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Local desingularization of quasi-excellent schemes

Final Report Summary - ILU (Local desingularization of quasi-excellent schemes.)

This project aims to improve our knowledge on resolution of singularities with a special accent on extending to general quasi-excellent schemes various results that are already known for algebraic varieties. Recall that, as a consistent theory, resolution of singularities is impossible for not quasi-excellent schemes, so the main aim of the project is to develop a desingularization theory in the most general possible context. Although it is conjectured that very strong desingularizations, such as functorial desingularization by successive blow ups with regular centers, should exist, even in the case of varieties (of positive characteristic) they are far beyond the reach of the contemporary theory. On the other side, weakened forms of desingularization can be obtained by current methods, for example, de Jong established desingularization of varieties by an alteration, and Gabber proved that any quasi-excellent scheme admits an alteration that resolves it locally along a valuation. The narrow aim of this project, independent of other teams, was to improve Gabber's results by controlling the generic degree of the alteration. In particular, the aim was to achieve that the alteration is purely inseparable in the positive characteristic case, and is of degree p^n in the mixed characteristic case. In addition, it was planned to check out the wider possibility of co-operation with other groups (first of all, with O.Gabber and L. Illusie) in order to combine our methods and/or explore other directions of desingularization of quasi-excellent schemes.

The starting point of the project was my preprint "Inseparable local uniformization", [arXiv:0804.1554v2] in which inseparable local uniformization was established for varieties. So, the first step was to polish the paper and get it published. This required two serious revisions of the paper (in particular, some results were improved in the final version at [arXiv:0804.1554v3] and it is 12 pages longer), and it is now published at Journal of Algebra 373 (2013), 65-119. Once the work on the above paper was completed, I started to work on local uniformization of Abhyankar valuations. This work is in process, though most of the needed results are already written down. Simultaneously with my work on the revisions of "Inseparable local uniformization" it was suggested to me by Luc Illusie to take part in writing down a volume on Gabber's results on weak desingularization of quasi-excellent schemes and applications to \'etale cohomology. This matched the wider perspective of the project and gave me an opportunity to establish co-operation with such world leading experts as Luc Illusie and Ofer Gabber. The tremendous work on preparing the volume has just been completed and it was published in a double Asterisque issue 363-364 (2014). \'Exposes VIII (103-162) and X (169-216) in the volume, are written by Illusie and me.

At some point, I joined to the project my postdoctoral fellow Dmitri Trushin and Ph.D. student Adina Cohen. Trushin has strong expertise in commutative algebra, and I suggested him to extend certain results of Elkik on algebraization of formal schemes to a formal scheme with a divisor. This is a difficult problem whose solution is needed for strong embedded desingularization of quasi-excellent schemes of characteristic zero. Trushin solved the suggested problem, and his paper will be published in Journal of Algebra and the preprint version can be found at [arXiv:1210.4176]. Based on his result, I found a way to substantially improve the results of my preprint [arXiv:0912.2570] on strong embedded desingularization. To incorporate this I split that paper in two parts, with the first one dealing with the case of varieties. Major part of the two resulting papers have already been written down but there is still some work to do in that direction.

The last direction that we studied in the framework of the project is a geometric description of wild ramification. Presence of the wild ramification in positive characteristic is the main obstacle for solving the local uniformization problem, so it is very desirable to understand its geometric aspects. In addition, this direction fit Cohen's expertise in arithmetics. Our research lasted two years and required a lot of training of my team (e.g. I gave a minicourse on Berkovich geometry). It has resulted in a joint paper [arXiv:1408.2949] and two separate papers by me [arXiv:1410.3079] and [arXiv:1410.6892]. In particular, a complete description of the wild ramification locus for morphisms of Berkovich curves and its relation to higher ramification theory is obtained in [arXiv:1410.6892]. All three papers are now being submitted for publication.

The main results achieved so far are as follows:
(1) The results on the inseparable local uniformization of varieties were published in a strong peer review journal (Journal of Algebra 373).
(2) Local uniformization of Abhyankar valuations was proved (the paper is in preparation).
(3) The co-operation with Luc Illusie resulted in two large \'exposes in Asterisque 363-364. The results of these \'exposes make essential use of two my earlier papers: "On stable modification of relative curves" and "Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case". In addition to writing down Gabber's results on l'-alterations, we found a stronger general result (see \'exposee X, section 3). In particular, we obtain a generalization of the semistable reduction theorem of Abramovich-Karu.
(4) Dmitry Trushin extended Elkik's algebraization to pairs. The preprint version is available at [arXiv:1210.4176].
(5) Strong embedded desingularization of quasi-excellent schemes of characteristic zero was deduced from Trushin's result (two papers in preparation).
(6) We studied the wild ramification locus for morphisms of Berkovich curves was studied by use of the different function in the paper [arXiv:1408.2949] by Cohen, Temkin and Trushin, and its complete description was then obtained in my paper [arXiv:1410.6892]. In addition, our joint paper is partially based on my results on norms of differential pluriforms on Berkovich spaces from the recent paper [arXiv:1410.3079].

Resolution of singularities is a classical problem of algebraic geometry which is very important and difficult. It has various applications and any progress in this area finds applications in a short time. For example, Gabber's results have powerful cohomological applications, and my results on desingularization of quasi-excellent schemes in characteristic zero were already used in works of Kedlaya, Mustata, Nicaise, and others since their publishing in 2008. The geometry of Berkovich curves was in the core of many recent papers, e.g. by Amini, Baker, Brugall\'e and Rabinoff, or Poineau and Pulita. However, the structure of finite morphisms between two curves remained obscure. By studying the wild topological ramification locus Cohen, Trushin and me completely settled this question. Therefore I am certain that the outcome of this project will have various applications in algebraic geometry, non-archimedean geometry, and adjacent areas of mathematics and physics.