Universal families of Gushel-Mukai fourfolds (original) (raw)

Unirationality of certain universal families of cubic fourfolds.

The aim of this short note is to de ne the universal cubic fourfold over certain loci of the moduli space. Then, by using some recent results of Farkas-Verra and more classical ones by Mukai, we prove that the universal cubic fourfolds over the Hassett divisors C14, C26 and C42 are unirational. As for the rationality of cubic fourfolds, this relies heavily on the existence of associated K3 surfaces, and the birational geometry of their universal families. Finally, we observe that for explicit in nitely many values of d, the universal cubic fourfold over Cd can not be unirational.

SOME MODULI OF n-POINTED FANO FOURFOLDS

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the n-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.

Cubic fourfolds containing a plane and a quintic del pezzo surface

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 associated to X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.

MODULI OF CUBIC FOURFOLDS AND REDUCIBLE OADP SURFACES

2024

In this paper we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic fourfolds X containing a plane P with other divisors Ci. Notably we study the irreducible components of the intersections with C12 and C20. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of P with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off P , or by finding examples of reducible one-apparent-double-point surfaces inside X. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.