We study several aspects of the geometry of Grothendieck’s Quot scheme Quot^l_S(E) of
length l coherent sheaf quotients of a locally free sheaf E of rank r on a smooth projective
surface S. This scheme has the following four fundamental properties:
(I) a quotient E'' of E induces a closed immersion of Quot^l_S(E'') into Quot^l_S(E);
(II) a closed subscheme Z of S induces a closed immersion of Quot^l_Z(E|_Z) into Quot^l_S(E);
(III) an invertible sheaf L on S induces an isomorphism Quot^l_S(E)->Quot^l_S(E x L);
(IV) the automorphism group of E acts naturally on Quot^l_S(E).

We study the singular locus of Quot^l_S(E), and conjecture that Quot^l_S(E) has rational
singularities. We prove this in the first nontrivial case Quot^2_S(E), and exhibit a modular
resolution of singularities of Quot^2_S(E), which we use to compute the cohomology of tautological sheaves on Quot^2_S(E), and obtain a sheaf-theoretic generalisation of a formula of Severi.

We then investigate the implications of (I) and (II) on the intersection theory of the scheme Quot^l_S(E). We prove that any locally free quotient E'' of E gives a relation between
the fundamental classes (ordinary or virtual) of Quot^l_S(E'') and Quot^l_S(E), with
an insertion given by the Euler class of a tautological sheaf. This allows us to compute
certain tautological integrals over Quot^l_S(O^r) in terms of tautological integrals over Quot^l_S(O). We show that if C in S is a canonical curve, [Quot^l_S(E)]^vir is the pushforward of (-1)^l [Quot^l_C(E|_C)]. We then combine these results with (III) and (IV) to obtain a structure theorem for tautological integrals over [Quot^l_S(E)]^vir. Using (III), we deduce from this the relation chi^vir(Quot^l_S(E))=chi^vir(Quot^l_S(O^r)) of virtual Euler characteristics.

We also study top intersections of the first Chern class of tautological sheaves, and express the corresponding generating series in terms of the Lambert W-function.