Birational geometry of quadrics

A central method in the theory of quadratic forms is the study of function fields of projective quadrics. In particular, it is important to ask when there is a rational map from one quadric over a field to another. This suggests the problem of determining when two quadrics are birational, which turns out to be much harder. The answer is known for quadratic forms of dimension at most 7 (thus for projective quadrics of dimension at most 5), by Ahmad-Ohm and Roussey among others [1, 16]. In particular, there is a conjectural characterization of which quadratic forms are ruled (meaning that the associated quadric is birational to P times some variety), Conjecture 1.1. The conjecture was known for quadratic forms of dimension at most 9 [17]. One result of this paper is to prove Conjecture 1.1 for odd-dimensional forms of dimension at most 17, and also for forms of dimension 10 or 14 (Theorem 4.1). We use in particular a new structure theorem on 14-dimensional forms (Theorem 4.2), generalizing Izhboldin’s theorem on 10-dimensional forms. Vishik gave an example of a 16-dimensional form with first Witt index 2 which is not divisible by a binary form; such a form should be ruled by the conjecture, but it fell outside previously known classes of ruled forms. Nonetheless, we give a new construction which shows that Vishik’s form is ruled (Theorem 7.2). In this paper we also solve the problem of birational classification for a significant class of quadratic forms, Pfister neighbors of dimension at most 16 (Corollary 3.2). (Among these forms, only the case of “special Pfister neighbors” was known before, which fails to include all Pfister neighbors in dimensions 9, 10, and 11.) We also give several generalizations for a broader class of quadratic forms: neighbors of multiples of a Pfister form (Theorems 3.1 and 6.3), strengthening Roussey’s theorem in this direction. We also give the first construction of a birational map between two non-isomorphic half-neighbors in section 5. This is genuinely different from any previously constructed birational map between quadrics. Thanks to Alexander Vishik for allowing me to include his example of a 16dimensional form with splitting pattern (2, 2, 2, 2) which is not divisible by a binary form (Lemma 7.1).