Most binary forms come from a pencil of quadrics

TL;DR

This paper shows that most binary forms over the rationals can be represented as discriminants of pairs of symmetric bilinear forms, connecting algebraic forms with hyperelliptic curve arithmetic and satisfying a local-global principle.

Contribution

It establishes a generic local-global principle for representing binary forms as discriminants of symmetric bilinear form pairs, extending to hyperelliptic and non-hyperelliptic curves.

Findings

Most binary forms over  are discriminant forms.

The local-global principle holds generically for these forms.

Connections to the arithmetic of hyperelliptic curves.

Abstract

A pair of symmetric bilinear forms A and B determine a binary form $f(x,y) = disc(Ax-By)$. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over $\mathbf{Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for non-hyperelliptic curves are also given.