Maximally algebraic potentially irrational cubic fourfolds

Laza, Radu

doi:10.1090/proc/15430

Maximally algebraic potentially irrational cubic fourfolds
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by Radu Laza

Proc. Amer. Math. Soc. 149 (2021), 3209-3220

DOI: https://doi.org/10.1090/proc/15430

Published electronically: May 18, 2021

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Abstract:

A well known conjecture due to Hassett asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ cannot 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” $\kappa _X\in \mathbb {Q}_+$, 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 by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett’s irrationality conjecture for cubic fourfolds.

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