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

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Abstracts:


Enrico Carlini, 27. January

Binary decomposition of polynomials

The Waring's problem for polynomials deals with the sums of powers presentation of a form f of degree d, i.e. an expression f=l_1^d+...+l_s^d where the l_i's are linear forms. In other terms, we are decomposing f as the sum of univariate forms, i.e. forms in 1 variable. It is interesting to generalize the problem looking for decompositions of forms as sums of binary forms, i.e. forms in 2 variables. Given a form of degree d, f in K[x_0,...,x_n], we look for expressions f=f_1+...+f_s where f_i in K[y_i,z_i] is a form of degree d and K[y_i,z_i] is a subalgebra of K[x_0,...,x_n]. Fixed n and d, it is interesting to know the minimal number of binary forms, namely s_min(n,d), needed for the decomposition of a generic form of degree d in n+1 variables. As in the sum of powers case, there is an expected value for s_min(n,d), namely s_exp(n,d), obtained by counting parameters. A natural question is whether s_min(n,d)=s_exp(n,d). We analyze the 3 variables case and, using apolarity, we are able to show the following Theorem: In the 3 variables case, s_min(d)=s_exp(d) iff d=1,2,3,4,5,6,8. [back to contents]

Matthieu Romagny, 3. February

Compact moduli for curves with level structure

There are deep relations between covers of algebraic curves, on one hand, and level structures on curves, on the other. Moduli spaces for both were used in order to prove the irreducibility of the moduli space of curves ${\cal M}_g$ (Hurwitz 1891, Deligne-Mumford 1969). Giving good (i.e. modular) compactifications for these moduli spaces encounters several well-known and well-studied problems. I will build such compactifications and discuss them; I will try to make it clear why algebraic stacks are the best-suited structures for these questions. [back to contents]

Arnfinn Laudal, 10. February

Noncommutative projective algebraic geometry

Noncommutative {\it algebraic} geometry is, in the litterature treated in many ways, as a direct copy of the Grothendieck Scheme Theory, as a purely categorical subject, or, as I propose, as an extension of the classical algebraic geometry, based on fields and 'valuations', as in R.Walker: Algebraic Curves, Dover, New York, 1962.
To me an object, in any geometry, should, if possible, be related to a geometric 'space' of 'points', that may be 'visualized'. The local properties of a point, corresponding to the valuation ring in the classical case, are contained in the noncommutative deformation functor of the point, and in its formal moduli, see e.g.:
Eivind Eriksen: An Introduction to Noncommutative Deformations of Modules. Warwick Preprint:2/2003.
The rest is, in the affine case, rather easy.
In this talk I shall look at the notion of non-commutative projective scheme, and show how it is related to Blowing ups of points in non-commutative affine schemes, and also to what is called MacKay-theory.
See:
O.A.Laudal: Non-commutative algebraic geometry. Preprint Series no.115, Max Planck Institute of Mathematics, (2000)
The next is a possibly more readable version of the last paper.
- http://www.math.uio.no/eprint/pure\_math/2002/21-02.html
And here is a paper on the modular isomorphism theorem for p-groups:
- http://www.math.uio.no/eprint/pure\_math/2002/19-02.html
I am currently optimistic in including all finite groups in the non-commutative 'algebraic geometry' that I propose.
See also,
-http://www.math.uio.no/eprint/pure\_math/2002/20-02.html [back to contents]

Christin Borge, 17. February

Solving the modular isomorphism problem

In November 2002, Arnfinn Laudal and I were able to show that (the isomorphism class of) a finite $p$-group $G$ is determined by its group algebra over the field of $p$ elements, hence solving the modular isomorphism problem.
In this talk I will present the proof, which combines methods from non-commutative geometry with group theoretic results. [back to contents]

Martin Gulbrandsen, 24. February

Towards a resolution of singularitites of the punctual Hilbert scheme of a surface

The punctual Hilbert scheme of a surface parametrizes subschemes of a given length with support at a given (nonsingular) point of a surface. It is singular in general, and we consider the partial resolution obtained as a certain component of the variety parametrizing complete flags of subschemes. We give a description of this variety, valid at least for length < 8, which shows that it is nonsingular for length < 5, but singular otherwise. [back to contents]

Kristian Ranestad, 3. March

Variety of polar simplices

This is a report on joint work with Frank-Olaf Schreyer on the Hilbert scheme compactification of the variety of presentations of a quadric as a sum of squares. We show that this compactification is singular when the quadric has rank n>5. [back to contents]

Eivind Eriksen, 24. March

Holonomic modules on some filtered rings and D-modules on quotient singularities

Let k be an algebraically closed field of characteristic 0, D an associative k-algebra with a positive, ascending, exhaustive filtration such that D^0=k and D^i has finite k-dimension for all i.
If gr D is commutative and Noetherian and D is a simple domain, we define the category of holonomic left D-modules and show that this category has good properties using Hilbert functions. In particular, we show that Bernstein's inequality holds, and that the category has finite length. This implies that the category can be studied using iterated extensions and noncommutative deformation theory.
We show that these results apply to some categories of D-modules on singular varieties, generalizing the smooth case. In particular, it applies to any quotient singularity of affine n-space by a finite group and many affine toric varieties. [back to contents]

Arvid Siqveland, 31. March

On The Structure of Non-commutative Regular Local Rings of Dimension Two

This talk is on a recent article by Michel Van Den Bergh and Martine Van Gastel where they conjecture that the center of a non-commutative complete regular local ring of global dimension two is a formal power series ring in two variables. They prove this conjecture in the special case of Ore extensions. [back to contents]

Rosa-Maria Miro-Roig

Complete Intersection Liaison and Gorenstein Liaison: New Results and Open Problems.

The purpose of my talk is to review some of the recent results on G-liaison confronting them with classical results in CI-liaison.
The idea of using complete intersections to link varieties has been used for a long time ago, going back at least to work of Noether, Severi and Macaulay. The development in the last four decades has been explosive. Many people has contributed to it and in codimension 2 case the picture is complete.
It is difficult to make a complete survey in a talk and I will make no attempt to do so. Instead I will try to convince the audience that G-liaison is a more natural approach if we want to carry out a program in higher codimension. [back to contents]