Castelnuovo bound for curves in projective 3-folds

Abstract

The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macr\`i-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field.