We study virtual invariants of Quot schemes parametrizing quotients of dimension at
most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture
that the generating series of virtual K-theoretic invariants are given by rational functions.
We prove rationality for several geometries including punctual quotients for all surfaces
and dimension 1 quotients for surfaces X with pg > 0. We also show that the generating
series of virtual cobordism classes can be irrational.
Given a K-theory class on X of rank r, we associate natural series of virtual Segre and
Verlinde numbers. We show that the Segre and Verlinde series match in the following
cases:
(i) Quot schemes of dimension 0 quotients,
(ii) Hilbert schemes of points and curves over surfaces with pg > 0,
(iii) Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes.
Moreover, for punctual quotients of the trivial sheaf of rank N, we prove a new symmetry
of the Segre/Verlinde series exchanging r and N. The Segre/Verlinde statements have
analogues for punctual Quot schemes over curves.