New cubic fourfolds with odd degree unirational parametrizations

TL;DR

This paper demonstrates the existence of a divisor in the moduli space of cubic fourfolds with a general member having a degree 13 unirational parametrization, and shows this divisor is uniruled.

Contribution

It introduces a new divisor in the moduli space of cubic fourfolds with explicit unirational parametrizations of degree 13, expanding understanding of their geometric properties.

Findings

Existence of a divisor with degree 13 unirational parametrizations

Explicit construction of examples via Hilbert schemes of rational scrolls

The divisor $\\mathcal{C}_{42}$ is uniruled

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.