On two rationality conjectures for cubic fourfolds

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.

10 pages; final version to appear in Math. Res. Lett

Cited by in corpus (9)

• The K3 category of a cubic fourfold
• Derived categories view on rationality problems
• Stability conditions in families
• Hyperkähler fourfolds and Kummer surfaces
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• On the double EPW sextic associated to a Gushel-Mukai fourfold
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• Sixfolds of generalized Kummer type and K3 surfaces
• On symplectic birational self-maps of projective hyperkähler manifolds of K3${}^{[n]}$-type