April 12 (Monday), 12-1 pm
Speaker: James Myer (CUNY)
Title: The alterations paradigm shift for the problem of resolution of singularities
Abstract: Heisuke Hironaka solved the problem of resolution of singularities for a variety over a field of characteristic zero in 1964, although the story goes that his proof was so complicated as to stump Alexander Grothendieck! Efforts have been made since then to simplify the proof and see if it can be made to work for positive characteristic, but to no avail.
Regardless of characteristic: The singularities of a curve are resolved in one fell swoop by the normalization, a powerful tool hailing from algebra made to work for us in geometry, by Jean-Pierre Serre’s Criterion for Normality, and we owe much to Oscar Zariski for teaching us about the normalization. Joseph Lipman dealt with surfaces in as much generality as one could hope for. Threefolds are rumored to have been handled by Vincent Cossart and Olivier Piltant, although their proof is quite long and there are some restrictions on the characteristic. Fourfolds and up are uncharted territory for the most part.
In his 1996 paper Smoothness, Semi-Stability, and Alterations, Johan de Jong introduces a paradigm shift for solving the problem of resolution of singularities by relaxing a resolution of singularities to what he calls an alteration. The distinction is that a resolution is a birational morphism (generically one-to-one), whereas an alteration is a generically finite morphism (generically finite-to-one). Every variety can be altered to a nonsingular variety regardless of the characteristic. In fact, de Jong’s technique paired up with Dan Abramovich’s geometric insight yields a proof that every variety over a field of characteristic zero admits a resolution of its singularities in a paper consisting of only twelve pages, see Smoothness, Semi-Stability, and Toroidal Geometry. de Jong’s technique relies on (at least) two ingenious ideas: The first is the statement that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves ~ intuitively, any variety can be modified to be a family of curves. The second is that the curves in the family of curves that results can be marked so as to become stable (in the sense of Pierre Deligne and David Mumford) so that it gives rise to morphism into the moduli space of stable curves, where we may take advantage of established facts about the moduli space of stable curves, including the fact that its compactification has curves with at worst nodal singularities. My talk has the goal of introducing Johan de Jong’s paradigm shift, and indicating, to the extent that I can, the idea of the proof that every variety may be altered to a nonsingular variety. An emphasis will be placed on the picture that demonstrates that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves.