Birational geometry of quadrics in characteristic 2

The theory of quadratic forms can be regarded as studying an important special case of the general problem of birational classification of algebraic varieties. A typical example of a Fano fibration in minimal model theory is a quadric bundle, which can be viewed without loss of information as a quadric hypersurface over the function field of the base variety. This description feeds the problem into the theory of quadratic forms, which seeks to classify quadrics over an arbitrary field up to birational equivalence. One can hope to extend the successes of quadratic form theory to broader areas of birational geometry. In the other direction, birational geometry emphasizes the interest of particular problems in the theory of quadratic forms. In the study of Fano fibrations in positive characteristic, one cannot avoid fibrations whose generic fiber is nowhere smooth over the function field of the base variety, such as “wild conic bundles” [17]. That can happen even when the total space is smooth. In this paper, we solve two of the main problems of birational geometry for quadrics which are nowhere smooth over a field. Such quadrics are called quasilinear; they exist only in characteristic 2. First, the “quadratic Zariski problem” [18] has a positive answer for quasilinear quadrics: for any quasilinear quadrics X and Y of the same dimension over a field such that there are rational maps from X to Y and from Y to X , X and Y must be birational (Theorem 6.5). Next, a quasilinear quadric over a field k is ruled, that is, birational to Z × P for some variety Z over k, if and only if its first Witt index is greater than 1 (Theorem 6.4). At the same time, we find that if a quasilinear quadric is ruled, then it is ruled over a lower-dimensional quadric Z. All these statements are conjectured for arbitrary quadrics, but they are wide open for other classes of quadrics, for example in characteristic not 2. The proofs begin by extending Karpenko and Merkurjev’s theorem on the essential dimension of quadrics to characteristic 2 [13]. The extension has already been made for smooth quadrics in the book by Elman, Karpenko, and Merkurjev [3]. Here we cover all quadrics in characteristic 2, smooth or not. As in other advances on quadratic forms over the past decade, the proofs use the Chow groups of algebraic cycles on quadrics and products of quadrics. Karpenko and Merkurjev’s theorem is a major achievement of the theory of quadratic forms. It includes Karpenko’s theorem that an anisotropic quadric with first Witt index equal to 1 is not ruled [11, Theorem 6.4], which we also extend here to characteristic 2. This is a strong nontriviality statement about the birational geometry of quadrics. As an application, we extend Kollár’s results on the birational geometry of conics (including non-smooth conics in characteristic 2) [14, section 4] to quadrics of any dimension (Corollary 3.3). Finally, besides proving the conjectured characterization of ruledness (Conjecture 6.1) for all quasilinear quadratic forms, we check it for nonquasilinear quadratic forms in characteristic 2 of dimension at most 6, using the results of Hoffmann and Laghribi (section 7). It is known for quadratic forms of dimension at most 9 in characteristic not 2 [20]. The inspiration for this work was Hoffmann and Laghribi’s series of papers showing that the classical theory of quadratic forms (usually restricted to characteristic not 2, or at most to nonsingular quadratic forms in characteristic 2) admits a rich generalization to singular quadratic forms in characteristic 2, including the extreme case of quasilinear