标题： 科斯特尔努沃关于三维射影流形中曲线的界 显示英文标题

标题： Castelnuovo bound for curves in projective 3-folds

摘要： 卡斯特尔诺沃界限猜想是由物理学家提出的，它预测了Picard数为一的卡拉比-丘3流形的Gopakumar-Vafa不变量的有效消失结果。以前，仅知道几个特殊情况，且所有证明都依赖于Bayer-Macrì-Toda的Bogomolov-Gieseker猜想。在本文中，我们证明了对于任何Picard数为一的卡拉比-丘3流形，该卡斯特尔诺沃界限猜想成立，除了一个线性项和有限多个次数，无需假设Bayer-Macrì-Toda的猜想。此外，我们还证明了Picard数为一的卡拉比-丘4流形的曲面计数不变量的有效消失定理。我们还将我们的技术应用于研究一些显式卡拉比-丘3流形上的低次数曲线。我们的方法基于一种通用的迭代方法，用于获得固定3流形中一维闭子概形的亏格的上界，该方法结合了经典技术与导出范畴上弱稳定性条件的墙交叉，并适用于任何代数闭域上具有最坏情况孤立奇点的射影3流形。 显示英文摘要

摘要： 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.