Spaces of rational curves in complete intersections

TL;DR

This paper proves the irreducibility and expected dimension of the space of smooth rational curves in general complete intersections under certain degree and dimension conditions, extending previous results.

Contribution

It generalizes prior work by establishing irreducibility of rational curve spaces in complete intersections with degree sum constraints.

Findings

Space of rational curves is irreducible under given conditions

Dimension of conics passing through a point is constant in general cases

Results extend Harris, Roth, and Starr's earlier work

Abstract

We prove that the space of smooth rational curves of degree $e$ in a general complete intersection of multidegree $(d_1, ..., d_m)$ in $\PP^n$ is irreducible of the expected dimension if $\sum_{i=1}^m d_i <\frac{2n}{3}$ and $n$ is large enough. This generalizes the results of Harris, Roth and Starr \cite{hrs}, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum_{i=1}^m d_i$ is small compared to $n$.