# Abstracts

Resolution of Singularities and Semistable Reduction

## Herwig Hauser `Desingularization' (Lecture)

Talk I: The problem of resolution of singularities

We describe and compare various versions of this problem (local/global, varieties/ideals/vector fields, strong/weak), sketch the historical development, specify the art and mention the still open problems.

Talk II: The main techniques for resolution

We introduce and discuss concepts like blowup, strict and total transform of varieties of ideals, transversality, upper semicontinous functions and stratifications.

Talk III: The proof of resolution in characteristic zero

Though it will not be possible to give all details, we intend to communicate the overall scheme of the proof, its logical structure and the main obstructions to extend it to positive characteristic.

Reading material:

We recommend the chapter on Blowups in Eisenbud-Harris [1], as well as the three Bulletin AMS surveys [2,3] on Hauser's homepage.

[1] David Eisenbud and Joe Harris. The geometry of schemes, vol. 197, GTM, Springer-Verlag, 2000.

[2] Herwig Hauser. The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand). Bull. Amer. Math. Soc. (N.S.), 40(3):323--403 (electronic), 2003.

[3] Herwig Hauser. On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for). Bull. Amer. Math. Scoc. (N.S.), 47(1):1--30, 2010.

**Qing Liu `Regular and canonical models of surfaces' (Lecture)**

Talk I: Normal surfaces (Tuesday, 09:00-10:30)

- Birational geometry of normal surfaces (Blowing-up; Birational maps; Divisors)

- Relative curves over Dedekind domains, projective models

Talk II: Regular models (Wednesday, 09:00-10:30)

- Dualizing sheaf, various numerical invariants

- Castelnuovo criterion and minimal models

- Models with normal crossings

Talk III: Canonical models (Thursday, 09:00-10:30)

- Contraction morphisms

- Canonical models, the case of stable reduction

- The case of elliptic curves (Weierstrass models)

Reading material:

We recommend Chinburg's article [4] and chapters 8--10 of [5].

[4] Ted Chinburg. Minimal models for curves over Dedekind rings. in `Arithmetic Geometry', edit. Cornell, Silverman

[5] Qing Liu. Algebraic geometry and arithmetic curves. Oxford.

**Werner Lütkebohmert `Semistable reduction' (Lecture)**

Talk I: Reduced Fiber Theorem (Tuesday, 11:00-12:30)

- Basic concepts of formal and rigid geometry

- Valuations corresponding to components of the special fiber

- Meaning of geometrically reduced fibers

- How to perform reduced fibers

-- Analytic method of Grauert--Remmert--Gruson

-- Elementary method of Epp

-- Natural method of Bosch--Lütkebohmert--Raynaud

Talk II: Stable Reduction Theorem (Wednesday, 11:00-12:30)

- Presentation of the theorem

- Formal structures, smooth points and double points

- Periphery of singular points

- Genus formula

- Divisors associated to $\ell$-torsion points

- End of the proof

Talk III: Stable Reduction Theorem and Uniformization of Jacobians (Friday, 11:00-12:30)

- Method of Abhyankar, Artin--Winters, Deligne--Mumford

- Regular model and semi-abelian reduction of the Jacobian

- Torus part of the Jacobian

- Uniformization of Jacobians

**Franz Kiraly `Wild quotient singularities of surfaces and their regular models' (Short Talk)**

In my talk, I will discuss the concept of quotient singularities and its meaning in the field of models of arithmetic surfaces.

In the first part of my talk, I will look at some easy examples of quotient singularities and the regularity criterion of Serre in the tame case. Then I will present recent results in the wild case of prime order.

In the second part, I will apply the theory of quotient to relate regular models of curves over different ground fields to each other. In particular, I will focus on $p$-cyclic monodromy actions on regular models and the minimal desingularization of their quotients.

**Sophie Schmieg `On the periphery of formal fibers' (Short Talk)**

In this talk I will discuss the structure of formal fibers and prove that the periphery of formal fibers are annuli.

**Osmanbey Uzunkol `Thetanullwerte, reciprocity law and CM-construction' (Short Talk)**

In the first part of my talk I am going to introduce the classical class invariants of Weber, and their generalizations, as quotients of values of `Thetanullwerte', which enables to compute them more efficiently than as quotients of values of the Dedekind $\eta$-function. Moreover, the result that most of the invariants introduced by Weber are actually units in the corresponding ring class fields will be introduced, which allows to obtain better class invariants in some cases, and to give an algorithm that computes the unit group of corresponding ring class fields.

In the second part, using higher degree reciprocity law I am going to introduce the possibility of generalizing the algorithmic approach of determining class invariants for elliptic curves with CM, to determining alternative class invariant systems for principally polarized simple abelian surfaces with CM.