Date: Tuesday, March 04, 2008
Title: Murphy's Law in algebraic geometry: Badlybehaved moduli spaces
Abstract: 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
higherdimensional 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 wellbehaved 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 complexminded 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.
Speaker: Ravi Vakil
Institution: Stanford University
