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

TL;DR

This paper develops a refined method to parametrize the Picard group of quadratic algebras over schemes where 2 is not a zero divisor, extending Wood's bijection with new invariants and orientation concepts.

Contribution

It introduces a classification of quadratic algebras using discriminant and parity, and extends orientation notions to non-free cases for better Picard group parametrization.

Findings

Classified quadratic algebras via discriminant and parity.

Extended orientation concepts to non-free quadratic algebras.

Provided examples illustrating obstructions and refinements.

Abstract

Let $S$ be a scheme such that $2$ is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over $S$, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over $S$ and isomorphism classes of pairs $(\mathcal{C},\mathcal{M})$ where $\mathcal{C}$ is a quadratic algebra over $S$ and $\mathcal{M}$ is an invertible $\mathcal{C}$-module. Unexpectedly, examples suggest that a refinement of Wood's bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over $S$; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.