  Consider a family of integral complex locally planar curves whose relative
Hilbert scheme of points is smooth. The decomposition theorem of Beilinson,
Bernstein, and Deligne asserts that the pushforward of the constant sheaf on
the relative Hilbert scheme splits as a direct sum of shifted semisimple
perverse sheaves. We will show that no summand is supported in positive
codimension. It follows that the perverse filtration on the cohomology of the
compactified Jacobian of an integral plane curve encodes the cohomology of all
Hilbert schemes of points on the curve. Globally, it follows that a family of
such curves with smooth relative compactified Jacobian ("moduli space of
D-branes") in an irreducible curve class on a Calabi-Yau threefold will
contribute equally to the BPS invariants in the formulation of Pandharipande
and Thomas, and in the formulation of Hosono, Saito, and Takahashi.
