The transcendental lattice of the sextic Fermat surface

Auel, Asher; Böhning, Christian; Bothmer, Hans-Christian Graf von

doi:10.4310/mrl.2013.v20.n6.a2

Cited by 6 publications

(10 citation statements)

References 23 publications

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“…The transcendental motive of a smooth projective surface over a field of characteristic 0 is integrally indecomposable. This is, of course, a motivic analog of the Hodge-theoretic indecomposability conjecture due to Kulikov, [24], which is known to be false for the Fermat sextic in P 3 , see [2].…”

Section: The Transcendental Motive Of the Fermat Sextic In Pmentioning

confidence: 94%

“…Now let us also look at the notion of integral essential (in)decomposability in dimension 2. Let S be a smooth projective connected surface over a field k. Recall that the motive M(S) decomposes in the standard Chow-Künneth way, as given by the formula (2). If M(S) is essentially decomposable, the corresponding integral decomposition of the diagonal induces the decomposition of the transcendental projector π 2 tr (S) and, accordingly, the decomposition of the transcendental motive M 2 tr (S) into two nonzero direct summands in C(k) Q .…”

Section: Essential (In)decomposability Of Motivesmentioning

confidence: 99%

“…According to Definition 8, integral (in)decomposability of the transcendental motive M 2 tr (S) is the same as essential (in)decomposability of the entire motive M(S), in case when we deal with smooth projective surfaces over the ground field. However, this extra piece of terminology can be useful in making analogies between the conjectural integral indecomposability of the transcendental motive M 2 tr (S), and the integral indecomposability of the transcendental Hodge structure of S, which is, in general, known to be false, see [2]. If M(S) is essentially decomposable, which is equivalent to saying that M 2 tr (S) decomposes integrally, then CH 0 (S) is essentially decomposable.…”

Section: Essential (In)decomposability Of Motivesmentioning

confidence: 99%

“…An explicit example of a curve satisfying the assumptions of Theorem A is the Fermat sextic C 6 in P 2 , see the proof of Proposition 7 in [7]. A refinement of the technique used in the proof of Theorem A gives us the following result, which should be compared with the main result in [2].…”

Section: Introductionmentioning

confidence: 99%

“…2 , see page 104 in loc.cit. The second Betti number for the surface E × C is 4g + 2 and the Picard number is 2g + 2 by Lemma 1 in[7].…”

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confidence: 99%

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Motivic obstruction to rationality of a very general cubic hypersurface in $\mathbb P^5$

Guletskii

2016

Preprint

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Let S be a smooth projective surface over a field. We introduce the notion of integral decomposability and, respectively, the opposite notion of integral indecomposability, of the transcendental motive M 2 tr (S). If the transcendental motive is indecomposable rationally, then it is indecomposable integrally. For example, M 2 tr (S) is rationally, and hence integrally indecomposable if S is an algebraic K3-surface whose motive is known to be finite-dimensional.In the paper we prove that M 2 tr (S) is integrally indecomposable when S is the self-product of a smooth projective curve having enough morphisms onto an elliptic curve with complex multiplication. This applies, for example, when S is the self-product of the Fermat sextic in P 2 . Some refinement of the same technique yields that M 2 tr (S 6 ) is integrally indecomposable, where S 6 is the Fermat sextic in P 3 . This suggests a conjecture saying that the transcendental motive of any smooth projective surface is integrally indecomposable. We prove in the paper that if this motivic integral indecomposability conjecture is true, and if the motive of any smooth projective surface is finite-dimensional, then a very general cubic hypersurface in P 5 is not rational. Contents 1. Introduction 1 2. Preliminaries and notation 4 3. Essential (in)decomposability of motives 9 4. The self-product of a curve mapped onto a CM elliptic curve 16 5. The transcendental motive of the Fermat sextic in P 3 32 6. Two motivic conjectures and cubic hypersurfaces in P 5 41 References 45

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Section: The Transcendental Motive Of the Fermat Sextic In Pmentioning

confidence: 94%

Section: Essential (In)decomposability Of Motivesmentioning

confidence: 99%

Section: Essential (In)decomposability Of Motivesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…2 , see page 104 in loc.cit. The second Betti number for the surface E × C is 4g + 2 and the Picard number is 2g + 2 by Lemma 1 in[7].…”

mentioning

confidence: 99%

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Motivic obstruction to rationality of a very general cubic hypersurface in $\mathbb P^5$

Guletskii

2016

Preprint

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Cubic Fourfolds, K3 Surfaces, and Rationality Questions

Hassett

2016

Lecture Notes in Mathematics

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This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Smooth cubic surfaces have been known to be rational since the 19th century [Dol05]; cubic threefolds are irrational by the work of Clemens and Griffiths [CG72]. Cubic fourfolds are likely more varied in their behavior. While there are examples known to be rational, we expect that most cubic fourfolds should be irrational. However, no cubic fourfolds are proven to be irrational.Our organizing principle is that progress is likely to be driven by the dialectic between concrete geometric constructions (of rational, stably rational, and unirational parametrizations) and conceptual tools differentiating various classes of cubic fourfolds (Hodge theory, moduli spaces and derived categories, and decompositions of the diagonal). Thus the first section of this paper is devoted to classical examples of rational parametrizations. In section two we focus on Hodge-theoretic classifications of cubic fourfolds with various special geometric structures. These are explained in section three using techniques from moduli theory informed by deep results on K3 surfaces and their derived categories. We return to constructions in the fourth section, focusing on unirational parametrizations of special classes of cubic fourfolds. In the last section, we touch on recent applications of decompositions of the diagonal to rationality questions, and what they mean for cubic fourfolds.

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