Singularities: Resolutions and Valuations, talk by Andrei Benguş-Lasnier

In this introductory talk, I will present some concepts surrounding singularities and why and how we look for their resolutions. Our main objects will be algebraic (or analytic) variety X over a field k, i.e., the zero set of a finite number of polynomial (or convergent power series) equations. In differential geometry we often encounter such X, but usually we wish them to be smooth: locally they look like an open subset of an Euclidean space. We are then equipped with many tools to study them: integration of differential forms, finiteness of cohomology groups, Chern classes, etc. However for arbitrary X these methods fall apart; even the concept of local charts falls apart as we do not have a standard model that can be glued together to form singular varieties. Some objects are more or less obligated to include singularities, like compactifications or moduli spaces of curves for instance.

Singularists work with these objects, either to study and classify them, or to find ways to counter the problems listed above, via resolutions: in broad terms we try to find another variety X′ that is smooth and a projection \(X’\xrightarrow{p}X\) , that is surjective over a Zariski open subset of X and then we can work with X′ instead.

I will try and give a glimpse of the history of the resolution of singularities, by insisting on the use of valuations to work out a local version of this problem, called local uniformization.