Resolution of Singularities: Games and Computations
Josef Schicho

The main reason for this lecture is a vague hope: the hope that a deeper understanding of the proof of existence of resolution of singularities of algebraic varieties in characteristic zero would eventually lead to a proof that also holds in the case of positive characteristics. Surely, new ideas will be needed, and the lecturer has them not; but the goal of this lecture is to pave the way by "trivializing" the characteristic zero proof.

By "deeper understanding", we mean two things. First, one can isolate the nested induction in the proof as a combinatorial play between two players. The first (our "local hero") attempts to improve the singularities, while the second (some "malevolent demon") tries to keep the singularity alive and difficult. Once the rules of this game are fixed, the proof can be cleanly separated into an algebraic part and a combinatorial part. The weave of nested inductions can be put entirely into the combinatorial part, which is considerably complex. But it is also purely elementary: in order to follow the rules of the game or to show that a winning strategy for the first player exists, one does not even need to know what a polynomial is.

Second, the construction of coefficient ideals or hypersurfaces of maximal contact is only locally possible and depends certain local choices, for instance a system of local coordinates whose differentials freely generate the module of K¨ahler differentials. It is a complication in most proofs that one has to show that the final resolution is independent of these local choices and the local results patch together to a global object. A deeper understanding would avoid any local choices in the first place, and introduce all constructions in a coordinateindependent way. In order to do this, we will work with global resolution objects, called "gallimaufries", which are capable of an intrinsic descent in dimension.

The following algebraic concepts will be needed: blowup of ideals, smooth schemes, Rees algebras, differential operators, projective morphisms, syzygies, Jacobean ideals. The plan is to introduce all necessary constructions and give computational examples. For computations which are too complex to be performed by hand, a computer algebra system (MAGMA) will be available at the summer school.

Literature. As a preparation of the course, study of fundaments in algebraic geometry is recommended, such as provided in [R. Hartshorne, Algebraic Geometry, Springer 1977].

The rules of the combinatorial game and most required algebraic concepts (e.g. gallimaufries) can also be found in [H. Hauser and J. Schicho, A Game for the Resolution of Singularities, ArXiv:1010.0163]. This preprint is therefore recommended as a reference manual for the lecture.

Knowledge of examples in positive characteristic where the characteristic zero proof does not work will be essential for attempts to solve the positive characteristic case. For these examples, and for attempts to prove the positive characteristic case, we refer to papers by Arapura/Bakhtary/Wlodarczyk, Benito/ Villamayor, Bravo/Villamayor, Cutkosky, Hauser, Kawawnue, Kawanue/Matsuki, Urabe, Villamayor, available at ArXiv, and to [H. Hironaka, A Program for Resolution of Singularities, in all Characteristics p > 0 and in all Dimensions], available from ICTP. It should be mentioned that the material in these papers will not be covered by this course.

Prerequisites: For the algebraic part, participants are assumed to be familiar with fundamental concepts in algebraic geometry: schemes, projective morphisms, syzygies. The combinatorial part requires no background.