On two rationality conjectures for cubic fourfolds

TL;DR

This paper explores conditions under which cubic fourfolds are rational by linking their associated K3 surfaces to the geometry of their Fano varieties of lines, revealing nuanced distinctions in these relationships.

Contribution

It establishes an equivalence between the existence of an associated K3 surface and the Fano variety being birational to a moduli space of sheaves, clarifying the geometric criteria for rationality.

Findings

Fano variety of lines is birational to a moduli space of sheaves on a K3 surface when an associated K3 exists.

Having the Fano variety birational to Hilb^2(K3) is a more restrictive condition.

The paper compares the moduli space loci where each of these conditions holds.

Abstract

Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface, but that having F birational to Hilb^2(K3) is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.