  We develop a theory of \emph{reduced} Gromov-Witten and stable pair
invariants of surfaces and their canonical bundles.
  We show that classical Severi degrees are special cases of these invariants.
This proves a special case of the MNOP conjecture, and allows us to generalise
the G\"ottsche conjecture to the non-ample case. In a sequel we prove this
generalisation.
  We prove a remarkable property of the moduli space of stable pairs on a
surface. It is the zero locus of a section of a bundle on a smooth compact
ambient space, making calculation with the reduced virtual cycle possible.

Algebraic Geometry;High Energy Physics - Theory