Gromov-Witten invariants of an algebraic surface S with a smooth connected canonical curve C are zero unless the curve class is a multiple of $$extract_itex$$$$C$$$$/extract_itex$$ . It is known by work of Lee-Parker that these invariants only depend on the sign (-1)x(OS)   and the restriction map from cohomology on S to cohomology on C.

We compute the stable pair invariants of the canonical bundle X of S for any curve class and any descendent insertions. By the GW/stable pairs correspondence this includes the GW invariants of S. Our calculation uses the cosection localisation method of Kiem-Li and extends the cosection of Chang-Kiem. This is joint work with R. P. Thomas.