We consider the question: ``How bad can the deformation space of an
object be?'' (Alternatively: ``What singularities can appear on a
moduli space?'') The answer seems to be: ``Unless there is some a
priori reason otherwise, the deformation space can be arbitrarily
bad.'' We show this for a number of important moduli spaces.
More precisely, up to smooth parameters, every singularity that can be
described by equations with integer coefficients appears on moduli
spaces parameterizing: smooth projective surfaces (or
higher-dimensional manifolds); smooth curves in projective space (the
space of stable maps, or the Hilbert scheme); plane curves with nodes
and cusps; stable sheaves; isolated threefold singularities; and more.
The objects themselves are not pathological, and are in fact as nice
as can be. This justifies Mumford's philosophy that even moduli
spaces of well-behaved objects should be arbitrarily bad unless there
is an a priori reason otherwise.
I will begin by telling you what ``moduli spaces'' and ``deformation
spaces'' are. The complex-minded listener can work in the holomorphic
category; the arithmetic listener can think in mixed or positive
characteristic. This talk is intended to be (mostly) comprehensible
to a broad audience.