Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Hilbert's theorem on positive ternary quartics: A refined analysis


Author: Claus Scheiderer
Journal: J. Algebraic Geom. 19 (2010), 285-333
DOI: https://doi.org/10.1090/S1056-3911-09-00538-4
Published electronically: July 28, 2009
MathSciNet review: 2580677
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Abstract: Let $ X$ be an integral plane quartic curve over a field $ k$, let $ f$ be an equation for $ X$. We first consider representations $ (*)$ $ cf=p_1p_2-p_0^2$ (where $ c\in k^*$ and the $ p_i$ are quadratic forms), up to a natural notion of equivalence. Using the general theory of determinantal varieties we show that equivalence classes of such representations correspond to nontrivial globally generated torsion-free rank one sheaves on $ X$ with a self-duality which are not exceptional, and that the exceptional sheaves are in bijection with the $ k$-rational singular points of $ X$. For $ k=\mathbb{C}$, the number of representations $ (*)$ (up to equivalence) depends only on the singularities of $ X$, and is determined explicitly in each case. In the second part we focus on the case where $ k=\mathbb{R}$ and $ f$ is nonnegative. By a famous theorem of Hilbert, such $ f$ is a sum of three squares of quadratic forms. We use the Brauer group and Galois cohomology to relate identities $ (**)$ $ f=p_0^2+p_1^2+p_2^2$ to $ (*)$, and we determine the number of equivalence classes of representations $ (**)$ for each $ f$. Both in the complex and in the real definite case, our results are considerably more precise since they give the number of representations with any prescribed base locus.


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Claus Scheiderer
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
Email: claus.scheiderer@uni-konstanz.de

DOI: https://doi.org/10.1090/S1056-3911-09-00538-4
Received by editor(s): November 13, 2007
Received by editor(s) in revised form: March 18, 2009
Published electronically: July 28, 2009
Dedicated: Dedicated to Jean-Louis Colliot-Thélène on the occasion of his 60th birthday

American Mathematical Society