New cubic fourfolds with odd-degree unirational parametrizations

Abstract

We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor ${\mathcal{C}}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that ${\mathcal{C}}_{42}$ is uniruled.