Note on double coverings and binary quadratic forms

TL;DR

This paper explores the equivalence between quadratic maps, double coverings, and divisors on projective bundles, providing general conditions and specific results for projective spaces, with applications to quadratic forms.

Contribution

It establishes the equivalence of three structures related to rank two vector bundles and double coverings, under minimal assumptions, and applies these results to quadratic forms on projective spaces.

Findings

Equivalence of quadratic maps, double coverings, and divisors on projective bundles.

Isomorphism of Picard groups for double coverings of P_n when n ≤ 3.

Application to quadratic forms on rank two bundles on P_n.

Abstract

Let E be a rank two vector bundle on a scheme X. The following three structures are shown to be equivalent : a) A primitive quadratic map q: E --> L, with values in an invertible module L. b) A double covering f: Y --> X endowed with an invertible module F on Y, plus an isomorphism from the direct image of F to E. c) An effective Cartier divisor on the projective space P(E), of degree two over X. The passages from one of these points of view to another are carefully settled in their greatest generality: we only need that 2 is invertible on X. We then restrict to the projective space P_n, and we prove that for any double covering Y of P_n, the homomorphism on Picard groups it induces is an isomorphism if n ? 3; we finally apply this result to quadratic forms on rank two vector bundles on P_n.