2021/11/26 by William Dallaporta, Dallaporta, William · 1 citation
Computer Science · Mathematics · #11E16 #11R29 #14C22 (Primary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2111.13422
openalex publication_date 2021/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (C,M) where C is a quadratic algebra over S and M is an invertible 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.