  Poisson structures vanishing linearly on a set of smooth closed disjoint
curves are generic in the set of all Poisson structures on a compact connected
oriented surface. We construct a complete set of invariants classifying these
structures up to an orientation-preserving Poisson isomorphism. We show that
there is a set of non-trivial infinitesimal deformations which generate the
second Poisson cohomology and such that each of the deformations changes
exactly one of the classifying invariants. As an example, we consider Poisson
structures on the sphere which vanish linearly on a set of smooth closed
disjoint curves.
