On two rationality conjectures for cubic fourfolds

Motivated by the question of rationality of cubic fourfolds, we show that a cubic $X$ has an associated $\mathrm{K}3$ 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 $\mathrm{K}3$ surface, but that having $F$ birational to $\mathrm{Hilb}^2 (\mathrm{K}3)$ is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.

Full Text (PDF format)

Accepted 26 September 2014

Published 25 May 2016