Oberseminar Zahlentheorie (SS 2013)

Werner Bley - Ulrich Derenthal - Cornelius Greither - Andreas Rosenschon - Tomasz Szemberg

Summer 2013, Wednesdays, 16:30, room B 251 (Mathematisches Institut, Theresienstr. 39, 80333 München)

Previous semester: Winter 2012/13 - next semester: Winter 2013/14

Abstracts

Christian Liedtke (TUM): Supersingular K3 Surfaces Are Unirational (24.4.13)

We show that supersingular K3 surfaces are related via purely inseparable isogenies. As an application, we deduce that they are unirational, which confirms conjectures of Artin, Rudakov, Shafarevich, and Shioda. To complete the picture, we prove Shioda-Inose type "sandwich" theorems for K3 surfaces of Picard rank at least 19 in positive characteristic.

Tomasz Szemberg (Cracow / LMU München): Symbolic powers, examples and counterexamples (8.5.13)

I will discuss relationship between symbolic powers of ideals of points in P² and geometry of linear systems on P² and its blow ups. In particular I will present a counterexample to a conjecture of Huneke to the effect that the third symbolic power of a planar points ideal need not to be contained in the second usual power of the ideal. Most results presented in the talk were obtained jointly with Marcin Dumnicki and Halszka Tutaj-Gasińska.

Alessandro Cobbe (LMU München): Steinitz classes of tame Galois extensions of number fields (15.5.13)

Let L/K be an extension of number fields and let O_L and O_K be the corresponding rings of integers. The structure of O_L as an O_K-module is determined by the degree of L/K and by an ideal class of Cl(K), called the Steinitz class of the extension. For any finite group G, we study the subset R_t(K,G) of Cl(K) of all the Steinitz classes of some tame Galois extensions L/K with Galois group isomorphic to G. This set is conjectured to be a subgroup of Cl(K). We will describe an explicit candidate for R_t(K,G) and prove that we always have an inclusion. Equality can be shown in a lot of cases.

Ben Green (University of Cambridge): On the Sylvester-Gallai theorem (27.5.13)

The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at least one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at least one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp. The talk will give an overview of this problem and the work towards its solution.

Slawomir Rams (Cracow / Universität Hannover): On quartics with many lines (5.6.13)

We show that a smooth quartic surface over algebraically closed field of characteristic different from 2,3 contains at most 64 lines. We discuss properties of quartics with lines of the second kind (joint work with Prof. M. Schuett (LU Hannover).

David Schmitz (Philipps-Universität Marburg): Big cone decompositions and Zariski chamber volume (12.6.13)

We recall some decompositions of the big cone of a smooth projective variety in terms of the mapping behavior of linear series and study their geometric properties. Furthermore, we introduce a metric description of these decompositions and present effective results in the case of the Zariski chamber decomposition on surfaces introduced by Bauer, Küronya, Szemberg.

Luca Caputo (Università di Pisa): Galois module structure of torsion modules in tame extensions of number fields (19.6.13)

Let N/E be a tame Galois extension of number fields and let J be a G-stable ideal of the ring of integers O_N of N. Then J defines a class (J) in the locally free class group Cl(Z[G]). Following an approach of Chase, I will show how to compute the difference (O_N-(J) by analysing the torsion module O_N/J, thus recovering a result of Burns. For particular ideals, such as the different of N/E and its square root, this can also be used in combination with Stickelberger's theorem to show that (O_N)=(J) in Cl(Z[G]) (this is related to results of M. Taylor and Erez). The same strategy allows one to prove that (O_N) is annihilated by the degree of N/E. If time permits, I will connect this approach to that of Fröhlich and M. Taylor. This is a joint work with Stéphane Vinatier.

Peter Scholze (Universität Bonn): Torsion in der Kohomologie von lokal symmetrischen Räumen (20.6.13)

Wir beweisen, dass es Galoisdarstellungen assoziiert zur mod p - Kohomologie der lokal symmetrischen Räume für GL_n gibt. Dies basiert auf neuen geometrischen Resultaten zur p-adischen Geometrie des Siegelschen Modulraums von prinzipal polarisierten abelschen Varietäten, insbesondere einer neuen Periodenabbildung, die erst definiert ist, wenn man zum inversem Limes über alle Level bei p übergeht.

Patrycja Luszcz-Swidecka (Cracow): Minkowski decomposition of Okounkov bodies on surfaces (3.7.13)

We recall the definition and some properties of Okounkov solids of divisors, concentrating mainly on divisors of surfaces. We present a decomposition of a big divisor corresponding to a decomposition of its Okounkov solid into simple simplices. Results presented in the talk were obtainedin a joint work with David Schmitz.

Halszka Tutaj-Gasinska (Cracow): Around Nagata's conjecture (10.7.13)

I will present some results and conjectures on the existence of forms in k[Pⁿ] vanishing on a given set with given multiplicities.

Sören Kleine (Universität Göttingen): Local behaviour of Iwasawa invariants (17.7.13)

Classical Iwasawa theory studies galois extensions of number fields whose galois groups are isomorphic to the additive group Z_p of p-adic integers, where p in Z denotes a fixed prime. To each such Z_p-extension K_∞/K, Iwasawa attached a triple of integers, the so-called Iwasawa invariants of K_∞/K. They describe the growth of the ideal class groups of the intermediate fields of K_∞/K. In this talk, we will consider the set of Z_p-extensions of a fixed number field K. We will introduce a topology on this set and ask for connections between the Iwasawa invariants of Z_p-extensions of K that are close together with respect to this topology. In particular, we will look for results concerning the local boundedness of Iwasawa invariants.