  We construct virtual fundamental classes on nested Hilbert schemes of points
and curves in complex nonsingular projective surfaces. These classes recover
the virtual classes of Seiberg-Witten theory as well as the (reduced) stable
theory, and play a crucial role in the reduced Donaldson-Thomas theory of
local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show
that certain integrals against the virtual fundamental classes of punctual
nested Hilbert schemes are expressed as integrals over the products of the
Hilbert scheme of points. We are able to find explicit formulas for some of
these integrals by relating them to Carlsson-Okounkov's vertex operator
formulas.
