The purpose of this class is to provide the participant with the necessary basics for starting her/his own research in resolution of singularities. Once the methods are explained and illustrated in many concrete situations, it will be shown how they can be used to build up a resolution proof in the known cases of zero characteristic, and, respectively, how to approach the still open positive characteristic case. Special emphasis will be laid on the comparison between the axiomatic description and the constructive realization of the various concepts and methods.
We will start with different views on the main tranformation of varieties in this context, blowups (this topic will also be covered in the other lectures, since we believe that distinct views on the same subject will provide a deeper understanding). There are at least seven possible definitions, and we will prove the respective equivalences carefully.
This will then be used to introduce the various transformations of varieties and ideals under blowup: strict, weak, controlled and total transforms. The significance of each will be made clear.
The next topic will be the choice of the centers of the blowup. This requires to stratify the variety according to the complexity of the singularities. The stratification is usually given by some uppersemicontinuous local invariants, and there are various ways of how to choose them: local multiplicity, characteristic pair, Hilbert-Samuel function, initial ideal, shape of tangent cone, etc.
Once the center is chosen, we may apply the respective blowup to the variety. The hope is that the singularities improve; to exhibit the improvement, some numerical measure of the intricacy of the singularities has to be developed. Again, one could refer to the invariants used to stratify the variety, but also other invariants are in principle possible. Any such numerical datum will be called a resolution invariant. It turns out that the main invariant, the local multiplicity, does not increase under blowup if the center is chosen appropriately (i.e., within the equimultiple locus). If it drops, we are done by induction. Otherwise, this leads us to the question of how to detect an improvement at the points of the transformed variety where the multiplicity has remained constant.
It is here that the argument ramifies depending on the characteristic of the ground field. In zero characteristic, it turns out that these points lie in a smooth hypersurface which coincides with the transform of a local hypersurface from below. This allows us to transcribe the resolution problem to these hypersurfaces (hypersurfaces of maximal contact), and to apply induction on the embedding dimension. The main technical tools for the descent in dimension are osculating hypersurfaces, Tschirnhaus transformations, coefficient ideals, retractions and projections. In the course of these constructions, other notions will become important and have be explained in the class: transversality and normal crossings, local coordinate systems, automorphism groups, flags of smooth subvarieties, subordinate coordinates, Newton polyhedra.
After this algebraic setup has been installed, there remains the delicate task to combine all the ingredients to build up the argument for the existence of resolution. This is a mostly logical problem based on a multiply interwoven induction: The descent in dimension has to be composed with the decrease (resp. constancy) of the invariant under blowup in the larger and smaller dimension. The argument is very elegant, but requires to keep track of various data (transversality, exceptional divisors, hypersurfaces, etc.). The logical part will be studied with more details in Schicho‚s lectures.
The resolution data are best collected in the notion of singular mobile (there are many other similar concepts, carrying different names such as idealistic exponents, basic objects, infinitesimal presentations, etc.). Mobiles are some enriched data structure associated to the various stages of a resolution process of a singular variety; they allow directly to define both the stratification of the variety and the resolution invariant for the next blowup.
It is then a combinatorial issue to complete the induction. In Schicho‚s lectures, the induction will be isolated as a purely combinatorial game between to players A and B, and the existence of resolution will be established by showing that the game has a winning strategy for player A.
A good acquaintance with the characteristic zero case will enable us to attack in the third week the case of positive characteristic. Even though it is still unsolved, there is machinery which will be useful for understanding the main phenomena. This will be exemplified in the known cases of the embedded resolution of curves and surfaces. The constructions will parallel methods explained in Villamayor's lectures, where a somewhat different approach will be pursued.
- Eisenbud-Harris, Geometry of Schemes. Springer. (Basics on blowups.)
- Hauser: The Hironaka Theorem on resolution of singularities. Bull. Amer. Math. Soc. 40 (2003), 323403. (Overall structure of the resolution proof in characteristic zero, see www.hh.hauser.cc.)
- Hauser: Three power series techniques. Proc. London Math. Soc. 88 (2004), 124. (Special characteristic independent techniques.)
- Kollar: Lectures on Resolution of Singularities. Princeton University Press. (Many proofs for curve resolution, general proof in characteristic zero.)
- Cutkosky: Lectures on Resolution of Singularities. Amer. Math. Soc. 2004. (Essentially Villamayor's proof for characteristic zero.)
- Zariski: Reduction of the singularities of algebraic three dimensional varieties. Ann. Math. 45 (1944), 472542. (Very stimulating reading, representing many of the main ideas.)