ETH :: D-MATH :: Zurich Colloquium in Mathematics

I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques that one would never have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic families of curves C on a complex surface S, nodal curves - those with the simplest possible singularities - appear in codimension 1. More generally those with d nodes occur in codimension d. In particular a d-dimensional linear family of curves should contain a finite number of such d-nodal curves. The classical problem - at least in the case of S being the projective plane - is to determine this number. The Göttsche conjecture states that the answer should be topological, given by a universal degree d polynomial in the four numbers C.C, c_1(S).C, c_1(S)^2 and c_2(S). There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler proof which was joint work with Martijn Kool and Vivek Shende.

Speakers:

Prof. Dr. Richard Thomas (Imperial College, London, UK)

6-mar-2012 (tue)

Burkhard Wilking

Structure of fundamental groups of manifolds with lower Ricci curvature bounds

K02 F 150

Speakers:

Prof. Dr. Burkhard Wilking (Universität Münster, Germany)

3-apr-2012 (tue)

Emmanuel Hebey

K02 F 150

Speakers:

Prof. Dr. Emmanuel Hebey (Université de Cergy-Pontoise, Paris, France)

24-apr-2012 (tue)

Endre Süli

K02 F 150

Speakers:

Prof. Dr. Endre Süli (University of Oxford, UK)

22-may-2012 (tue)

Brendan Hassett

Fibrations in rational surfaces and their sections

K02 F 150

Abstract:

A fibration is a surjective morphism from a smooth projective variety to a smooth curve, all defined over a field k. Assume the fibers are rational surfaces. Then every fibration admits a section, when k is algebraically closed. There is a conjectural framework for deciding whether there is a section when k is finite, expressed in terms of the Brauer group and the existence of local sections.

Our approach to these questions hinges on understanding the geometry of the scheme parametrizing all sections of our fibration, especially in contexts where the rational surfaces are relatively simple, e.g., quadric surfaces and intersections of two quadric hypersurfaces. The main application is the existence of sections provided the fibration is sufficiently general, in a sense that can be made precise.

(Joint with Yuri Tschinkel)

Speakers:

Brendan Hassett (Rice University, Houston, TX, USA)

Alle Interessentinnen - insbesondere auch die Studierenden beider Hochschulen - sind herzlich eingeladen. Im Anschluss an das Kolloquium findet ein Apéro statt.

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