  The open subvariety $\overline{M}_g^{\leq k}$ of $\overline{M}_g$
parametrizes stable curves of genus $g$ having at most $k$ rational components.
By the work of Looijenga, one expects that the cohomological excess of
$\overline{M}_g^{\leq k}$ is at most $g-1+k$. In this paper we show that when
$k=0$, the conjectured upper bound is sharp by showing that there is a
constructible sheaf on $\overline{H}_g^{\leq k}$ (the hyperelliptic locus)
which has non-vanishing cohomology in degree $3g-2$.
