Special sessions

Blowups and resolution of singularities
A mini course by Herwig  Hauser, University of Vienna

Blowups are a central technique in algebraic and differential geometry to transform “singular” objects (varieties, vector fields, foliations,...) into “regular” or “smooth” ones. Aside from their applicability they offer a beautiful theory on their own with many different viewpoints and mutual interactions: geometric, algebraic, computational, axiomatic.

In the course we will start from a geometric outset, the Mobius strip, well known already to children when they are asked to cut a twisted paper strip along its equator and then get a doubly twisted strip. Surprisingly enough, one may also contract the equator of the Mobius strip (which is, essentially, a circle) to a point and obtain as a result the standard Euclidean plane. Inverting this collapsing process, i.e., exploding a point of the plane to a circle to recover the Mobius strip is the first instance of a blowup.

We will show in the course how to characterize this process algebraically – yielding a general definition which enables explicit computations – and axiomatically, by a universal property – providing elegant proofs. Once we have seen many examples and obtained a thorough understanding of blowups we will start to use them to desingularize varieties and vector fields. This involves a very elementary step, the substitution of the variables in a polynomial or a differential form by specific monomials.

The point is then to show that these substitutions, when chosen very carefully, help to reduce the complexity of the situation and lead, when repeated sufficiently often, to a complete resolution of the singularities. The detailed proofs of two most prominent theorems, Hironaka’s resolution of the singularities of algebraic varieties in characteristic zero, and the theorem of Camacho-Sad about the existence of separatrices of complex planar vector fields, will be out of reach, but we will make the overall strategy of the proofs very clear: it is based on a genuine web of mutually interwoven inductions which show that the complexity of the singularities diminuishes in each blowup.

The course is designed so as to be accessible for Master and PhD students while still being interesting for Faculty members. Only little prior knowledge is required (polynomials, rings and ideals, some differential geometry, ...): The focus is laid on the interplay between the study of concrete examples and the use of axiomatic arguments.

Reading Material

• C. Camacho, P. Sad: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. 115 (1982), 579-595.
• H. Hauser: Blowup and Resolution. Clay Mathematics Institute Summer School 2012, Obergurgl. CMI series, Amer. Math. Soc. 2014. Available at www.hh.hauser.cc.
• H. Hauser: The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. 40 (2003), 323-403.
• H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79 (1964), 109-203 & 205-326.
• J. Kollár: Lectures on resolution of singularities. Princeton University Press 2007.

Permanent link to this information: https://geolis.math.tecnico.ulisboa.pt/lecture_series?sgid=107