Department of Mathematics, University of Oslo
Seminar in Algebra and Algebraic Geometry, Spring 2004

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Christin Borge, 26. January

The Schur multiplier

It is now 100 years since Issai Schur introduced multipliers in order to study projective representations of groups (I. Schur: 'Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen', J. reine angew. Math. 127 (1904), 20-50).
To celebrate this, we look at some of the (equivalent) descriptions we have of the Schur multiplier. We will concentrate on the 'covering group-approach', which has been found useful in getting simple proofs of results on the multiplier.
In particular, we will look at covering groups of direct products, which lead to nilpotent products, an interesting construction... [back to contents]

Andreas Knutsen, 2. February

Exceptional curves

We first review the basic notions of Brill-Noether theory of curves, which is loosely the study of special line bundles on curves, i.e. line bundles $A$ with $h^1(A) >0$.
A central property of a curve $C$ is its gonality $gon C$, which is the smallest degree of a line bundle $A$ with $h^0(A)=2$, alternatively the smallest degree of a map from the curve to $\mathbf{P}^1$. This measures how different the curve is from $\mathbf{P}^1$ in a different way than its genus.
At the same time one has the notion of Clifford index $Cliff C$, which is the smallest value of $\deg A - 2(h^0(A)-1)$ taken over all line bundles $A$ with both $h^0(A) >1$ and $h^1(A) >1$.
One easily sees that $gon C \geq Cliff C+2$ and for the general curve one has equality. The curves for which equality does not hold are called exceptional and are conjectured to be extremely rare. Easy examples are smooth plane curves and intersections of two cubics in $\mathbf{P}^3$.
In the talk we present a conjecture of Eisenbud-Lange-Martens-Schreyer about exceptional curves and some basic results about them. [back to contents]

Jan Oddvar Kleppe, 9. February

Deformations of graded algebras. Parametrization and Hilbert schemes.

Let A be a graded algebra of a polynomial k-algebra R. We will look to tangent and obstruction spaces of deforming A as a graded/non-graded (flat) algebra and to some natural 'Hilbert scheme' GradAlg^H(R) parametrizing all graded quotients of R with a fixed Hilbert function H. The theory works nicely for closed subschemes of the projective n-space P^n as well (in which case the parameter scheme is Grothediecks usual Hilbert scheme with Hilbert polynomial p). We will prove a few results about the dimension and the smoothness of GradAlg^H(R) and Hilb^p(P^n) in the case A itself is a quotient of some other graded quotient B of R (i.e. in the case we have two surjections R->B and B->A) which turn out to have several applications. [back to contents]

Martin Gulbrandsen, 23. February

Low-dimensional moduli spaces of sheaves on an abelian surface

We give an introduction to the theory of moduli spaces of (stable) sheaves on projective varieties in general. Then we consider in more detail sheaves on an abelian surface. In particular we indicate how the moduli space of rank two sheaves may be compared with the Hilbert scheme of points. Using this we will show how to count the number (when finite) of rank two stable sheaves with fixed determinant and Euler characteristic. [back to contents]

George Hitching, 8. March

Moduli of symplectic vector bundles over algebraic curves

We give a brief introduction to symplectic vector bundles over a smooth curve X and describe the semistable boundary of the moduli space for the rank 4 case. We give a criterion for a certain extension of vector bundles to be symplectic.
Now suppose that X is of genus 2. We describe work in progress on the construction of (the finitely many) symplectic vector bundles of rank 4 which admit nonzero maps from every line bundle of degree 1 over X. [back to contents]

Silke Lekaus, 22. March

Tannakian categories associated to vector bundles

A Tannakian category is an abelian category with a tensor product which is isomorphic to the category of finite-dimensional representations of an affine group scheme. Based on the article 'On the representations of the fundamental group' by Madhav Nori we define Tannakian categories associated to vector bundles. We compute the corresponding group schemes in some cases. [back to contents]

Pål Hermunn Johansen, 29. March

The implicitization problem

Given a generally finite map p:P^{n-1}-->P^n, we want to write the closure of the image of p as Z(F). This is known as the implicitization problem. This can be solved in all generality by using Gröbner bases computations, but since these methods are terribly slow, algorithms based on resultants in more widely used in applications. This talk start with an introduction to these methods, but the main focus will be on an algorithm by Busé and Jouanolou using approximation complexes. [back to contents]

Eivind Eriksen, 19. April

Noncommutative deformations of sheaves of modules

Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F_p a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def_F: a_p -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result:
If X has a good A-affine open cover U, and F_i is a quasi-coherent left A-module for all i. Then Def_F has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F_i) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism.
In particular, if X is a scheme over k, and F_i is coherent for all i, then H^n(U,F_j,F_i) is isomorphic to Ext^n_A(F_j,F_i) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme. [back to contents]

Dan Isaksen, 24. May

Compositions of quadratic forms in characteristic p

A 'sums of squares' formula over a field F is a certain kind of polynomial identity over F; they are examples of composition formulas for quadratic forms. A classical theorem of Hopf and Stiefel says that if such a formula exists over a field of characteristic 0, then certain binomial coefficients must be even. I will explain what 'sums of squares' formulas are and why they are interesting. Then I will describe the Stiefel-Hopf theorem. Finally, I will show how Dan Dugger and I have used motivic cohomology to prove that the same result holds in characteristic p. [back to contents]