Explicit computations with cubic fourfolds, Gushel-Mukai fourfolds, and their associated K3 surfaces

TL;DR

This paper demonstrates computational methods using Macaulay2 to construct and analyze special cubic and Gushel-Mukai fourfolds, revealing new examples and their associated K3 surfaces, including some rational cases.

Contribution

It introduces a Macaulay2 package for explicit computations with special fourfolds and their associated K3 surfaces, enabling new constructions and rationality checks.

Findings

Constructed new examples of special cubic and Gushel-Mukai fourfolds.

Identified some fourfolds as rational.

Calculated associated K3 surfaces using explicit unirationality.

Abstract

We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.