  Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a
level-2 structure. We prove here that there is always a non hyperelliptic
element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$.
We prove also that for all $g\geqslant3$, each component of the hyperelliptic
locus in $\mathcal{M}_{g,2}$ is a connected component of the intersection of
$g-2$ thetanull divisors.
