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 (d₁,...., d_m) in Pⁿ is irreducible of the expected dimension if Σ_i=1^m d_i < 2n/3 and n is large enough. This generalizes the results of Harris, Roth and Starr, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if Σ_i=1^m d_i is small compared to n.