{"title":"Rational Curves on Coindex 3 Fano Varieties","authors":"Eric Jovinelly, Fumiya Okamura","doi":"arxiv-2409.00834","DOIUrl":null,"url":null,"abstract":"We describe the moduli space of rational curves on smooth Fano varieties of\ncoindex 3. For varieties of dimension 5 or greater, we prove the moduli space\nhas a single irreducible component for each effective numerical class of\ncurves. For varieties of dimension 4, we describe families of rational curves\nin terms of Fujita's $a$-invariant. Our results verify Lehmann and Tanimoto's\nGeometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the\ncomplex numbers.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00834","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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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