Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Abstract

Any algebraic surface in \({\mathbb {P}^{n}(\mathbb {C})}\) which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of \({\mathbb {P}^{n}(\mathbb {C}).}\) In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in \({\mathbb {P}^{4}(\mathbb {C})}\) ruled in cubics.