On the genus of a curve in a projective $3$-fold

TL;DR

This paper establishes a Castelnuovo-type bound for the arithmetic genus of curves on factorial threefolds in projective space, with implications for conjectures on Gopakumar-Vafa invariants in Calabi-Yau varieties.

Contribution

It improves a recent bound on the genus of curves in threefolds and connects this to conjectures about Gopakumar-Vafa invariants in Calabi-Yau threefolds.

Findings

Derived a Castelnuovo's bound for the arithmetic genus of curves in factorial threefolds.

Extended the bound to Calabi-Yau threefolds, advancing understanding of Gopakumar-Vafa invariants.

Provided conditions under which the bound applies to projective varieties with isolated singularities.

Abstract

Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a projective subscheme of dimension $1$, degree $d\gg n$, and arithmetic genus $p_a(C)$. Improving a recent result by Liu, we exhibit a Castelnuovo's bound for $p_a(C)$. In the case $X$ is Calabi-Yau, our bound gives a step forward for a certain conjecture concerning the vanishing of Gopakumar-Vafa invariants of $X$.