Program
Time  Wednesday 14 July  Thursday 15 July  Friday 16 July 
9:15  10:15  Laza  Kondo  
10:30  11:30  Ito  Artebani  
11:45  12:45  Kuwata  Ekedahl  
Lunch break  
14:00  15:00  Shioda  Elkies  
15:30  16:30  Kumar  Liedtke  
16:45  17:45  Sarti  Schoen 
Abstracts
 Michela Artebani (Concepcion)  Cox Rings of K3 surfaces
Abstract: Let X be a smooth projective surface X over C with finitely generated Picard group Pic(X). The Cox ring of X is the graded ring
R(X) := Sum_{[D] in Pic(X)} H^{0}(X,O_{X}(D)). The ring R(X) is known to be a polynomial ring if and only if X is a toric surface and an explicit description of the ring is available for Del Pezzo surfaces. In general, it is even dificult to decide if the Cox ring of a surface is finitely generated. In this talk I will present a joint work with J. Hausen and A. Laface about Cox rings of K3 surfaces. I will explain that the Cox ring of a K3 surface is finitely generated if and only if its effective cone is polyhedral. Moreover, I will show how to compute the Cox ring of some K3 surfaces which either have Picard number two or are double covers of rational surfaces. An example of both types is the generic K3 surface with Picard lattice isometric to U(2), in this case:R(X) = C[a_{1}, a_{2}, b_{1}, b_{2}, c]/(c^{2}  f_{4,4}(a, b)), where deg(a_{i}) = (1,0), deg(b_{i}) = (0,1) and deg(c) = (2, 2).  Torsten Ekedahl (Stockholm)  Automorphisms of blown up K3surfaces
Abstract: For K3surfaces there is a very satisfactory cohomological description of their automorphism groups; any automorphism of the Hodge structure of the second cohomology group which also preserves the nef cone comes from an automorphism. One may ask if this is part of a general pattern. I shall show that it is false in general for K3surfaces blown up at a point.
 Noam Elkies (Harvard)  Pseudoelliptic K3 surfaces and certain isotropic codes
Abstract: It is known that in characteristic 2 or 3 there are "pseudoelliptic surfaces": rational maps on a surface X whose generic fiber has arithmetic genus 1 but geometric genus 0. When X is a (necessarily supersingular) K3 surface, we show that the description of its pseudoelliptic fibrations in terms of the NeronSeveri lattice naturally leads to an isotropic code C over Z/pZ, of length 20 for p=2 (with doubly even weights) and 10 for p=3, and with dim(C*/C) equal twice the Artin invariant. Conversely each C yields a family of K3 surfaces of the correct dimension. We describe some of these families explicitly and conjecture what they should be in general.
 Hiroyuki Ito (Hiroshima)  Classification of elliptic K3
surfaces with p^{n}torsion sections
Abstract: This is a joint work with Christian Liedtke. We clssify elliptic K3 surfaces with p^{n}torsion sections in characteristic p. In this talk, we give the explicit description of these surfaces. Especially, we study them in connection with the families arising from deformations of ratinal double points.
 Shigeyuki Kondo (Nagoya)  The supersingular K3 surface with
Artin invariant 1 in characteristic 2 (revisited)
Abstract: Dolgachev and I gave several ways to construct the supersingular K3 surface with Artin invariant 1 in characteristic 2. In this talk, I shall add another two ways. This is a joint work with Toshiyuki Katsura.
 Abhinav Kumar (MIT)
 All the elliptic fibrations on a generic Jacobian Kummer surface
Abstract: Kuwata and Shioda pose the following question in their paper, "Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface": Determine the essentially different elliptic parameters, Weierstrass equations and MordellWeil lattices for the Kummer surface of the Jacobian of a genus two curve. Earlier, Oguiso had described all the essentially different elliptic divisors on the Kummer surface of a general product of elliptic curves, and Kuwata and Shioda turn these into explicit equations. I will talk about recent work which answers the above question completely for Km(Jac(C)), where C is a generic genus two curve.

Masato Kuwata (Chuo University) 
3torsion of the Jacobian of a curve of genus 2 and the
MordellWeil lattice of a rational elliptic surface
Abstract: Let f(x) be a polynomial of degree 5 or 6 with coefficients in a number field k and without multiple roots. Let C be the curve of genus 2 defined by y^{2}=f(x). The Galois group G_{k}=Gal(k/k) acts on the group of 3torsion points in the Jacobian J(C) and the action is compatible with the Weil paring. Thus we have a homomorphism ρ_{C,3}:G_{k} → GSp(4,F_{3}). Meanwhile, using the polynomial f(x), we can define an elliptic curve over k(x) by the Weierstrass equation E : Y^{2} = X^{3} + f(x). This is a rational elliptic surface whose MordellWeil lattice is isomorphic to the root lattice of type E_{8}. Hence, we have ρ_{E}:G_{k} → W(E_{8}), where W(E_{8}) is the Weyl group. In this talk we discuss the relationship between these two Galois representations, and we construct a polynomial of degree 40 whose Galois group is isomorphic to PGSp(4,F_{3}). (This work has been done in collaboration with Kiichiro Hashimoto.)
 Radu Laza (Stony Brook)  The arithmetic and geometry of degenerations of K3 surfaces
Abstract: An important (and still open) question in algebraic geometry is to to find a geometric compactification for the moduli of polarized K3 surfaces. In this talk, I will survey some recent approaches to this problem. My focus will be on explaining the interplay between arithmetic, combinatorics, and geometry in the study of degenerations of K3 surfaces.
 Christian Liedtke (Stanford)  The Double Cover of an
Enriques Surface
Abstract: Every Enriques surface possesses a canonically defined flat double cover. In characteristic not equal to 2, this cover is etale and the covering surface is a K3 surface. Projective models of the K3 surfaces that arise this way have been obtained by Cossec, using SaintDontat's analysis of linear systems on K3 surfaces. On the other hand, in characteristic 2, these covers may be K3 surfaces, but are in general only integral, possibly even nonnormal, Gorenstein surfaces with trivial dualizing sheaves ("K3like"). In this talk, I will explain how some of Cossec's results can been carried over to characteristic 2, even in case where the cover is not a K3 surface.
 Alessandra Sarti (Poitiers)  Elliptic fibrations and automorphisms of K3 surfaces
An important tool in the study of automorphisms of K3 surfaces is the use of elliptic fibrations. In fact by using torsion sections one can produce symplectic automorphisms of finite order (which are automorphisms preserving the holomorphic 2form). Then using the geometry, one can give an explicit description of the invariant lattice and its orthogonal complement, which are important objects in the study of the moduli space of K3 surfaces with symplectic automorphism.
On the other hand many K3 surfaces with non symplectic automorphism are also elliptic, again the geometry is very explicit and allows to study properties, such as the topology of the base locus. In the talk I will also describe K3 surfaces admitting a symplectic and a nonsymplectic automorphism, showing that in certain cases the generic K3 surface with a nonsymplectic automorphism of a finite order also admits a symplectic automorphism of the same order.
 Chad Schoen (Duke)  Torsion in the cohomology of
desingularized fiber products of elliptic
surfaces.
Abstract: Torsion in the ladic cohomology of the 3 folds of the title is described. Representing torsion cohomology classes by algebraic cycles will be discussed.
 Tetsuji Shioda (Rikkyo/RIMS Kyoto) 
Cubic surfaces via MordellWeil lattices, revisited.
Abstract: We discuss the classical topics of the 27 lines on a cubic surface from the viewpoint of MWL. A rational elliptic surface (RES) with a section has MWL of type E_{6} iff there is a unique reducible fibre with three irreducible components, i.e. of Kodaira type I_{3} or IV. We treated the "additive case" (with IV fibre) in 1990's, and we give the corresponding results in the "multiplicative" case with I_{3}. We obtain an excellent family of RES with the Weyl group W(E_{6}) as the Galois group, and its consequence on cubic surfaces. A major difference from the additive case is the explicit formula of the coefficients of the family as the W(E_{6})invariants, not of polynomials, but of Laurent polynomials, in splitting variables.
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Organizers
Remke Kloosterman (HumboldtUniversität zu Berlin)Matthias Schütt (Leibniz Universität Hannover)
Last modified: Mon Jun 28 10:18:36 CEST 2010