Maximally Algebraic potentially irrational Cubic Fourfolds

TL;DR

This paper identifies a class of cubic fourfolds with maximal algebraicity index, characterized by Eckardt points, which are prime candidates for irrationality and serve as key examples for testing major conjectures in algebraic geometry.

Contribution

It introduces the concept of an algebraicity index to classify cubic fourfolds and shows that those with Eckardt points maximize this index, highlighting their significance in studying irrationality.

Findings

Cubic fourfolds with Eckardt points maximize the algebraicity index.

These fourfolds are the most algebraic potentially irrational examples.

They serve as important test cases for major conjectures in the field.

Abstract

A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic $2$-cycles on them, it is natural to ask for the most algebraic cubic fourfolds $X$ to which this conjecture is still applicable. In this paper, we show that for an appropriate `algebraicity index' $\kappa_X$, there exists a unique class of cubics maximizing $\kappa_X$, not having an associated $K3$ surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in [LPZ17]). Arguably, they are the most algebraic potentially irrational cubic fourfolds, and thus a good testing ground for the Harris, Hassett, Kuznetsov conjectures.