Non-free curves on Fano varieties

TL;DR

This paper proves a key prediction of Geometric Manin's Conjecture regarding non-free curves on Fano varieties, showing that such curves are contained in a proper subset, and highlights differences in studying morphism spaces versus sections.

Contribution

It provides the first proof of a prediction of Geometric Manin's Conjecture about the structure of non-free curves on Fano varieties and explores their geometric properties.

Findings

Non-free curves are contained in a proper closed subset of the Fano variety.

The paper verifies Batyrev's expectation about non-dominant components.

It identifies key differences between morphism spaces and section spaces of Fano fibrations.

Abstract

Let $X$ be a smooth Fano variety over $\mathbb{C}$ and let $B$ be a smooth projective curve over $\mathbb{C}$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \mathrm{Mor}(B, X)$ parametrizing curves which are non-free and have large anticanonical degree. Following ideas of our previous work, we prove the first prediction of Geometric Manin's Conjecture describing such irreducible components. As an application, we prove that there is a proper closed subset $V \subset X$ such that all non-dominant components of $\mathrm{Mor}(B, X)$ parametrize curves in $V$, verifying an expectation put forward by Victor Batyrev. We also demonstrate two important ways that studying $\mathrm{Mor}(B,X)$ differs from studying the space of sections of a Fano fibration $\mathcal{X} \to B$.