Most binary forms come from a pencil of quadrics

Creutz, Brendan

doi:10.1090/bproc/24

Author: Brendan Creutz
Journal: Proc. Amer. Math. Soc. Ser. B 3 (2016), 18-27
MSC (2010): Primary 11D09, 11G30; Secondary 14H25, 14L24
DOI: https://doi.org/10.1090/bproc/24
Published electronically: December 6, 2016
MathSciNet review: 3579585
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Abstract: A pair of symmetric bilinear forms $A$ and $B$ determine a binary form $f(x,y) := \operatorname {disc}(Ax-By)$. 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 $\mathbb {Q}$ are discriminant forms. This is related to the arithmetic of the hyperelliptic curve $z^2 = f(x,y)$. Analogous results for nonhyperelliptic curves are also given.

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