The (Generalized) Kontsevich Matrix Model is known
to be relevant for describing 2d topological minimal
models coupled to gravity. We shall build a solution
for this model in terms of resolvents (loop operator
averages) starting from Virasoro constraints on partition
function (i.e. Ward identities for the corresponding matrix
integral). It turns out that resolvents can be considered
as meromorphic multi-differentials on a certain auxiliary
Riemann surface (complex spectral curve). Different
phases of the theory correspond to different
Riemann surfaces. We shall discuss various features
of this construction, in particular, the so-called p-q duality
in terms of resolvents and corresponding Riemann
surfaces in the example of dualities between (2,3)
and (3,2) models.