Isotropy of quadratic forms over the function field of a quadric in characteristic 2

Hoffmann, Detlev W.; Laghribi, Ahmed

doi:10.1016/j.jalgebra.2004.02.038

Cited by 33 publications

(27 citation statements)

References 22 publications

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“…The characteristic 2 version below has been shown in [18]. Before that, a somewhat weaker version has been proved in [31].…”

Section: Theorem 42mentioning

confidence: 93%

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Quadratic forms and Pfister neighbors in characteristic 2

Hoffmann

Laghribi

2004

Trans. Amer. Math. Soc.

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Abstract. We study Pfister neighbors and their characterization over fields of characteristic 2, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form ϕ which is dominated by q. From this, we derive an analogue in characteristic 2 of a result by Knebusch saying that, in characteristic = 2, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic 2 for singular forms. As an application, we characterize certain forms of height 1 in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

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“…The characteristic 2 version below has been shown in [18]. Before that, a somewhat weaker version has been proved in [31].…”

Section: Theorem 42mentioning

confidence: 93%

“…), and we will state several deep results used in the proofs, some of them rather recent (e.g., [6], [7], [18]). …”

Section: Pfister Neighbors In Characteristic 2 4021mentioning

confidence: 99%

Quadratic forms and Pfister neighbors in characteristic 2

Hoffmann

Laghribi

2004

Trans. Amer. Math. Soc.

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“…A subform of a quadratic form means any linear subspace with the restricted quadratic form. This differs from Hoffmann-Laghribi's terminology, where a form α is called a subform of β if it is an orthogonal summand of β [8,Definition 2.8]. A subquadric is the projective quadric associated to a subform.…”

Section: Notationmentioning

confidence: 99%

Birational geometry of quadrics in characteristic 2

Totaro

2008

J. Algebraic Geom.

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A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety; in that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics, which remain open for quadrics in characteristic not 2: when are two quadrics birational, and when is a quadric ruled over the base field? The proofs begin by extending Karpenko and Merkurjev's theorem on the essential dimension of quadrics to arbitrary quadrics (smooth or not) in characteristic 2.Corollary 5.3. Let q and r be anisotropic quadratic forms over any field k such that dim r > 2 n for some integer n. If q has dimension at most 2 n , then q is not isotropic over k(r). If q has dimension 2 n + 1, then q is isotropic over k(r) if and only if r is isotropic over k(q).Proof. This is immediate from Theorem 5.2 together with Hoffmann-Laghribi's theorem that i 1 (r) ≤ dim(r) − 2 n [8, Lemma 4.

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“…Réciproquement, si B est une forme bilinéaire qui vérifie les conditions de (2), alors B F(B) ∼ C F(B) . Puisque dim B > 2 d > dim C, on obtient par [Hoffmann and Laghribi 2006] que la forme C F(B) est anisotrope. Ainsi, (B F(B) ) an C F(B) .…”

Section: Les Formes Bilinéaires Bonnesunclassified

“…En effet, dans le cas (1)(ii) on a B F(B) ∼ (αC) F(B) . Puisque dim C = 2 d < dim B, la forme C F(B) est anisotrope [Hoffmann and Laghribi 2006]. Ainsi, (B F(B) ) an (αC) On n'a pas une caractérisation des formes bilinéaires indiquées dans l'assertion (1)(i) du théorème 5.10, c'est-à-dire, les formes B anisotropes qui sont bonnes de hauteur 2 telles que dim B soit une puissance de 2 strictement supérieureà 2 deg(B)+1 .…”

Section: Les Formes Bilinéaires Bonnesunclassified

Sur le déploiement des formes bilinéaires en caractéristique 2

Laghribi¹

2007

Pacific J. Math.

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This article deals with the standard splitting of bilinear forms in characteristic 2. The first part is devoted to the study of bilinear Pfister neighbors (the definition of such a bilinear form is slightly different from the classical definition of a Pfister neighbor quadratic form). In the second part, we introduce the degree invariant for bilinear forms and we prove that for any integer d ≥ 0, the d-th power of the ideal of even dimensional bilinear forms coincides with the set of bilinear forms of degree ≥ d (this is a positive answer to the analogue of the degree conjecture for quadratic forms).In the third part, we classify good bilinear forms of height 2, and we give information on the possible dimensions of bilinear forms of height 2 which are not necessarily good.

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Isotropy of quadratic forms over the function field of a quadric in characteristic 2

Cited by 33 publications

References 22 publications

Quadratic forms and Pfister neighbors in characteristic 2

Quadratic forms and Pfister neighbors in characteristic 2

Birational geometry of quadrics in characteristic 2

Sur le déploiement des formes bilinéaires en caractéristique 2

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