Let X⊂ℙr 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⊂X be a projective subscheme of dimension 1, degree d≫n, and arithmetic genus pa(C). Improving a recent result by Liu, we exhibit a Castelnuovo’s bound for pa(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.
Di Gennaro, V., Rapagnetta, A., Sabatino, P. (2026). On the genus of a curve in a projective 3 -fold [Altro] [10.48550/arXiv.2601.17852].
On the genus of a curve in a projective 3 -fold
Di Gennaro, Vincenzo;Rapagnetta, Antonio;Sabatino, Pietro
2026-01-25
Abstract
Let X⊂ℙr 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⊂X be a projective subscheme of dimension 1, degree d≫n, and arithmetic genus pa(C). Improving a recent result by Liu, we exhibit a Castelnuovo’s bound for pa(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.| File | Dimensione | Formato | |
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