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...
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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.
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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.
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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.
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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.
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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.
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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.
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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:
Thm:
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.
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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.
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