Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

Actions of Cremona groups on CAT(0) cube complexes.

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In this talk, based on a common work with Christian Urech, we will construct such complexes on which Cremona groups of rank n act. We will then see which kind of results on these groups we can obtain.

]]>Prelog Chow rings and stable rationality in semistable degenerations

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Birational models of terminal sextic double solids

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Maximum Likelihood Estimation of Toric Fano Varieties

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Dual complexes of degenerations and Berkovich geometry

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Birational Kleinian groups and birational structures

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Graded Rings and Birational Geometry

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The main purpose of the talk is to give a short introduction to the theory and describe some of its applications.

]]>ℚ-homology planes satisfying the Negativity Conjecture

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On continuous automorphisms of Cremona groups

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Finite subgroups of tame polynomial automorphisms

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Mathematical Models of Cancer Evolution

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Irrationality of quotient varieties

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Birational transformations of tetragonal conic bundles

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In this talk, I will first discuss application of Sarkisov program to the rationality problem of algebraic varieties having conic bundle structures.

Then I concentrate on some special, so-called

The second part of the talk is based on the joint work in progress with V. Shokurov. ]]>

Real algebraic curves in real minimal del Pezzo surfaces

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The essential skeletons of pairs and the geometric P=W conjecture

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Symmetries of foliation: transverse action

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I will briefly recall a criterion for the finiteness of the transverse action in the case of algebraically integrable foliations (i.e. foliations whose leaves coincide with the fibres of a fibration). Then I will explain how the presence of certain transverse structures on the foliation allow to recover the same result; in this case, one can study the monodromy of such a structure (which is defined in an analogous way as that of a more familiar (G,X)-structure) and apply factorization results in order to reduce the problem to subvarieties of quotients of the product of unit discs, whose geometry is now quite well understood.

]]>Equations with complex analytic coefficients

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Almost homogeneous curves and surfaces

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In this talk, I will explain how to classify the pairs (X,G) where X is a curve or a surface and G is a smooth and connected algebraic group acting on X with a dense orbit.

For curves, I will mainly focus on the regular ones, defined over an arbitrary field. Over an algebraically closed field, the "natural" notion of non-singularity is "smoothness". However, over an arbitrary field, the weaker notion of "regularity" is more suitable. I will recall the difference between those two notions and show that there exist regular homogeneous curves which are not smooth.

For surfaces, I will restrict to the smooth ones, defined over an algebraically closed field. The situation is more complicated than for curves. Moreover, new phenomena and several difficulties appear in positive characteristic, and I will highlight them. ]]>

Hyperkähler manifolds and their Betti numbers

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Positivity, Graphs and Unknotting

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Around a big mapping class group

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Talks to present past and current work.

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Talks to present past and current work.

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Immanuel van Santen will speak about „Embeddings and tame automorphisms in affine geometry",

Anne Lonjou will explain some link between "Cremona group and geometric group theory", and

Jérémy Blanc will present us "Birational geometry of surfaces and threefolds".

]]>Talks to present past and current work.

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Julia will speak about "A_k-singularities of plane curves of fixed bidegree".

Egor will answer to "What transformation groups in algebraic, differential and metric geometry have in common?".

Philipp will speak about "Algebraic Statistics: Gaussian Mixtures and Beyond".

]]>Abelian quotients of the Cremona groups

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Automorphism groups of Danielewski surfaces (TALK CANCELLED)

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Complex plane curves, their intersection with round spheres, and knot concordance

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Noetherianity up to conjugation

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Isomorphisms between complements of unicuspidal curves in the projective plane

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Rational simple connectedness for Fano varieties

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In the current joint project with Laurent Gruson and Nicolas Perrin, we study some examples of Fano varieties in low dimension via explicit birational methods. ]]>

Some affine plane bundles over the punctured affine plane

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The famous Dolgachev-Weisfeiler conjecture predicts that such fibrations are in fact isomorphic to the trivial bundle. We will show that this holds true in some particular examples. For instance, we will recover a result of Drew Lewis which states that the $\mathbb{A}^2$-fibration induced by the second Vénéreau polynomial is trivial.

Our proof is inspired by a previous work of Kaliman and Zaidenberg and consists in first showing that the considered fibrations have a fiber bundle structure when restricted over the punctured affine plane.

This is a joint work in progress with Jérémy Blanc. ]]>

Automorphisms of pointless surfaces

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We will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces. ]]>

Non-rationality of conic-bundles over IP^3

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Birational geometry of pairs

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Signature morphisms of the Cremona group of the plane

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Fibre-like Fano manifolds: a bestiary

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Characterisation of varieties by their automorphisms

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Let X be a quasi-affine irreducible variety such that Aut(X) \simeq Aut(A

(1) X is a Q-acyclic open subset of a smooth affine rational variety, and dim(X) is a most equal to n;

(2) X is toric and dim(X) is at least equal to n. After giving a brief history on some related results that concern the characterisation of geometric objects via their automorphisms, we give the key ideas of the proof of our main result. ]]>

On birational transformations of P3 of low degree

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Commensurating actions of birational groups

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Small G-varieties

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An affine G-variety X is called

A striking consequence is the following result.

Theorem. Let n > 4. Then a smooth $\SL_n$-variety of dimension d < 2n-2 is an $\SL_n$-vector bundle over a smooth variety of dimension d-n. There are also interesting applications to actions of the affine group $\Aff_n$. This was the starting point of this joint work with with Andriy Regeta and Susanna Zimmermann. ]]>

Cremona group and hyperbolic spaces

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Del Pezzo fibrations in positive characteristic

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Some remarks on Poonen's conjecture

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We are interested in cycles of φ

Igusa quartic and and Wiman-Edge sextics

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Arnaud Beauville proved that all but four of these quartic threeffolds are irrational. Later Cheltsov and Shramov proved that the remaining threefolds in this pencil are rational. In this talk, I will give an alternative prove of both these results. To do this, I will describe Q-factorizations of the double cover of the four-dimensional projective space branched over the Igusa quartic, which is known as Coble fourfold. Using this, I will show that $\mathbb{S}_6$-symmetric quartic threefolds are birational to conic bundles over quintic del Pezzo surfaces whose degeneration curves are contained in the pencil studied by Wiman and Edge.

This is a joint work with Alexander Kuznetsov and Constantin Shramov (Moscow). ]]>

From noncommutative motivic measures to subgroups of the Cremona group

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I will introduce the Grothendieck ring of triangulated categories, and show how, using Bondal-Larsen-Lunts motivic measure, a subgroup of such ring will define a subgroup of the group Bir(X) of birational self-maps of X. A main example is given by the filtration via the motivic dimension, which induces a filtration on Bir(X). As a consequence, in the case X=P

Geometry of the moduli of parabolic bundles on elliptic curves

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The former moduli space is itself related to the moduli space of rank 2 parabolic bundles over a 5-punctured P

Betti numbers and pseudo effective cones in 2-Fano varieties

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Hyperbolicity of moduli spaces of abelian varieties

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Orthogonal tensor decomposition from an algebraic perspective

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(Joint work with Ada Boralevi, Emil Horobet, and Elina Robeva.) ]]>

The space of Kähler metrics on singular varieties

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This is based on a joint work with Vincent Guedj. ]]>

On projectivity of the moduli space of stable surfaces in characteristic p > 5

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A knot theorists approach to singularities and their deformations

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Algebraic aspects of hyperbolic volume

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The main part deals with similar features and questions about hyperbolic volume of (non-)arithmetic Coxeter orbifolds of higher dimensions. We discuss the volume problem for certain Coxeter pyramids. This family gives rise to interesting (non-)arithmetic reflection groups whose commensurability classification has recently been performed (joint work with R. Guglielmetti and M. Jacquemet).

This is a report about ongoing work. ]]>

On the number of minimal models of a smooth threefold of general type

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Moduli space of cubic surfaces and their anti canonical divisors

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Tropical compactifications, Mori Dream Spaces and Minkowski bases

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This is a work in progress with Elisa Postinghel. ]]>

Conjugacy classes of n-tuples in semi-simple Jordan algebras

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Regularization of Rational Group Actions - part 2

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Regularization of Rational Group Actions - part 1

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On the Wright's complex

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Jordan groups and birational automorphisms of algebraic varieties

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The existence of morphisms from a Calabi-Yau manifold

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Isomonodromic deformations of the five punctured sphere arising from complex plane quintics

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On a classical question about Chern numbers

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PSL(2,C), the exponential and some new free groups

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Torus equivariant K-stability

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A^1-contractibility of Koras-Russell threefolds

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Automorphism groups of affine algebraic surfaces preserving an A1-fibration

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In particular, we derive an example of a surface with infinite discrete automorphism group. ]]>

Automorphism groups of positive entropy of normal projective varieties

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A new approach to the classification of singular Fano-Mori 3-folds.

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Additive group actions on algebraic varieties

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In this talk we present some recent results about additive group actions on non-necessarily affine algebraic varieties that generalize the usual description of additive group actions on affine varieties via locally nilpotent derivations. In particular, we provide a characterization of additive group actions on a wide class of algebraic varieties in terms of a certain type of integrable vector fields. ]]>

Noncommutative invariant algebras and weighted homographic actions

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In the particular case of homography groups, we will give in this talk an overview of some results about this topic involving transvectants, modular forms and Rankin-Cohen brackets. ]]>

Cohomological action of automorphisms of compact Kaehler threefolds

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Quadratic maps with a periodic critical point of period 2

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Separating invariants and local cohomology

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(Joint with Jack Jeffries) ]]>

Universal Groebner bases and Cartwright-Sturmfels ideals

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These ideas will then be applied to the study of universal Groebner bases of ideals of minors, which admit such a grading. The main tools and concepts are Z^{m} gradings, multigraded Hilbert series, universal Groebner bases, initial and generic initial ideals. I will introduce and study two families of multigraded ideals with the property that any two ideals in the same family that have the same Hilbert series also have the same generic initial ideal. I will show that ideals of minors belong to one of these families, and derive some result about their universal Groebner bases.

Invariance of Plurigenera for foliations on surfaces

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Effective bounds on positive characteristic singular surfaces

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Real algebraic surfaces with many handles in (CP^1)^3

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Some properties of the group GL_2(C[X_1, . . . , X_m]) and some applications to the polynomial automorphism group Aut(A^n) of the affine space A^n

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On the fibres of Mori fibre spaces

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A symplectic version of the Chevalley restriction theorem for polar representations

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Family of exotic affine spheres

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On the geometry of matrix multiplication

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On Cremona Representations

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In this talk I will recall some properties of Cr

Birational equivalence of divisors and T-varieties

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Special reductive groups over an arbitrary field

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Affine n-Space is Characterized by its Automorphisms

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We also show that every torus appears as Aut(X) for a suitable affine variety X, but that Aut(X) cannot be isomorphic to a semisimple group. In fact, if Aut(X) is finitedimensional and if X \not\simeq

Concerning the structure of Aut(

Finally, we prove that every non-trivial homomorphism φ: SAut(

Explicit counterexamples to the Laurent Cancellation Problem

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Classical Invariant Theory and Category O

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Lifting real algebraic curves to real algebraic knots

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In this talk I will explain how to associate a hyperplane arrangement to a nodal planar rational real algebraic curve. This arrangement describes the space of nonsingular liftings and allows us to calculate the homology (and thus, in particular, the number of liftings up to rigid isotopy). We will show that, up to degree 5, this hyperplane arrangement is a rigid isotopy invariant (of planar curves) and can provide real algebraic analogues of the classical Reidemeister moves. Obstructions in the case of higher degrees will be discussed. The talk should be accessible to nonspecialists. ]]>

Group actions, entire curves and rational points

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This is discussed in connection with algebraic groups and prinicipal bundles. More precisely, let X be a projective manifold, G an algebraic group and E a G-principal bundle on X, all defined over some number field K. Then E admits a Zariski

dense set of L-rational points for some finite extension field iff X does. This corresponds to the homotopy lifting property whose complex analytic analogue allows to lift entire curves. ]]>

Geometric Lang-Vojta Conjecture in P^2

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In the (split) function field case the conjecture predicts (weak) algebraic hyperbolicity for log-general type varieties.When the completion of the variety is the projective plane the conjecture is known both if the divisor at infinity consits of four lines in general position (Brownawell-Masser and,independently, Voloch) and for a conic and two lines with five singular points (Corvaja and Zannier). With different methods Chen and Pacienza-Rousseau proved that the conjecture holds in the hyperbolic case, i.e. the complement of a very generic curve of degree at least 5.

In the talk, after an introduction to this fascinating subject, we will show how to prove the conjecture in general for the complement of a very generic curve of degree at least four.The proof relies on a deformation argument applied to a conic and two lines and on the theory of logarithmic stable maps as defined by Abramovich-Chen (and independently by Gross and Siebert) which extends usual stable maps to the logarithmic category (in the sense of Kato and Illusie). ]]>

Residual periodicity in arithmetic dynamics

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Generalisations of the tame automorphisms over a domain of positive characteristic

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Given a domain R of characteristic p >0, there exist two subgroups of the group GAn(R) of polynomial automorphisms which are natural generalisations of the linear group GL_{n}(R).

1) The subgroup of additive automorphisms, i.e. automorphisms with n components satisfying f(x+y) =f(x)+f(y) where x and y are set of n variables.

2) The subgroup of automorphisms with a Jacobian matrix in GL_{n}(R).

The subgroup generated by the translations and automorphisms of the type 1 (resp. 2) is called geometrically affine (resp. differentially affine). This group, together with the triangular automorphisms, generates the subgroup of geometrically tame (resp. differentially tame) automorphisms. We study these groups in dimension 2. We prove that they are different and endowed with a nice structure of amalgamated product.

]]>Double centralizers of unipotent elements in simple algebraic groups.

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On Cremona contractibility of plane curves

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This is a joint work in progress with Ciro Ciliberto (Univ. Roma "Tor Vergata"). ]]>

Holomorphically Equivalent Algebraic Embeddings

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Theorem. If f, g : X → **C**^{m} are algebraic embeddings and 2 dimX+ 1≤m, then there exists a holomorphic automorphism φ of **C**^{m} such that φ◦f=g.

In fact, the proof is based on an idea of Kaliman, with which he proved that two algebraic embeddings of **C** into **C**^{3} are holomorphically equivalent. In the course of this talk, we discuss this idea. Moreover, we provide examples of algebraic embeddings into **C**^{m} that are holomorphically non-equivalent.

Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus

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Parametrized automorphisms of manifolds with the density property

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Conjugacy classes of special automorphisms of the affine spaces

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Holomorphic Automorphisms of the Koras-Russell Cubic

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The density property for complex manifolds - a strong form of holomorphic flexibility

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On the Cancellation Problem for algebraic tori

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Moduli spaces of (G,h)-constellations

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Given a reductive group G acting on an affine scheme X, a Hilbert function h, and a stability condition θ, we explain how to construct the moduli space M of θ-stable (G,h)-constellations on X, which is a common generalization of the invariant Hilbert scheme after Alexeev and Brion and of the moduli space of θ-stable G-constellations for finite groups introduced by Craw and Ishii. The main tools for this construction are the geometric invariant theory and the invariant Quot schemes. Moreover, the moduli space M is naturally equipped with a morphism μ: M → X//G which turns to be a “nice” desingularization of the quotient X//G in many situations.

]]>Prime-order birational diffeomorphisms of the sphere

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Jordan properties of groups of birational automorphisms

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The Demazure roots of a spherical embedding

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Joint work with Alexander Perepechko. ]]>

Hyperbolic manifolds of small volume

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On orbits of the automorphism group on an affine toric variety

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In the second part we introduce the total coordinates on

Automorphisms of the Lie algebra of vector fields on affine n-space

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As an immediate consequence, we get the following result due to Kulikov. If every injective endomorphism of the Lie algebra VF(

Preperiodic points over global fields

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We consider dynamical systems on the projective line given by rational functions. Let $\phi$ be an endomorphism of the projective line defined over a global field K. We prove a bound for the cardinality of the set of K-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an S-unit equation theorem in 2 variables.

]]>Closed ind-subgroups with the same Lie algebras

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The tame automorphism group of affine 3-space is not closed

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The absolute continuous spectrum of skew products of compact Lie groups

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T

have purely absolutely continuous spectrum in the subspace associated to

Lie subalgebras of Vector Fields and Jacobian Conjecture

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(a) The Jacobian Conjecture holds in dimension 2;

(b) All Lie subalgebras

(c) All Lie subalgebras

Lifting automorphisms of adjoint representations

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We also showed that one can choose a biholomorphic lift Ψ such that Ψ(gv)=σ(g)Ψ(v), g ∈ G, v ∈ V, where σ is an automorphism of G. This leaves open the following questions: Can one lift holomorphic automorphisms of Z? Which automorphisms lift if V is not large? We answer the first question in the affirmative and also answer the second question. ]]>

Finiteness for Plücker varieties

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I will sketch a proof that any "bounded" Plücker variety is defined set-theoretically by equations of bounded degree, and in fact by finitely many equations up to symmetry. So far, this statement was unknown even for the first secant variety of the Grassmannian. The talk is based on joint work with Rob Eggermont, and inspired by Snowden's work on Delta-varieties. ]]>

Lie Algebra generated by LNDs on surfaces {xy=p(z)}

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Plane Cremona transformations of fixed degree

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Essential dimension of fibered categories

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Quasimorphisms and defect spaces

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Extensions of principal G_a-bundles over the punctured plane

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the punctured affine plane

Examples of rigid surfaces with infinitely transitive cylinders

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The Field of Definition of Point-Sets in P^1

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Geometry and Invariants of the Affine Group

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Transitivity of automorphism groups of Gizatullin surfaces

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**Conjecture (Gizatullin):** The big orbit of a Gizatullin surface *V* coincides with its smooth locus, i. e. O = *V*_{reg} .

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’s 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.

]]>The Lüroth problem and the Cremona group

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After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example, and its application to the study of finite simple groups of birational automorphisms of

Maximum-Likelihood duality for determinantal varieties

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(

with

In recent work by Hauenstein, Rodriguez, and Sturmfels the maximum-likelihood degree of determinantal varieties was studied. Extensive computations using numerical algebraic geometry led to the conjecture that the maximum-likelihood degree of the variety of rank-r matrices whose entries add up to 1 equals that of the variety of corank-(r-1) matrices whose entries add up to 1. I will present a proof of that conjecture, and variations of it for symmetric and skew-symmetric matrices. Joint work with Jose Rodriguez. ]]>

Real differential forms and currents in p-adic geometry

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In this talk, I will begin with a presentation of that of Berkovich. It roughly consists of 'adding plenty of points' to the usual

Then I will try to illustrate the following slogan: 'to see the good analog of a complex object in the *p*-adic world, one often has to work with Berkovich spaces', through three examples: spectral theory; dynamical systems; and the theory of real (p,q)-forms and related notions (integrals, boundary integrals, curvature forms of metrized line bundles) that as been recently developped in the framework of Berkovich spaces by Chambert-Loir and myself, and that I will try to describe in some detail.

Infinite transitivity on universal torsors

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On automorphisms of the affine Cremona group as an abstract group and as an ind-group

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In a next step we only consider those automorphisms of $\mathcal{G}_n$ that also respect its additional algebraic structure as an ind-group. It turns out that these are exactly the inner automorphisms of $\mathcal{G}_n$. To prove this I will follow an idea recently presented by Belov-Kanel and Yu, which uses tame approximation.

These are results from my Master's thesis under the supervision of Hanspeter Kraft.

]]>Two 45 minutes talks: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators" and "The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra"

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In 1968, Dixmier posed six problems for the algebra of polynomial differential operators, i.e. the Weyl algebra. In 1975, Joseph solved the third and sixth problems and, in 2005, I solved the fifth problem and gave a positive solution to the fourth problem but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra 'behaves' as a finite field extension. In more detail, the first problem/conjecture of Dixmier asks: is it true that an algebra endomorphism of the Weyl algebra is an automorphism? In 2010, I proved that this question has an affirmative answer for the algebra of polynomial integro-differential operators. In my talk, I will explain the main ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

**The group of automorphisms of the algebra ****of one-sided inverses of a polynomial ****algebra**

The algebra *S*_{n} of one-sided inverses of a polynomial algebra *P _{n}* in

The group of automorphisms

When is the blow-up of points or curves in the projective space a weak Fano threefold?

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I will describe the theorem, give the proof, and describe the generalisation to dimension 3, by considering blow-ups of points and curves in the projective space. We can get similar descriptions, the conditions of generality are now in terms of multisecant lines, conic or twisted cubics.

Joint work with Stéphane Lamy.

]]>Invariant Hilbert schemes and resolutions of quotient singularities

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In this talk, we will see that *H* is a smooth variety when the dimension of *V* is small, but that *H* is generally singular. When *H* is smooth, the Hilbert-Chow morphism *H* -> *X*//*G* is a canonical resolution of the singularities of the categorical quotient *X*//*G* (=Spec(*k*[*X*]^{G})). Then it is natural to ask what are the good geometric properties of this resolution (for instance if it is crepant).

To finish, we will mention some analogue results in the symplectic setting, that is to say by letting *p*=*q* and replacing *X* by the zero fiber of the moment map. The quotients that we get by doing this are isomorphic to the closures of nilpotent orbits, and the Hilbert-Chow morphism is a resolution of their singularities (sometimes a symplectic one).

Hyperbolic volume, estimates, and simplices

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Preperiodic points for rational maps

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Algebraic and geometric presentations of algebraic groups over local fields

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Well-behaved separating algebras

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Endomorphisms of Varieties (joint work with Rafael Andrist)

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Valuative analysis of the dynamics of superattracting germs in C^2

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I shall present a joint work with William Gignac, on the growth of attracting rates for iterates of a superattracting germ in C

If time allows, I will present other applications and examples. ]]>

On the Connectedness of Forcing Schemes

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The T-graph of the Hilbert scheme of points in the plane

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to an action of the torus on the Hilbert scheme of points in the plane.

The T-graph is the graph whose vertices are the fixed points under the

torus action (corresponding to monomial ideals), where two such vertices are connected by an edge if there exists a one-dimensional torus orbit whose closure contains the corresponding fixed points. I will describe some conditions for the existence of an edge between two given vertices.

This is joint work with Diane Maclaga.

]]>The finite subgroups in the plane Cremona group over field of complex numbers

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nonsingular del Pezzo surfaces and conic bundles.

The description by equations of nonsingular Del Pezzo surfaces in weighted projective spaces and description of their automorphism groups is a very old completed problem. The description of the ''minimal'' finite subgroups of automorphism groups of nonsingular del Pezzo surfaces was done by I.V. Dolgachev and V.A. Iskovskikh in 2007. I constructed a method that allows us to describe by equations in weighted projective spaces the nonsingular conic bundles with an action of ''minimal'' subgroup of automorphisms . I applied the method in case, when the ''minimal'' subgroup of automorphisms is nonsolvable.

]]>The representation type of a projective variety

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ACM sheaves also provide a criterium to determine the complexity of the underlying variety. More concretely, this complexity can be studied in terms of the dimension and number of families of indecomposable ACM sheaves that it supports, namely its \emph{representation type}. Along this line, a variety that admits only a finite number of indecomposable ACM sheaves (up to twist and isomorphism) is called of

On the other extreme of complexity, we would find the varieties of

Euler characteristic and Moebius inversion for finite categories

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This talk is based on joint work with Tom Leinster (Glasgow). ]]>

A generalized Newton's formula and Macdonald symmetric functions

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Integral closure and affine varieties with a torus action

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In this talk, we provide some examples whenever

Consider the group

We classify the

The problem of normalization for complexity zero case is well known (monomial or toric case). For the complexity one, the normalization of

Assume that

On h-vectors of matroids

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T.B.A.

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How primary decomposition of monoid congruences and binomial ideals is wrong

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Ideally one would want to carry out algebraic computations, such as primary decomposition of binomial ideals, entirely in this combinatorial language. We will present such a calculus, enabling one to compute by looking at pictures of monoids.

]]>Minimal representatives of even liaison classes

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In this talk we introduce a theoretical definition for 'minimal' representatives in any even linkage class. We show that these ideals

exist under reasonable assumptions on the linkage class, and, in general, if they exist they are essentially unique.

We then show that these ideals minimize homological invariants (e.g. Betti numbers, multiplicity, etc.) and they enjoy the best homological and local properties among all the ideals in their even linkage class. This justifies why they are, in some sense, the `best' possible ideals in the even linkage class.

We provide several classes of ideals that are the minimal representatives of their even linkage classes (including determinantal ideals) and, if time permits, show an easy application to produce more evidence towards the Buchsbaum-Eisenbud-Horrocks Conjecture. ]]>

Generalized Inversion Statistics

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On the annihilators of rational functions in the Lie algebra of derivations of k[x, y]

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Let us consider the Lie algebra W

of subalgebras in W

Let us consider natural action of the Lie algebra W_{2}(K) on the field of rational functions K(x,y). Recall that every derivation D

of W_{2}(K) of the ring K[x,y] can be uniquely extended to a derivation of the field K(x,y). It is natural to consider for a

fixed rational function u in K(x,y)\K the set A_{W_{2}}(u) of all derivations D of W_{2} such that D(u)=0. This set is called the annihilator

of u in W_{2}(K). It is a Lie subalgebra of W_{2}(K) and at the same time a K[x,y]-submodule of the K[x,y]-module W_{2}(K).

We show that it is a free K[x,y]-module of rank 1 and describe centralizers of elements and the maximal abelian subalgebras of the

Lie algebra A_{W_{2}}(u).

Holomorphic automorphisms of Danielewski surfaces

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In case of a Danielewski surface the so-called overshear group is dense in the group of holomorphic automorphisms. We describe the group structure of the overshear group with the help of Nevanlinna theory. ]]>

Hilbert depth and Stanley decomposition

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Stanley decompositions of multigraded modules M over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the

We test our new notion on the syzygy modules of the residue class field of K[X

Related ideals are the powers of the irrelevant maximal ideal. For them the (standard graded) Hilbert depth can be computed

precisely. ]]>

T.B.A.

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Irreducible characters of spin wreath products

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T.B.A.

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T.B.A.

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Infinite-dimensional non-positively curved symmetric spaces of finite rank

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We will look at some infinite dimensional symmetric spaces of non-positive curvature which have a remarkable property : they have finite rank. There exists a positive integer p such that any isometrically embedded Euclidean space has dimension at most p.

The talk will be focused on the properties of these spaces and some group actions which come from (non-unitary) infinite-dimensional representations.

]]>Vertex operator realizations of Jack symmetric functions

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Contents:

1. partitions and symmetric functions

2. Some history and questions about symmetric functions

3. Vertex operator realization of rectangular Jack functions

4. A special case of Stanley's conjecture and the realization of general Jack functions

5. Open questions ]]>

Singularities of moduli spaces of principal bundles on curves

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Finite separating sets and quasi-affine quotients

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Koszul algebras and their syzygies

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The main idea I will discuss is that the syzygies of Koszul algebras have some properties in common with the syzygies of algebras defined by monomials

of degree two. ]]>

Factorial algebraic group actions and categorical quotients

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We begin with a survey of known results concerning categorical quotients.

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we characterize existence of categorical quotient in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete, varieties with finitely generated Cox ring via lifting to the universal torsor. ]]>

Galois groups for inseparable field extensions

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T.B.A.

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Cohomology of line bundles on the cotangent bundle of a complete homogeneous space

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On the problem of connectedness for the Hilbert schemes of space curves

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It is an open question whether H

I will discuss the state of affairs about this question, and briefly describe work in progress (with the help of Macaulay 2) showing that curves of type (a,a+4) on a smooth quadric surface are in the connected component of extremal curves; this problem was raised in Hartshorne's papers "On the connectedness of the Hilbert scheme of curves in P

On some geometric properties of orbital varieties

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Torsion in the symmetric algebra and images of rational maps

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Their approach was first put on firm mathematical bases by David Cox and several collaborators. The key point in this approach is to control the torsion in the symmetric algebra. This is performed using a construction of Herzog, Simis and Vasconcelos, that gives information on the equations of Rees algebras.

]]>Upgrades and Downgrades of p-divisors

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LVMB manifolds and triangulations of spheres

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The talk will mainly deal with a generalization due to Bosio of the LVM manifolds, emphazing the combinatorial aspect of the LVM manifolds. These new manifolds are known as LVMB manifolds. In particular, our aim will be to show the very strong connection between LVMB manifolds, toric varieties and triangulations of spheres.

]]>Shelling, constructing, counting

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Separating invariants for the basic $G_a$-actions

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Parametrizations of Ideals in K[x,y]

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Gorenstein projections and diptych varieties

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Holomorphic foliations on CP2 and Geometric Invariant Theory

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Birational constructions of automorphisms of affine 3-folds

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Projective Varieties of Low Degree

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Normal singularities with torus actions

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Additive group actions in positive chracteristic

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Algebraic approximations of diffeomorphisms of surfaces

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On rigidity of low degree del Pezzo fibrations

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Using polynomial invariants to compute geometric predicates

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A tropical proof of the Brill-Noether theorem

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Quivers and saturation for tensor products multiplicities for classical groups

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Isomorphism classes of Gorenstein local rings via Macaulay's inverse system

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Beispiel eines Sl_2-Hilbertschemas

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Versal actions with a twist

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A birational characterization of affine varieties with trivial ML invariant

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Picard groups of homogeneous varieties

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Algebraic actions on the affine plane

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Limits of metabelian groups

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Rational points on cubic surfaces

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Was ist ein Faserbündel?

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Some examples coming from the study of the Danielewski hypersurfaces

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Essential dimension of algebraic tori

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Aspects of class field theory for global function fields

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Normal subgroup generated by a plane polynomial automorphism

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Gorestein Liaison and determinantal schemes

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Some aspects of A^1-bundles

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On the Bhatwadekar-Dutta-Berson-Vénéreau Polynomials

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An open problem on the coordinate ring of a variety with group action

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Spherical orbit closures in simple projective spaces and their normalization

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A sufficient criterion for the asymptotic stability of depths of ideal transformed Rees-modules

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