Rational curves on hypersurfaces

TL;DR

This paper investigates how the properties of rational curves on a projective variety and its divisors influence the uniruledness and rational connectedness of those divisors, with criteria based on positivity conditions.

Contribution

It provides new criteria linking the positivity of divisors and the behavior of rational curves to the uniruledness and rational connectedness of divisors within projective varieties.

Findings

Criteria for uniruledness of divisors

Criteria for rational connectedness of divisors

Influence of positivity of divisors on rational curves

Abstract

Let $(X,D)$ be a pair where $X$ is a projective variety. We study in detail how the behavior of rational curves on $X$ as well as the positivity of $-(K_X+D)$ and $D$ influence the behavior of rational curves on $D$. In particular we give criteria for uniruledness and rational connectedness of components of $D$.