Motives of Quadrics with Applications to the Theory of Quadratic Forms

Vishik, Alexander

doi:10.1007/978-3-540-40990-8_2

Cited by 54 publications

(107 citation statements)

References 17 publications

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“…Proof: It follows from [14,Corollary 4.14] that there exists 1 t h(q) such that j t−1 (q) a(N), (n − b(N)) < j t (q), and …”

Section: Applications Of the Main Theoremmentioning

confidence: 99%

“…Let q be nondegenerate quadratic form of dimension (n + 2), and Q be respective (smooth) projective quadric of dimension n. From [14] one can see that motivic decomposition of quadrics with Z-coefficients carries the same information as the one with Z/2-coefficients, and MDT (Q) can be reconstructed out of…”

Section: The General Casementioning

confidence: 99%

“…Let Q be smooth projective quadric of dimension n over the field k of characteristic not 2, and M(Q) be its motive in the category Chow(k) of Chow motives over k (see [14], or Chapter XII of [2]). Over the algebraic closure k, our quadric becomes completely split, and so, cellular.…”

Section: Introductionmentioning

confidence: 99%

“…Over the algebraic closure k, our quadric becomes completely split, and so, cellular. This implies that M(Q| k ) becomes isomorphic to a direct sum of Tate motives: , where Λ(N) ⊂ Λ(Q) (see [14] for details). We say that λ, µ ∈ Λ(Q) are connected, if for any direct summand N of M(Q), either both λ and µ are in Λ(N), or both are out.…”

Section: Introductionmentioning

confidence: 99%

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Excellent connections in the motives of quadrics

Vishik¹

2011

Ann. Sci. École Norm. Sup.

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In this article we prove the Conjecture claiming that the connections in the motives of excellent quadrics are minimal ones for anisotropic quadrics of given dimension. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalises the well-known Binary Motive Theorem. Moreover, we have the description of Tate-motives involved. This, in turn, gives another proof of Karpenko's Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.

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“…Proof: It follows from [14,Corollary 4.14] that there exists 1 t h(q) such that j t−1 (q) a(N), (n − b(N)) < j t (q), and …”

Section: Applications Of the Main Theoremmentioning

confidence: 99%

Section: The General Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Excellent connections in the motives of quadrics

Vishik¹

2011

Ann. Sci. École Norm. Sup.

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“…We use Chow groups as in [3] to prove this, although the argument could also be formulated in terms of motivic decompositions as in [18]; the equivalence between the two languages is described in [3, Theorem 94.1]. Let Q be the projective quadric of dimension 12 associated to q.…”

Section: The Ruledness Conjecture and The Structure Of 14-dimensionalmentioning

confidence: 99%

Birational geometry of quadrics

Totaro¹

2009

Bul. Soc. Math. France

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Abstract. -We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14. The proof uses a new structure theorem for 14-dimensional forms, generalizing Izhboldin's theorem on 10-dimensional forms. We also show that Vishik's 16-dimensional form is ruled.Résumé (La géométrie birationnelle des quadriques). -Nous construisons de nouvelles applications birationnelles entre quadriques sur un corps. Diverses formes quadratiques sont considérées: les voisines de Pfister, les voisines des multiples d'une forme de Pfister, et les demi-voisines. Un corollaire est la détermination des quadriques ré-glées (c'est-à-dire birationnelles au produit de la droite projective et d'une variété) en certaines dimensions. Nous décrivons complètement les quadriques réglées lorsque la forme quadratique est de dimension impaire inférieure à 17, de dimension paire infé-rieure à 10, ou de dimension 14. La preuve utilise un nouveau théorème de structure sur les formes de dimension 14, généralisant le théorème d'Izhboldin sur les formes de dimension 10. Nous montrons également que la forme de Vishik de dimension 16 est réglée.Texte reçu le 2 avril

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Lifting of Coefficients for Chow Motives of Quadrics

Haution

2010

Quadratic Forms, Linear Algebraic Groups, and Cohomology

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Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal H X (k) of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of X. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of H, such as the Chow group. When H is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When H is the algebraic cobordism, we give a new proof of a theorem of Levine and Morel. Finally we discuss some splitting properties of geometrically unirational field extensions of small transcendence degree.

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Motives of Quadrics with Applications to the Theory of Quadratic Forms

Cited by 54 publications

References 17 publications

Excellent connections in the motives of quadrics

Excellent connections in the motives of quadrics

Birational geometry of quadrics

Lifting of Coefficients for Chow Motives of Quadrics

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