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arXiv · 2601.17852

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

Vincenzo Di Gennaro·Antonio Rapagnetta·Pietro Sabatino
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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 $X$.

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Vincenzo Di Gennaro, Antonio Rapagnetta, Pietro Sabatino. 2026-01-25. On the genus of a curve in a projective $3$-fold. https://arxiv.org/abs/2601.17852

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