# Algebra and Geometry Seminar 2014/2015

## Organizers

Lenny Taelman
Email: Lenny.Taelman AT gmail.com

Mingmin Shen
Email: M.Shen AT uva.nl

## Schedule Spring 2015

Time: 5 February 2015, 16:00 -- 17:00; Room F3.20
Speaker: Lie Fu (Lyon)

Title: On the motivic hyperkähler resolution conjecture
Abstract:
This is a joint work with Zhiyu Tian. Ruan's conjecture on hyperkähler resolution says that the orbifold cohomology ring of an smooth projective orbifold is isomorphic to the cohomology ring of its hyperkähler resolution of singularities. I would like to formulate the motivic analogue of this conjecture on the Chow ring (or more generally the Chow motive as an algebra in the category of Chow motives) of hyperkähler varieties. The particular interesting cases that I want to discuss are the Hilbert-Chow morphisms for the symmetric product of K3 surfaces and the similar generalized Kummer varieties. This work could be seen as a continuation of the recent work of C. Vial.

Time: 7 May 2015, 16:00 -- 17:00; Room F3.20
Speaker: Gerard van der Geer (UvA)

Title: The cycle classes of divisorial Maroni loci
Abstract:
To a general curve of genus g with a linear system of degree d one can associate a (d-1) tuple of integers that describe the type of scroll spanned by the fibres of the linear system on the canonically embedded curve. This gives rise to the so-called Maroni stratification of the Hurwitz space H(d,g). We determine the cycle classes of Maroni divisors in the compactified Hurwitz spaces. This is joint work with Alexis Kouvidakis.

Time : 21 May 2015, 16:00 -- 17:00; Room F3.20
Speaker: Bernhard Köck (Southampton)

Title: Euler characteristics and epsilon constants of curves over finite fields - some wild stuff
Abstract:
Given a smooth projective curve over a finite field equipped with an action of a finite group, we first introduce the corresponding Artin L-function and a certain equivariant Euler characteristic. The main result is a precise relation between the epsilon constants appearing in the functional equations of Artin L-functions and that Euler characteristic. This generalises a theorem of Chinburg from the tamely to the weakly ramified case.

## Schedule Fall 2014

Time: 11 September 2014, 16:00 -- 17:00; Room B0.209
Speaker: Mingmin Shen

Title: On the rationality problem of cubic fourfolds

Abstract:
The rationality problem of cubic fourfolds is one of the open problems in algebraic geometry. We have many examples of rational cubic fourfolds. It is widely believed that a very general cubic fourfold is irrational but so far we have no proof of this. People has been attacking this problem from different points of view, such as Hodge structures, derived categories, Chow groups and Grothendieck ring of varieties. In this talk, I will survey our current knowledge about this problem and discuss some recent progresses.

Time: 25 September 2014, 16:00 -- 17:00; Room B0.209
Speaker: Lenny Taelman

Title: Non-additive functors and Gronthendieck groups

Abstract:
Let F:A\to B be a left-exact functor. Under suitable finiteness conditions, the alternating sum \sum (-1)^i [ R^iF(X) ] in the Grothendieck group K_0(B) only depends on the class [X] in K_0(A) and one obtains a homomorphism K_0(A) \to K_0(B). In his 2014 master's thesis Niels uit de Bos has constructed for a large class of not necessarily additive functors F an induced function K_0(A) \to K_0(B). In general this induced map is not a homomorphism. Interesting examples include multilinear' functors such as exterior and symmetric powers. In this lecture I will explain the results of Niels, and show how simplicial techniques play a major role.

Time: 23 October 2014, 16:00 -- 17:00; Room B0.209
Speaker: Eugen Hellmann

Title: Trianguline representations and eigenvarieties

Abstract:
I will talk about joint work with C. Breuil and B. Schraen in which we study families of Galois representations that occur on eigenvarieties. Using a patching module as in recent work of Caraiani, Emerton, Gee, Geraghty, Paskunas and Shin one can define a local analogue of an eigenvariety and compare this to a space of local Galois representations defined in terms of (phi,Gamma)-modules. I will discuss this construction and the relation with modularity lifting theorems.

Time: 13 November 2014, 16:00 -- 17:00; Room F3.20, Science Park 107 (meeting/seminar room of KdVI)
Speaker: Evgeny Shinder

Title: The Fano variety of lines and rationality question for cubics

Abstract:
We prove a relation in the Grothendieck ring of varieties between a cubic hypersurface of arbitrary dimension and its Fano variety of lines. Using this relation we compute the Hodge structure of the Fano variety. Finally assuming the Cancellation conjecture we deduce implications on irrationality of a very general cubic fourfold. Joint work with Sergey Galkin.

Time: 14 November 2014
!!!!!Special Event!!!!!: Belgian-Dutch Algebraic Geometry Day. Speakers include: Minhyong Kim, Mingmin Shen, Jakob Stix.
Time: 20 November 2014, 16:00 -- 17:00; Room F3.20, Science Park 107 (meeting/seminar room of KdVI)
Speaker: Giulia Battiston

Title: The p-curvature conjecture in positive characteristic

Abstract:
While the p-curvature conjecture of Grothendieck and Katz, which lives in mixed characteristic, remains still widely unsolved, André stated and proved its analogous for the equicharacteristic zero situation. In a similar spirit, Esnault and Langer proved a weak form of the conjecture in positive characteristic. In this talk we will explain how the strong form of the conjecture holds over uncountable fields of positive characteristic and its complete failure if the field is countable.

Time: 18 December 2014, 16:00 -- 17:00; Room F3.20, Science Park 107 (meeting/seminar room of KdVI)
Speaker: Christian Liedtke

Title: Good reduction of K3 surfaces

Abstract:
By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface admits an `RDP model'' over the that field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.