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]
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]
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]
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]
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]
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]