# Workshop on algebraic surfaces

Venue: Leibniz Universität Hannover, Main building (Welfenschloss), room b302

Dates: March 1/2, 2018

## Speakers

Kenji Hashimoto (Tokyo)
DongSeon Hwang (Ajou/Warwick)
Xin Lu (Mainz)
Yuya Matsumoto (Nagoya)
Zsolt Patakfalvi (Lausanne)
Stefan Schreieder (LMU München)

## Schedule

Thursday

14:00-15:00 - Matsumoto
15:30-16:30 - Lu
17:00-18:00 - Patakfalvi

Friday

09:30-10:30 - Hwang
11:00-12:00 - Hashimoto
12:15-13:15 - Schreieder

## Titles and abstracts

Global sections of some special elliptic surfaces (Hashimoto)

We discuss how to reconstruct an elliptic K3 surface from the data of singular fibers. The problem is reduced to counting of global sections of some special elliptic surfaces.

Cascades of log del Pezzo surfaces and SSDB (Hwang)

There have been numerous attempts to classify log del Pezzo surfaces. In this talk, I will quickly summarize the known attempts to this goal and report my recent work on the classification of log del Pezzo surfaces of Picard number one. The result is obtained by generalizing the notion of ‘cascades’ of nonsingular del Pezzo surfaces, based on the approach initiated by Miyanishi and Zhang. At the end of the talk, I will present the computer program (under construction) realizing this approach.

$$\mu_p$$- and $$\alpha_p$$-actions on K3 surfaces in characteristic $$p$$ (Matsumoto)

There are three group schemes of order $$p$$ in characteristic $$p$$, namely $$\mathbb{Z}/p\mathbb{Z}$$, $$\mu_p$$ and $$\alpha_p$$. We consider their actions on K3 surfaces in characteristic $$p$$, with emphasis on the latter two. To be precise, we consider not only smooth K3 surfaces but also K3 surfaces with rational double point singularities (RDPs)''. We will give (partial) answers to the following questions: For which $$p$$ such actions exist? How can we distinguish actions whose quotients are again K3 surfaces (with RDPs)? Which kind of RDPs may appear on the quotient surface or on the original surface?

Geometric vs. non-geometric singularities (Patakfalvi)

For a variety X defined over an imperfect field k, there can be a discrepancy between its singularity properties over k, and its singularity properties over the algebraic closure of k. The later singularity properties are also frequently called geometric. For example, it frequently happens that a surface over k is normal but over the algebraic closure of k it becomes non-normal. First, I give some examples on this phenomenon. Second, I present a general canonical formula relating the canonical bundle of X to the canonical bundle of the normalized base-extension to the algebraic closure of k. Third, I explain how this implies that normal but geometrically non-normal Gorenstein Del-Pezzo surfaces do not exist in characteristic greater than 3. In particular, this implies the generic smoothness of terminal 3-fold Mori-fiber spaces in characteristic greater than 7. Lastly, if time permits, I briefly mention a few other applications too. This work is joint with Joe Waldron.

The Kuga-Satake construction under degeneration (Schreieder)

We extend the Kuga--Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga-Satake abelian varieties over the punctured disc. This is joint work with A. Soldatenkov.

Strict Arakelov inequality for families of curves (Lu)

For a one-dimensional semi-stable family of curves, the Arakelov inequality gives an upper bound on the Hodge bundle in terms of the genera of the base curve and the general fiber together with the number of singular fibers. Later it's proved that the number of singular fibers appearing above can be replaced by the number of singular fibers with non-compact Jacobians. It is expected that such an Arakelov inequality should be strict when the fibers are of high genera. In this talk, I would like to report some progress on this problem. This is a joint work with K. Zuo.

## Organisers

Simon Brandhorst
Víctor Gonzalez Alonso
Matthias Schütt