  We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not
sums of squares (sos). Such a form has at most ten real zeros. We give a
complete and explicit characterization of all sets
$S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd
non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only
real nodes there is a certain natural divisor class $\tau_X$ of degree~$9$, and
$S$ is the real zero set of some psd non-sos sextic if, and only if, there is a
unique cubic $X$ through $S$ and $S$ represents the class $\tau_X$ on $X$. If
this is the case, there is a unique extreme ray $\mathbb{R}_+q_S$ of psd
non-sos sextics through $S$, and we show how to find $q_S$ explicitly. The
sextic $q_S$ has a tenth real zero which for generic $S$ does not lie in $S$,
but which may degenerate into a higher singularity contained in $S$. We also
show that for any eight points in $\mathbb{P}^2(\mathbb{R})$ in general
position there exists a psd sextic that is not a sum of squares and vanishes in
the given points.
