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

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.