##
Murphy's Law in algebraic geometry: Badly-behaved deformation spaces

Ravi Vakil, Stanford

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
ugly.'' Hence many of the most important moduli spaces in algebraic
geometry are arbitrarily singular, justifying a philosophy of Mumford.

More precisely, every singularity of finite type over **Z** (up to
smooth parameters) appears on the Hilbert scheme of curves in projective
space, and the moduli spaces of: smooth projective general-type surfaces
(or higher-dimensional varieties), 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: the
curves are smooth, the surfaces have very ample canonical bundle, the
stable sheaves are torsion of rank 1, the singularities are normal and
Cohen-Macaulay, etc.

Thus one can construct a smooth curve in projective space whose
deformation space has any given number of components, each with any given
singularity type, with any given non-reduced behavior along various
associated subschemes. Similarly one can give a surface over
**F**_{p} that lifts to p^{7} but not p^{8}. (Of course the results
hold in the holomorphic category as well.)