Quadratic forms and Pfister neighbors in characteristic 2

Abstract

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.

Isotropy of 5-dimensional quadratic forms over the function field of a quadric in characteristic 2

“…Another fact related to the domination relation that we will use is the following result knows as the "Completion Lemma": Proposition 2.3. ([5,Lem. 3.9]) Let ϕ and ψ be nonsingular F -quadratic forms and…”

“…([5, Th. 4.2]) Let ϕ and ψ be F -quadratic forms such that ϕ is anisotropic and nonsingular and ψ is nondefective.…”

Similitude des multiples des formes d’Albert en caractéristique 2

“…On renvoie à [1] et [7] pour plus de détails sur certaines notions qu'on va utiliser sur les formes bilinéaires et quadratiques. Rappelons tout de même que deux formes bilinéaires (ou quadratiques) ϕ et ψ sont dites semblables si ϕ αψ pour α ∈ F * convenable.…”

“…Une forme quadratique (ou bilinéaire) de Pfister ϕ vérifie la propriété de multiplicativité, ce qui signifie que ϕ αϕ pour tout scalaire α ∈ D F (ϕ) ; de plus, si ϕ est isotrope, alors elle est hyperbolique (ou métabolique). Comme conséquence du théorème de sous-forme [7,Th Dans ce théorème on exclut la condition de similitude entre les formes bilinéaires elles-mêmes car, en général, deux formes bilinéaires de même dimension qui sont Witt-équivalentes ne sont pas nécesssairement isométriques. On démontre ce théorème de la même façon que le théorème 3.…”

On fibrations whose geometric fibers are nonreduced

“…Mori and Saito [21] studied them in the context of fibrations, using the name wild hypersurface bundle, in connection with the minimal model program. Hoffmann [13] studied such forms from an algebraic perspective, viewing them as generalizations of quadratic forms in characteristic 2; the latter were studied, for example, in [14]. It might be fruitful to combine algebraic and geometric approaches.…”