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# Mathematical Research Letters

## Volume 23 (2016)

### Number 1

### On two rationality conjectures for cubic fourfolds

Pages: 1 – 13

DOI: https://dx.doi.org/10.4310/MRL.2016.v23.n1.a1

#### Author

#### Abstract

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.

Accepted 26 September 2014

Published 25 May 2016