  We study the family of piecewise linear differential systems in the plane
with two pieces separated by a cubic curve. Our main result is that 7 is a
lower bound for the Hilbert number of this family. In order to get our main
result, we develop the Melnikov functions for a class of nonsmooth differential
systems, which generalizes, up to order 2, some previous results in the
literature. Whereas the first order Melnikov function for the nonsmooth case
remains the same as for the smooth one (i.e. the first order averaged function)
the second order Melnikov function for the nonsmooth case is different from the
smooth one (i.e. the second order averaged function). We show that, in this
case, a new term depending on the jump of discontinuity and on the geometry of
the switching manifold is added to the second order averaged function.
