Conference BirPol 5
Algebraic Geometry in Dijon
17-20 September 2013
Université de Bourgogne ~ DIJON
Program
The Conference will be centered around the following three mini-courses
-
Automorphisms of T-varieties
by Alvaro LIENDO
University of Talca, Chile
Abstract
A T-variety is a normal complex variety endowed with a faithful action of
the algebraic torus T=(C*)n. The complexity of a
T-variety is the codimension of a generic orbit of the torus. The
best known examples of T-varieties are toric varieties, i.e.,
T-varieties of complexity 0, i.e., varieties endowed with a
T-action having an open (dense) orbit.
In the first part of this mini-couse, we present the well known
combinatorial description of toric varieties via some polyhedral
complexes called fans.
Based on this description, we present some results, originally
obtained by Demazure in the smooth case, describing the additive group
actions on a toric variety X that are normalized by the torus
action. In the case of a complete toric variety, this allows us to
describe the full automorphism group.
In the second part of this mini-course we present a description of
T-varieties of complexity 1 that generalizes the combinatorial
description of toric varieties. This description is given in terms of
a complete curve C, a finite set of points on C and a
polyhedral complex for each of these points.
Finally, we show some generalizations of Demazure results to the case
of T-varieties of complexity 1. If time allows, we will also present
a combinatorial description of T-varieties of arbitrary complexity
and we will give some perspectives on the possibility and difficulties
of extending these results beyond complexity 1.
-
Planar cuspidal curves and the logarithmic Minimal Model Program
by Karol PALKA
IMPAN Warsaw, Poland
Abstract
Consider ider a morphism of a complex projective line into a projective plane π: P1 → P2 which is 1-1 on closed points. The rational curve E=π(P1), which is homeomorphic to P1 in the Euclidean topology, is cuspidal, i.e. its singularities are locally analytically irreducible. All planar rational cuspidal curves arise this way. Classifying them up to a change of coordinates on P2 is an interesting and difficult problem, whose affine version has been solved by Abhyankar-Moh-Suzuki and Lin-Zaidenberg. Many conjectures have been made.
The classical Coolidge-Nagata conjecture predicts that E is Cremona-equivalent to a line. The rigidity conjecture of Flenner-Zaidenberg states that in a typical case, i.e. when P2\E is of general type, E is projectively rigid and has unobstructed deformations. Zaidenberg Finiteness Conjecture says that possible boundaries of snc-completions of P2\E satisfy a strong combinatorial finiteness conditions.
Recently, we obtained some progress approaching these problems by analysing the minimal resolution of singularities of (P2,E) in terms of the logarithmic Minimal Model Program applied in a suitable way. The lectures will be an introduction to the method used and in general to methods of the theory of non-complete algebraic surfaces from the log-MMP point of view.
Mini-course Notes
-
Kähler manifolds, harmonic maps and the Cremona group
by Pierre PY
IRMA Strasbourg, France
Abstract
We will describe some classical results (due to Donaldson, Corlette, Siu... among others) used to study the linear representations of fundamental groups of compact Kähler manifolds. In particular we
will explain how to associate a harmonic map to a linear representation and how the Bochner-Siu-Sampson formula allows to build holomorphic objects from harmonic ones. Finally, using these techniques we will explain how to describe (at least partially) homomorphisms from fundamental groups of Kähler manifolds to the Cremona group.
and a series of research talks/short presentations:
-
Dynamical degrees of birational transformations of surfaces
Jérémy BLANC
Universität Basel, Switzerland
Abstract
The dynamical degree λ(f) of a rational transformation f measures the exponential growth rate of
the degree of the formulae that define the n-th iterate of f. We study the set of all dynamical degrees
of all birational transformations of projective surfaces, and the relationship between the value of λ(f)
and the structure of the conjugacy class of f.
For instance, the set of all dynamical degrees of birational transformations of the complex projective
plane is a closed, well ordered set of algebraic numbers.
Joint work with Serge Cantat.
-
The Strong Factorial Conjecture
Éric EDO
Université de Nouméa, New Caldonia
Abstract
We present an unexpected link between
the Factorial Conjecture (A. van den Essen, D. Wright, W. Zhao)
and Rigidity Conjecture (J.-P. Furter).
The Factorial Conjecture in dimension m asserts that if a polynomial f in
m variables Xi over the complex numbers C is such that L(fk)=0 for all k ≥1, then f=0, where L is the C-linear map from
C[X1,...,Xm] to C defined by L(X1l1... Xmlm)=l1!...lm!. The Rigidity Conjecture asserts that a univariate polynomial map a(X) with complex coefficients of degree at most m+1 such that a(X)≡ X mod X2, is equal to X if m consecutive coefficients of the formal inverse (for the composition) of a(X) are zero. (Joint work with A. van den Essen)
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Transitivity of automorphism groups of Gizatullin surfaces
Sergei KOVALENKO
Ruhr-Universität Bochum, Germany
Abstract
Gizatullin surfaces are normal affine surfaces completable by a zigzag, i. e. by a linear chain of smooth rational curves. An equivalent characterization of such surfaces V , except for the surface C*xC*, is that the automorphism group acts with a big orbit O, i. e. V\O is finite.
Considering some examples of smooth Gizatullin surfaces like the affine plane A2, the Danielewski surfaces V={xy-P(z)=0} ⊆ A3 or the Danilov-Gizatullin surfaces Vk+1, being the complement of an ample section σ of self-intersection k+1 of a Hirzebruch surface Σr, it follows that the big orbit O coincides
with the smooth locus Vreg. Gizatullin formulated in his pioneer works the following conjecture: The big orbit of a smooth Gizatullin surface V coincides with V .
We show that the action of the automorphism group of a smooth Gizatullin surface with a distinguished and rigid extended divisor is not transitive in general. Thus such surfaces represent counterexamples to Gizatullin conjecture. For such surfaces we give an explicit orbit decomposition of the natural action of the automorphism group. Moreover, the automorphism group of such smooth Gizatullin surfaces can be represented as an amalgamated product of two automorphism subgroups.
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Tame automorphisms of affine 3-folds acting on 2-dimensional complexes
Stéphane LAMY
Université Toulouse III, France
Abstract
When considering transformation groups of rational surfaces, the most obvious cases being the group Aut(C2) of polynomial automorphisms of the plane, or the Cremona group Bir(P2) of birational selfmaps of the projective plane, an ubiquitous property seems to be the existence of spaces with non positive curvature on which these groups act, leading to results such as the Tits alternative or the non-simplicity.
I will present an ongoing project with C. Bisi and J.-P. Furter where we try to extend these results in higher dimension: precisely we construct a CAT(0) hyperbolic square complex on which the tame group of a 3-dimensional affine quadric acts, and we deduce from this construction various properties of the group (linearizability of finite subgroups, Tits alternative, non-existence of free abelian subgroups where all non-trivial elements have dynamical degree > 1...).
We also propose a similar construction for the case of Tame(C3), whose properties are still to be explored...
-
Cyclic covers of T-varieties
Charlie PETITJEAN
IMB Dijon, France
Abstract
A T-variety is a normal affine variety endowed of an effective action of an algebraic torus T.
By virtue of a result due to Altmann and Hausen, every such variety can be described in term of a so-called
proper-polyhedral divisor on a semi-projective variety which plays the role of a quotient of X by T.
In this talk we consider T-varieties which are further equipped with an action of a finite group
G commuting with that of T. We will explain the relation between the corresponding representations
for X and X/G. As an illustration, we determine the presentations of a family of C*-varieties
of complexity 2: Koras-Russell threefolds and some of their cyclic covers.
-
TBA
Bachar ALHAJJAR
IMB Dijon, France
Abstract
Comming soon ....
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Bir(P2) endowed with the euclidean topology is compactly presentable
Susanna ZIMMERMANN
Universität Basel, Switzerland
Abstract
By a recent result of Blanc-Furter, there exists a topology on Bir(P2) - the Euclidean topology - such that
the induced topology on any algebraic subgroup is the Euclidean topology on this subgroup.
Endowed with this topology, Bir(P2) is Hausdorff, and for every integer d ≥ 2,
Bir(P2) ≤ d is closed and locally compact. We show that, endowed with this topology,
Bir(P2) is compactly presentable, i.e. that there exists a compact subset K such that Bir(P2)
is generated by K with relations of bounded length. To show this, we first prove that that Bir(P2) is
boundedly generated by Aut(P2), Aut(F0) and Aut(F2) and then use that these groups
are compactly presentable.
Schedule
The conference will start on Tuesday September 17th at 13h00 and will end on Friday September 20st at noon.
| Tuesday |
Wednesday |
Thursday |
Friday |
| |
PALKA II
8:30-10:00
|
PY III
8:30-10:00
|
LIENDO III
9:00-10:30
|
PY II
10:30-12:00 |
PALKA III
10:30-12:00 |
EDO
10:45-11:45 |
| Lunch |
PY I
13:00-14:30
|
LIENDO I
14:00-15:30
|
LIENDO II
14:00-15:30
|
|
PALKA I
14:45-16:15
|
KOVALENKO
16:00-17:00
|
PETITJEAN
15:45-16:15
|
BLANC
16:45-17:45
|
ZIMMERMANN
17:00-17:30
|
LAMY
16:30-17:30
|
| |
AL HAJJAR
17:45-18:15
|
18:30 ...
Dinner |
The time table and the abstracts are also available as a
pdf file.
Practical Informations
Access
The workshop will take place at the
Mathematical Institute of Burgundy (IMB) in Dijon.
More information, including a map of the campus, on how to reach us can be found
here .
Accomodation
Participants who register after
August 26th are asked to arrange their accommodation is Dijon by themselves.
Here is a selection of convenient hotels in town:
Contacts
For more information, please contact:
Adrien DUBOULOZ
Lucy MOSER-JAUSLIN
Our sponsors
