Rational Curves on Coindex 3 Fano Varieties

TL;DR

This paper studies the structure of rational curves on coindex 3 Fano varieties, proving irreducibility of their moduli spaces in high dimensions and confirming a conjecture for these varieties.

Contribution

It provides a detailed description of the moduli space of rational curves on coindex 3 Fano varieties and verifies the Geometric Manin's Conjecture for these cases.

Findings

Moduli space has a single irreducible component for each curve class in dimension ≥5.

Families of rational curves characterized by Fujita's $a$-invariant in dimension 4.

Verification of Lehmann and Tanimoto's Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties.

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.