Some loci of rational cubic fourfolds

Abstract

In this paper we investigate the divisor C14 inside the moduli space of smooth cubic hypersurfaces in P 5 , whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in P 5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in C14 is rational. Combining our proof with the Hodge theoretic definition of C14, we deduce that on a smooth cubic fourfold every class T ∈ H 2,2 (X, Z) with T 2 = 10 and T · h 2 = 4 is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in C14.

Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

“…In [Nue15] there are similar descriptions for the values 12 ≤ d ≤ 44, d = 42, while for d = 42 one can consult [Lai16]. Every [X] ∈ C 14 has been proved to be rational by showing that it contains either a smooth surface with one apparent double point or one of its small degenerations, see [BRS15] and also Theorem 2 here for a different proof. The extension of this geometrical approach to rationality for other (admissible) values d appeared to be impossible due to the paucity of surfaces with one apparent double point, which essentially can be used only for d = 14.…”

New examples of rational Gushel-Mukai fourfolds

“…This condition can be expressed by saying that the rational cubic fourfolds are parametrized by a countable union d C d of irreducible hypersurfaces inside the 20-dimensional coarse moduli space of cubic fourfolds, where d runs over the so-called admissible values (the first ones are d = 14, 26, 38, 42, 62). The rationality for the cubic fourfolds in C 14 has been classically proved by Fano in [Fan43] (see also [BRS19]), while in [RS19a] it has been proved the rationality in the case of C 26 and C 38 . Very recently, in [RS19b] it has been also proved the rationality in the case of C 42 .…”

Special cubic birational transformations of projective spaces

“…The admissible values are the even integers δ > 6 not divisible by 4, by 9 and nor by any odd prime of the form 2 + 3m, so that the first admissible values are 14, 26, 38. The rationality for C 14 was shown in the classical works by Morin and Fano (see [Mor40,Fan43]; see also [BRS15]), and in the recent paper [RS17], Russo and ourselves showed the rationality for C 26 and C 38 . The decisive step of our discovery was to find a description for C δ in terms of some surface S contained in the generic [X ] ∈ C δ and which admits (for some e ≥ 1) a congruence of (3e − 1)-secant rational curves of degree e, that is, through a general point p ∈ P 5 there passes a unique rational curve C p of degree e which is (3e − 1)-secant to S; see [RS17] for precise and more general definitions.…”