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arXiv · 1805.04063

Maximally Algebraic potentially irrational Cubic Fourfolds

Radu Laza
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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.

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Radu Laza. 2018-05-10. Maximally Algebraic potentially irrational Cubic Fourfolds. https://arxiv.org/abs/1805.04063

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