Restricted tangent bundles for general free rational curves

TL;DR

This paper studies the structure of tangent bundles restricted to general free rational curves on smooth projective varieties, revealing their filtrations are closely related and providing finite control in Fano cases, with applications to Manin's Conjecture.

Contribution

It establishes that the Harder-Narasimhan filtration of tangent bundles along free rational curves closely matches the global filtration, and in Fano varieties, it shows these restricted bundles are governed by finite data, advancing understanding of rational curves.

Findings

Harder-Narasimhan filtration of $T_X|_C$ approximates the restriction of the global filtration.

In Fano varieties, restricted tangent bundles are controlled by finite data.

Application to Peyre's 'freeness' formulation of Manin's Conjecture.

Abstract

Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder-Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre's "freeness" formulation of Manin's Conjecture in the setting of rational curves.