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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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