## Illinois Journal of Mathematics

### Cubic fourfolds and spaces of rational curves

#### Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138390

Digital Object Identifier
doi:10.1215/ijm/1258138390

Mathematical Reviews number (MathSciNet)
MR2085418

Zentralblatt MATH identifier
1081.14007

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes)