  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.
