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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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