EVENT DETAILS AND ABSTRACT

Title: Murphy's Law in Algebraic Geometry: Badly-behaved deformation spaces
Speaker: Professor Ravi Vakil
Speaker Info: Stanford University
Brief Description:
Special Note:
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: the moduli space of smooth projective (``general type'') surfaces (or higher-dimensional varieties); the moduli space of 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.

The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. In any case, this talk is intended to be (mostly) comprehensible to a broad audience.