Cubic fourfolds and spaces of rational curves
Let \(X\) be a nonsingular cubic fourfold \(X\subset\mathbb{P}^5\) and denote by \(M_e\) the scheme of rational curves of degree \(e\) in \(X\). The authors prove that, for \(X\) general and \(e\geq 5\) an odd integer, the space \(M_e\) is non-uniruled. Moreover for \(e\geq 6\) an even integer, they prove that the generic fiber dimension of the maximally rationally connected fibration of \(M_e\) is at most one. The authors use different compactifications of \(M_e\) by means of Hilbert schemes, Chow varieties and Kontsevich moduli spaces of stable maps. They prove also that the scheme \(M_e\) is an irreducible variety of dimension \(3e+ 1\).
scientific article; zbMATH DE number 2138865