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Rational curves on hypersurfaces of low degree, II

Published online by Cambridge University Press:  01 December 2004

Joe Harris  and
Jason Starr
Joe Harris
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USAharris@math.harvard.edu
Jason Starr
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAjstarr@math.mit.edu
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Abstract

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This is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general hypersurface $X \subset\mathbb{P}^n$ of degree d. In Part I (J. reine angew. Math. 571 (2004), 73–106) it is proved that, if $d<({n+1})/{2}$, then for each e the space of rational curves is irreducible, reduced and has the expected dimension. In this paper it is proved that, if $d^2 + d + 1 \leq n$, then for each e the space of rational curves is a rationally connected variety; in particular it has negative Kodaira dimension.

Keywords

Kontsevich moduli spacestable maprationally connected variety14C05 (primary)14E08 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 1 , January 2005 , pp. 35 - 92
Copyright
Foundation Compositio Mathematica 2005