Department of Mathematics, University of Oslo
Seminar in Algebra and Algebraic Geometry, Spring 2007

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Eduardo Esteves, 12. February

Abel maps for singular curves

The Abel map embeds a nonsingular projective curve in a projective algebraic group, the so-called Jacobian variety of the curve. Using the group structure we can consider higher versions of the Abel map, which carry a lot of information about the projective geometry of the curve. If the curve varies in a family, so do its Jacobian variety and the Abel map. So it is natural to ask what happens when the family degenerates to a singular curve, for instance, to a Deligne--Mumford stable curve. We will see in this talk how to construct an analogue of the Abel map that ``nearly'' embeds a Gorenstein curve in a generalization of the Jacobian variety. This is joint work with Caporaso (Roma Tre) and Coelho (IMPA). [back to contents]

Oliver Labs, 12. March

Surfaces with solitary points

A solitary (ordinary double) point is a singularity which can be described locally by the equation $x^2+y^2+z^2=0$. In earlier papers we discussed the possible maximum number of complex ordinary double points or real ones with local equation $x^2+y^2+z^2=0$ on a surface of given degree $d$.
It turns out that the question on the maximum number of solitary points is very different to both of the others. In fact, it is related to the topology of real smooth surfaces. In the talk, we will give some theoretical upper bounds as well as lower bounds resulting from constructions. [back to contents]

Lars H. Halle, 19. March

Galois actions on models of curves and abelian varieties

Let A_K be an abelian variety defined over the fraction field K of a discrete valuation ring R. The Néron model A of A_K is a canonical extension of A_K to a smooth commutative group scheme over R. It is the 'best' possible such extension, and has important applications in Algebraic and Arithmetic Geometry.
Let K be complete, and let K'/K be a finite, separable and tamely ramified extension of fields. I will explain how one may then study A in terms of the Néron model A' of A_K', together with the induced action on A' by the Galois group G of the extension K'/K. This is particularly useful in the case where A' is semi-abelian. However, all tamely ramified extensions give interesting information about A, especially if one considers the whole tower of tame extensions of K. I will in particular discuss the case when A_K is the Jacobian of a smooth curve. [back to contents]

Stephanie Yang, 21. May

Linear systems of plane curves with multiple base points

In this talk I'll address the problem of computing the dimension of the space of plane curves of degree $d$ with multiple base points in general position. One can estimate this dimension using linear algebra. A conjecture of Harbourne-Hirschowtiz gives geometric meaning to when the actual dimension exceeds the expected dimension. I will describe this conjecture and show that it holds if the base points are of multiplicity 7 or less. [back to contents]

Cheryl Praeger, 24. May

Groups, probability, algorithms: computing with `giants'

Mathematicians often want to recognise, or compute with, groups that are too large even to list all of their elements. One solution is to examine a random sample of group elements and from this make inferences about the group structure. The lecture will present some of the issues that arise, and give examples in the context of recently developed algorithms. The lecture will be aimed at a wide audience. [back to contents]

George Hitching, 31. May

Vector bundle extensions and decoding of algebraic-geometric codes

We present a theorem of Trygve Johnsen (math.AG/9608018), showing that syndromes of error vectors of strongly algebraic-geometric codes can be interpreted as extensions of line bundles over curves, and that correctable errors correspond to extensions which are unstable vector bundles. We discuss a generalisation of this result to certain codes produced from scrolls of arbitrary rank. This is joint work with Johnsen (math.AG/0705.2817). [back to contents]