Hilbert schemes of rational curves on Fano hypersurfaces

TL;DR

This paper investigates the geometry of rational curves on generic Fano hypersurfaces, establishing their dimension, integrality, and rational connectedness properties for certain degrees and hypersurface parameters.

Contribution

It provides explicit dimension formulas and rational connectedness results for the Hilbert schemes of rational curves on Fano hypersurfaces, extending understanding of their geometric structure.

Findings

$R_d(X, h)$ is an integral local complete intersection of specified dimension.

Under additional conditions, $R_d(X, h)$ is rationally connected.

Results apply to generic hypersurfaces with degrees between 4 and $n-1$.

Abstract

In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the Hilbert scheme parameterizing irreducible rational curves of degree $d$ on $X$. We obtain that (1) If $4\leq h\leq n-1$, $R_d(X, h)$ is an integral, local complete intersection of dimension \begin{equation} (n+1-h)d+n-4. \end{equation} (2) If furthermore $(h^2-n)d+h\leq 0$ and $h\geq 4$, in addition to part (1), $R_d(X, h)$ is also rationally connected.