Workshop on Algebraic Surfaces


Date and location:     Feb 3, 2012, 10:00 - 17:00 in room B305 at Leibniz Universität Hannover (Welfenschloss/main building)

Schedule: 10:00 - 11:00 Cecilia Salgado (MPI Bonn) - Zariski density of rational points on del Pezzo surfaces of degree one (abstract)
11:30 - 12:30 Ananyo Dan (HU Berlin) - Non-reduced components of the Noether-Lefschetz locus (abstract (pdf))
14:00 - 15:00 Bas Heijne (Groningen/Hannover) - Computing sections on elliptic (Delsarte) surfaces (abstract)
15:30 - 16:30 Ronald van Luijk (Leiden) - Computability of Picard numbers (abstract)

Registration:     There is no formal registration, but we would appreciate if you send an email to schuett [--] math.uni-hannover.de in case you plan to attend the workshop.

Directions:     file under Algebraic and Complex Geometry

Inquiries (about accommodation etc): Nicole Rottländer (sekretariat-c [--] math.uni-hannover.de) will be pleased to assist you.

Sponsors:     Institute of Algebraic Geometry, Leibniz Universität Hannover
    ERC Starting grant 279723 (SURFARI)

Organised by:     Matthias Schütt

Abstracts

Zariski density of rational points on del Pezzo surfaces of degree one
Cecilia Salgado (MPI Bonn)

Abstract: Let k be a non-algebraically closed field and X a projective algebraic surface defined over k. Unlike in the case of curves, it is an open problem to know whether the k-rational points of X are Zariski-dense. Despite the amount of research done in this field, the problem is, surprisingly, not yet completely solved for rational surfaces. It is expected that the answer is positive for this class. In this lecture, I will treat the important subclass of del Pezzo surfaces. While the cases of degree at least three and two - under the assumption of the existence of a k-rational point outside a certain set of points - is solved, the case of degree one is still (mainly) open. I will briefly talk about the known results and focus on the problem for del Pezzo surfaces of degree one. The basic idea is that a point P that is different from the base point of the anti-canonical linear system admits a multi-section through it. If P is sufficiently general, then the existence of infinitely k-rational points in this multi-section assures that the k-rational points in X are Zariski dense. This is work in progress with Ronald van Luijk.


Computing sections on elliptic (Delsarte) surfaces.
Bas Heijne (Groningen/Hannover)

Abstract: In this talk I will focus on elliptic surfaces. I will present a classification of elliptic Delsarte surfaces. After that I will discuss several ways which can be used to search for sections on elliptic surfaces. Most of the examples used will be elliptic Delsarte surfaces.


Computability of Picard numbers
Ronald van Luijk (Leiden)

Abstract: The Neron-Severi group of a variety is the group of its divisor classes modulo algebraic equivalence. The rank of this group is called the Picard number of the variety. After giving a short review of methods that compute the Picard number in certain cases, I will sketch an idea to prove that the Picard number is computable in general. This is joint work in progress with Damiano Testa and Bjorn Poonen.