Illinois Journal of Mathematics

Cubic fourfolds and spaces of rational curves

Jason Starr and A. J. de Jong

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For a general nonsingular cubic fourfold $X\subset \PP^5$ and $e\geq 5$ an odd integer, we show that the space $M_e$ parametrizing rational curves of degree $e$ on $X$ is non-uniruled. For $e \geq 6$ an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of $M_e$ is at most one, i.e., passing through a very general point of $M_e$ there is at most one rational curve. For $e < 5$ the spaces $M_e$ are fairly well understood and we review what is known.

Article information

Illinois J. Math., Volume 48, Number 2 (2004), 415-450.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C05: Parametrization (Chow and Hilbert schemes)
Secondary: 14E08: Rationality questions [See also 14M20]


de Jong, A. J.; Starr, Jason. Cubic fourfolds and spaces of rational curves. Illinois J. Math. 48 (2004), no. 2, 415--450. doi:10.1215/ijm/1258138390.

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