9:10 Opening
9:15-10:15 Jean-Louis Colliot-Thélène (CNRS / Université Paris-Sud): In einem Zahlkörper ist das Komplement der Menge der n-ten Potenzen eine Diophantische Menge / The set of non-n-th powers in a number field is Diophantine
In joint work with J. Van Geel, we prove: For any natural integer n, the complement of the set of n-th powers in a number field k is the image of the set of k-rational points of some k-variety X under some k-morphism from X to the affine line. For n=2, this is a result of B. Poonen (2009). His proof uses local-global theorems (CT, Coray, Sansuc, 1980) for rational points on Châtelet surfaces. The proof for n arbitrary combines Poonen's method and local-global theorems (CT, Swinnerton-Dyer, Skorobogatov, 1994, 1998) for zero-cycles on higher dimensional analogues of Châtelet surfaces.
11:00-12:00 Arend Bayer (Edinburgh): Birational geometry of hyperkähler varieties
I will explain a description of the birational geometry of hyperkähler varieties deformation equivalent to Hilbert schemes of K3 surfaces. The description is based on using wall-crossing for the case of moduli spaces of sheaves on a K3 surface, deformation theory for rational curves, and Markman/Verbitsky's global Torelli statement. This is based on joint work with Macri, and with Hassett and Tschinkel.
15:00-16:00 Daniel Loughran (Bristol): Good reduction of Fano varieties
In 1983, Faltings proved the famous Mordell conjecture on the finiteness of the set of rational points on curves of higher genus. Along the way he proved numerous other finiteness statements, including the Shafarevich conjecture, which states that there are only finitely many curves of higher genus over a number field which have good reduction outside any given set of prime ideals. In this talk we shall consider analogues of the Shafarevich conjecture for certain classes of Fano varieties given as complete intersections in projective space. This is joint work with Ariyan Javanpeykar.
16:30-17:30 Hélène Esnault (FU Berlin): 0-cycles for surfaces over local fields
We study the Chow group of 0-cycles of smooth projective varieties (mainly surfaces) over local and strictly local fields. (joint with Olivier Wittenberg)
9:15 Bus from LMU to TUM
10:00-11:00 Takeshi Saito (Tokyo): Characteristic cycle and the Euler number of a constructible sheaf on a surface
We define the characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. The intersection number with the 0-section equals the Euler number and the total dimension of vanishing cycles at an isolated characteristic point is also computed as an intersection number.
11:30-12:30 Cecília Salgado (Rio de Janeiro): All del Pezzo surfaces of degree two over finite fields are unirational
A consequence of the Segre-Manin theorem is that a del Pezzo surface of degree two is unirational over its base field as long as it possesses a general rational point defined over the field in question. In this work, joint with D. Testa
and A. Várilly-Alvarado, we show that all del Pezzo surfaces of degree two over a finite fields are unirational with at most three possible exceptions. Recently, Festi and van Luijk showed that these three last surfaces are also unirational. I will discuss the arguments involved in our proof.
15:00-16:00 Masayuki Kawakita (Kyoto): A connectedness theorem over the spectrum of a formal power series ring
The relative Kodaira vanishing on excellent schemes is annoyingly unknown besides the work on surfaces by Lipman. We study the connectedness lemma by Shokurov and Kollar, which is an important geometric application of the vanishing theorem in birational geometry, over the spectrum of a formal power series ring. Investigating further in dimension 3, we prove the existence and normality of the smallest lc centre. It is applied to obtain the ACC for minimal log discrepancies greater than 1 on non-singular 3-folds.
16:30-17:30 Adrian Langer (Warschau): The Bogomolov-Miyaoka-Yau inequality in positive characteristic
I will talk bout Bogomolov's inequality for Higgs bundles in positive characteristic. As a corollary I will prove the Bogomolov-Miyaoka-Yau inequality for surfaces in positive characteristic (despite existing counterexamples...).
9:15-10:15 Per Salberger (Göteborg): Counting rational points on del Pezzo surfaces
Let X be a non-singular del Pezzo surface over Q of degree d≥3 and H: X(Q)→N be the natural height function given by an anticanonical embedding X⊂Pd. Let U be the complement of the lines on X and N(U,B) be the number of rational points of height at most B. We have then that N(U,B)=Oε(B12/(d+4)+ε) for d≤8, which is the best known general bound for d≤5. We discuss in our talk the proof in the case of cubic surfaces.
10:45-11:45 Alice Garbagnati (Milano): Kummer surfaces and K3 surfaces with a symplectic action of (Z/2Z)4
The aim of this talk is to describe the families of the K3 surfaces which admit G=(Z/2Z)4 as group of symplectic automorphisms and of the K3 surfaces which are the quotients of K3 surfaces by G. Both these families specialize to the families of Kummer surfaces. As a consequence one obtains that several results which were already known for Kummer surfaces hold more in general for these families of K3.
In particular, we will prove that every K3 surface with Picard number 16 and admitting a symplectic action of G admits also an Enriques involution. Moreover, we will prove that every K3 surface with 15 nodes is in fact obtained as quotient of a K3 surface by G.
The results presented are obtained in collaboration with Alessandra Sarti.
12:00-13:00 Sir Peter Swinnerton-Dyer (Cambridge): Solubility and insolubility of certain diagonal quartic surfaces
9:15-10:15 Matthias Schütt (Hannover): The maximum number of lines on smooth quartic surfaces
In 1943, Benjamino Segre published the claim that a smooth quartic complex surface contains at most 64 lines. His arguments, however, contain serious errors and gaps. In my talk, I will discuss joint work with Slawek Rams which gives a complete proof of Segre's claim over any field of characteristic other than 2 and 3. A special focus will be put on the exceptional characteristics.
11:00-12:00 Jakob Stix (Heidelberg): Simply transitive quaternionic lattices and a non-classical fake quadric
Square complexes with only one vertex allow to combinatorially construct a plethora of (combinatroial) lattices in the automorphism group of a product of two trees with constant valency. These lattices act freely and vertex-transitively. On the other hand, quaternionic arithmetic lattices, by acting on products of the respective Bruhat-Tits trees, again yield (arithmetic) lattices on such products of trees. These lattices are easier to understand group theoretically (by a linear embedding), however, there is no a priori control on the number of orbits on the set of vertices.
The talk deals with lattices in the intersection of both constructions, which therefore possess the pleasant properties of both sides: a finite realisation of a K(Γ,1) for the lattice Γ, hence a finite explicit presentation, and via rigid analytic uniformization the construction of a smooth surface with c12 = 2c2 in characteristic p. For p=3 the resulting surface is a (non-classical) fake quadric.
15:00-16:00 Ivan Cheltsov (Edinburgh): Cylinders in del Pezzo surfaces
For a projective variety X and an ample divisor H on it, an H-polar cylinder in X is an open ruled affine subset whose complement is a support of an effective Q-divisor Q-rationally equivalent to H. This notion links together affine, birational and Kahler geometries. I will show how to prove existence and non-existence of cylinders in smooth and mildly singular del Pezzo surfaces. This will answer an old question of Zaidenberg and Flenner about additive group actions on the cubic Fermat affine threefold cone. This is a joint work with Park and Won.
16:30-17:30 Gerard van der Geer (Amsterdam): Cycle classes of a stratification on the moduli of K3 surfaces
Moduli spaces in positive characteristic can be more accessible that their characteristic 0 counterparts because of the existence of stratifications. We calculate the cycle classes of the strata defined by the height and Artin number on the moduli of K3 surfaces in positive characteristic. The formulas generalize Deuring's famous formula for the number of supersingular elliptic curves. This is joint work with Katsura and with Torsten Ekedahl.
9:15-10:15 Alexei Skorobogatov (Imperial College London): The Brauer group of K3 surfaces: recent progress and open problems
A consequence of the recent proof of the Tate conjecture for K3 surfaces is the finiteness of the prime-to-p torsion subgroup of the transcendental Brauer group of K3 surfaces in odd characteristic p. This is a joint work with Yuri Zarhin. I will also discuss how in some cases one can compute the Brauer group, construct Brauer classes and calculate the Brauer-Manin obstruction. Examples will be mostly taken from a joint work with Evis Ieronymou on diagonal quartic surfaces.
11:00-12:00 Duco van Straten (Mainz): Elliptic modular surfaces via Langlands correspondence and congruence differential equations
Beyond hypergeometric or more generally rigid local systems, it is hard to characterise the local systems that have a geometrical origin. On the level of differential equations it is the problem of the accessory parameters.
I will describe joint work in progress with V. Golyshev, which shows that a Langlands approach via an explicit description of the SL2-Hecke-algebra given by Kontsevich, combined with the new idea of a "congruence sheaf" leads to a practical approach in the rank 2 case. This leading to a new approach to the "Apery-Beukers-Zagier" operators of the Beauville list of elliptic modular surfaces.