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

TL;DR

This work sharpens Castelnuovo-type bounds for the arithmetic genus of 1-dimensional subschemes on projective 3-folds with Picard group generated by the hyperplane class. It first proves a Noether-type bound for curves on contained surfaces and then extends the bound to curves on 3-folds via an iterative method that blends wall-crossing arguments with classical Noether techniques. The main result is a universal bound pa(C) ≤ $\frac{d^2}{2n} + \frac{d}{2n}(2\pi_X - 2 - n) + 4\nu^6$ for curves of large degree, with explicit constants $\nu = \dfrac{4}{3}n(n+1)$ and $M(n)$, and $\pi_X$ the sectional genus; in the Calabi–Yau case, a sharper corollary yields pa(C) ≤ $\frac{d^2}{2n} + d/2 + n^4$ when $d$ exceeds a threshold. These bounds imply that extremal curves tend to lie in hyperplane sections for large degree, providing progress toward Castelnuovo-type conjectures and connecting to Gopakumar–Vafa invariants.

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 $X$.

Theorems & Definitions (38)

• Theorem 1.1: Halphen-Gruson-Peskine
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof