Very free rational curves in Fano varieties

TL;DR

This paper studies the normal bundles of rational curves in Fano varieties and similar spaces, establishing their rational connectedness and providing new computational methods for normal bundles in product varieties.

Contribution

It introduces a method to compute normal bundles of rational curves in complete intersections and proves the separable rational connectedness of broad classes of Fano varieties.

Findings

Normal bundle computations for rational curves in complete intersections.

Separable rational connectedness of general Fano complete intersections.

A new approach to compute normal bundles in product varieties.

Abstract

Let $X$ be a projective variety and let $C$ be a rational normal curve on $X$. We compute the normal bundle of $C$ in a general complete intersection of hypersurfaces of sufficiently large degree in $X$. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and normal bundles on each factor.