On some invariants of cubic fourfolds

TL;DR

This paper investigates specific invariants of cubic fourfolds, including Hodge numbers of certain line loci and bounds on the degree of irrationality of their Fano schemes, contributing to algebraic geometry understanding.

Contribution

It provides explicit calculations of Hodge numbers for lines of second type and establishes an upper bound on the irrationality degree of Fano schemes of lines.

Findings

Hodge numbers of the locus of lines of second type are computed.

An upper bound of 6 is established for the degree of irrationality.

Results apply to general cubic fourfolds and their Fano schemes.

Abstract

For a general cubic fourfold $X \subset \mathbb{P}^5$, we compute the Hodge numbers of the locus $S \subset F$ of lines of second type. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.