Spaces of rational curves in complete intersections

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 $n$.

Cited by in corpus (4)

• Geometric Manin's Conjecture and rational curves
• Rational curves on smooth hypersurfaces of low degree
• Positivity Results for spaces of rational curves
• On the asymptotic enumerativity property for Fano manifolds