Positivity Results for spaces of rational curves

TL;DR

This paper studies the positivity properties of spaces of rational curves on very general hypersurfaces, revealing conditions under which these spaces lack rational curves and possess differential forms.

Contribution

It provides new results on the absence of rational curves in certain loci and constructs differential forms on compactifications of these spaces.

Findings

No rational curves meet the smooth embedded curves for small e.

No rational curves in the locus swept out by lines when n ≤ d.

Existence of differential forms on compactified spaces for specified e, n, and d.

Abstract

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$. We investigate positivity properties of the spaces $R_e(X)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e(X)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n \leq d$, there are no rational curves in the locus $Y \subset X$ swept out by lines. And we exhibit differential forms on a smooth compactification of $R_e(X)$ for every $e$ and $n-2 \geq d \geq \frac{n+1}{2}$.