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Spaces of rational curves on complete intersections
Published online by Cambridge University Press: 26 March 2013
Abstract
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We prove that the space of smooth rational curves of degree 
$e$ on a general complete intersection of multidegree 
$(d_1, \ldots , d_m)$ in 
$\mathbb {P}^n$ is irreducible of the expected dimension if 
$\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and 
$n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], 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 
$n$.
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