  We determine the asymptotic behavior in the limit of large Higgs fields of
the sectional curvatures of the natural $L^2$ hyperk\"ahler metric $G_{L^2}$ of
the moduli space $\mathcal M$ of rank-$2$ Higgs bundles on a Riemann surface
$\Sigma$ away from the discriminant locus. It is shown that their leading order
part is given by a sum of Dirac type contributions on $\Sigma$, for which we
find explicit expressions.
