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• W3113015872 abstract "A well known conjecture due to Hassett asserts that a cubic fourfold<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>whose transcendental cohomology<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T Subscript upper X><mml:semantics><mml:msub><mml:mi>T</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:annotation encoding=application/x-tex>T_X</mml:annotation></mml:semantics></mml:math></inline-formula>cannot be realized as the transcendental cohomology of a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>K3</mml:annotation></mml:semantics></mml:math></inline-formula>surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2><mml:semantics><mml:mn>2</mml:mn><mml:annotation encoding=application/x-tex>2</mml:annotation></mml:semantics></mml:math></inline-formula>-cycles on them, it is natural to ask for the most algebraic cubic fourfolds<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>to which this conjecture is still applicable. In this paper, we show that for an appropriate “algebraicity index”<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa Subscript upper X Baseline element-of double-struck upper Q Subscript plus><mml:semantics><mml:mrow><mml:msub><mml:mi>κ<!-- κ --></mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo>∈<!-- ∈ --></mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>Q</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>kappa _Xin mathbb {Q}_+</mml:annotation></mml:semantics></mml:math></inline-formula>, there exists a unique class of cubics maximizing<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa Subscript upper X><mml:semantics><mml:msub><mml:mi>κ<!-- κ --></mml:mi><mml:mi>X</mml:mi></mml:msub><mml:annotation encoding=application/x-tex>kappa _X</mml:annotation></mml:semantics></mml:math></inline-formula>, not having an associated<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mn>3</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>K3</mml:annotation></mml:semantics></mml:math></inline-formula>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." @default.
• W3113015872 created "2020-12-21" @default.
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• W3113015872 date "2021-05-18" @default.
• W3113015872 modified "2023-10-05" @default.
• W3113015872 title "Maximally algebraic potentially irrational cubic fourfolds" @default.
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• W3113015872 doi "https://doi.org/10.1090/proc/15430" @default.
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