Some Moduli of $n$-pointed Fano Fourfolds

TL;DR

This paper develops a method to prove the unirationality of moduli spaces of certain fourfolds containing special rational surfaces, with applications to specific loci of cubic and Gushel-Mukai fourfolds.

Contribution

It introduces a general approach for establishing the unirationality of moduli spaces of n-pointed fourfolds with special rational surfaces, applied to recent loci in the literature.

Findings

Proves unirationality of specific moduli spaces of fourfolds.

Applies the method to codimension 1 loci of cubic and Gushel-Mukai fourfolds.

Provides new insights into the structure of these moduli spaces.

Abstract

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.