positive characteristic when the Jacobian is non-smooth.

Provides bounds that depend on the characteristic of the base field and the genus.

Abstract

Let $f:X \to \mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $g\geq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}\geq 5$ if $k=\mathbb C$ and $g\geq 5$; we also prove that $s_{nc}\geq 4$ if ${\rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.