Department of Mathematics, University of Oslo
Seminar in Algebra and Algebraic Geometry, Autumn 2003

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Silke Lekaus, 15. September

Unipotent flat bundles and Higgs bundles over compact Kaehler manifolds

We characterize those unipotent representations of the fundamental group \pi_1(X,x) of a compact Kaehler manifold X, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of \pi_1(X,x) that can be realized as the monodromy of a flat connection on the holomorphically trivial bundle. [back to contents]

Ignacio de Gregorio, 22. September

An Introduction to Frobenius Manifolds

The notion of Frobenius Manifolds appears in two seemingly unrelated branches of mathematics: quantum cohomology and unfoldings of functions with isolated singularities. The very definition of Frobenius manifolds is intricate and involves a number of different objects. Among them one can highlight a multiplication of tangent vector fields and a flat connection coming from a non-degenerate bilinear pairing (metric), satisfying a number of compatibility conditions. In this talk we will introduce all the different objects and how they appear in each of the branches. This also leads to a rather general formulation of the notion of 'mirror objects' in Mirror Symmetry. [back to contents]

Alessandra Dragotto, 6. October

Locally Cohen-Macaulay surfaces in P^4

In this talk I'll present some partial results on degenerations of surfaces whose general fibre is a smooth projective algebraic surface and whose central fiber is a reduced, connected surface T in P^4. Furthermore T is assumed to be a locally Cohen-Macaulay surface whose irreducible components are a finite number of planes and a smooth surface.
A first result reads as follows
If T_1 a reducible locally Cohen-Macaulay surface in P^4 of degree d < 11 whose irreducible components are a plane and a smooth rational surface then T is a degeneration of a smooth surface T_1. In particular
(i) for d < 5, T_1 is a rational surface;
(ii) for 4< d <11, T_1 is a non special rational surface or a K3 surface.
During the seminar I'll try to explain the main techniques involved and I'll discuss an example in degree 6. [back to contents]

Carla Novelli, 13. October

Connections between the geometry of a projective variety and of an ample section

We compare the geometry of an ample section of a projective variety with the geometry of the variety itself by the way of Mori theory. In particular we compare the two Kleiman--Mori cones, that is the closure of the cones of effective $1$-cycles modulo numerical equivalence.
We apply this result to the case in which the section is a surface with Kodaira dimension $0$ or $1$, and to the case in which the section is a Fano manifold $($of high index$)$. [
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Alessandra Sarti, 14. October

Polyhedral groups and pencils of K3-surfaces with maximal Picard number

The talk is about a result of a joint work with Prof. W. Barth of the Univerity of Erlangen. I describe three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each pencil has Picard number $19$ and each pencil contains four surfaces with Picard number $20$. These surfaces are obtained as the minimal resolution of quotients $X/G$, where $G\subset SO(4,\mathbb{R})$ is some finite subgroup and $X\subset \mathbb{P}_3(\mathbb{C})$ denotes a $G$-invariant surface. The singularities of $X/G$ come from fix points of $G$ on $X$ or from singularities of $X$. In any case the singularities on $X/G$ are $A-D-E$ surface singularities. The rational curves which resolve them give almost all the generators of the Neron-Severi group of the minimal resolution. [back to contents]

Steven L. Kleiman, 20. October

Bounding the Tjurina number of a curve

Consider a plane curve of given degree. How singular can it be? In 1997, du Plessis and Wall found gaps in the list of possible Tjurina numbers. In fact, the possiblities lie in precise intervals indexed by another number, the least degree of a tangent polynomial vector field.
This talk will report on a generalization to curves in n-space, proved with Esteves, following a different approach. The Tjurina number must be replaced by the lambda-invariant of Buchweitz and Greuel, but the two invariants agree on plane curves and other local complete intersections. [back to contents]

Andreas Knutsen, 27. October

On the classification of threefolds through hyperplane sections

The classical way to study algebraic varieties is to introduce numerical invariants, study their possible values and then look at families of varieties with such invariants. However this method gets more and more difficult as the dimension grows, already for dimension $3$. Therefore, a lot of interest has recently been devoted to finding methods to ``recognize'' a variety from the geometry of its subvarieties (e.g. Mori theory).
Around the late eighties Zak and L'vovskii gave sufficient conditions for a projective variety $X \subset P^N$ so that it is not hyperplane section of another one, except a cone. More or less simoultaneously Wahl introduced the Gaussian/Wahl maps and these provided a very good tool to apply Zak's theorem when $X$ is a curve or when one knows $X$ and its curve section well, for example if it is a canonical curve general in moduli, like in the case of smooth Fano threefolds and more generally Mukai varieties. On the other hand these methods do not appear to be very effective when the curve secion of $X$ is not a "general" canonical curve (like in the cases of singular Fano threefolds).
In this talk I will present a new approach to the problem of classifying threefolds, still based on Gaussian/Wahl maps, that allows to apply Zak's theorem in a more general context, for example in the interesting case of surfaces with large Picard group. This is joint work with A. Lopez, L. Giraldo and R. Mu\~noz.
I show how to apply this to the specific (open) problem of classifying Enriques-Fano threefolds (that is a threefold whose general hyperplane section of is an Enriques surface), and to the case of threefolds whose hyperplane sections are pluricanonical embeddings of surfaces of general type. The method appears to be promising also in the study of singular Fano threefolds. [back to contents]

Johannes Kleppe, 3. November

Polynomials that split additively

First I will define precisely what I mean by an additive spilt of a homogeneous polynomial. As a basic example, a form f of degree d in k[x,y] splits additive if it equals x^d + y^d after a suitable base change. I will explain how to find any (regular) splitting that a given f has, and I will say something about degenerations of such f. If time permits, I will compute the resolution of the associated Artinian Gorenstein quotient R/ann f, the dimension of a 'splitting' subfamily of PGor(H) and the dimension of the tangent space at the point correspondig to f, assuming the corresponding information is known for the additive components of f. [back to contents]

Roy Mikael Skjelnes, 10. November

The good component of the Hilbert scheme

The Hilbert scheme $H[n]$ of $n$-points on a quasi-projective scheme $X$ is known to exist, but what is it? For smooth curves A. Grothendieck and P. Deligne showed that the Hilbert scheme is the $n$-fold symmetric product of $X$. When $X$ is the complex affine plane Mark Haiman showed that a certain blow-up of the $n$-fold symmetric product yields the Hilbert scheme $H[n]$.
In the talk I will present a joint work with Torsten Ekedahl where we generalize Haimans construction to arbitrary quasi-projective $X$.
Over the Hilbert scheme we have the universal family parameterizing closed subschemes that are flat, finite and of rank $n$. Let $U$ denote the open subscheme of the Hilbert scheme that parametrizes closed subschemes that are etale. The good component, or the principal part, $P[n]$ is the schematically closure of the open etale part $U$.
We show that the good component $P[n]$ is a certain blow-up of the $n$-fold symmetric quotient of $X$. The center of the blow-up we describe by giving generators of its defining ideal.
For smooth surfaces $X$ the good component equals the Hilbert scheme. Thus we get a blow-up construction for the Hilbert scheme of points on smooth surfaces, including a description of the universal family. [back to contents]

Trygve Johnsen, 24. November

Grassmann- og Schubertkoder

Det er velkjent hvordan en lager såkalte Goppa-koder med utgangspunkt i algebraisk-geometriske kurver. På lignende måter kan en produsere feilopprettende koder fra andre algebraisk-geometriske varieteter. Vi vil her ta for oss en konstruksjon basert på Grassmann-varieteter og Schubert-undervarieteter og oppsummere noen kjente resultater for slike. [back to contents]