Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms

The first part of this article aims at classifying quadratic algebras over an arbitrary scheme: this goal is achieved by using two invariants, the Wood-discriminant and the parity. Next, we obtain a parametrization of the Picard group of a given quadratic algebra by twisted quadratic forms, provided 2 is not a zero divisor in our base scheme. For that purpose, we start from Wood's set-theoretical bijection (2011) generalizing the classical result over Z. To remedy the fact that some Picard classes are identified in this bijection, we extend the notion of orientation of quadratic algebras to the non-free case, and we use our classification to conclude. All along the paper, we illustrate various notions and obstructions with a wide range of examples.