同指数 3 法诺变体上的有理曲线

arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-01 DOI:arxiv-2409.00834

Eric Jovinelly, Fumiya Okamura

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引用次数: 0

摘要

我们描述了指数为 3 的光滑法诺变种上有理曲线的模空间。对于维数为 5 或更大的有理曲线，我们证明了模空间对于每一个有效数类曲线都有一个不可还原的分量。对于维数为 4 的有理曲线，我们用藤田的 $a$ 不变式描述了有理曲线族。我们的结果验证了莱曼和谷本的几何马宁猜想，即所有复数上的光滑同指数 3 法诺变种的几何马宁猜想。

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Rational Curves on Coindex 3 Fano Varieties

We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For varieties of dimension 4, we describe families of rational curves in terms of Fujita's $a$-invariant. Our results verify Lehmann and Tanimoto's Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the complex numbers.

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arXiv - MATH - Algebraic Geometry

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