Spring 2023
Date | Speaker | Title | Abstract |
---|---|---|---|
April 27 (Thursday) Alte Uni, Seminarraum -201 |
University of Jena |
On the automorphism group of an affine variety | I will discuss properties of the automorphism group of an affine variety. In particular I will discuss some fundamental differences between such a group and algebraic groups as well as groups of birational transformations. |
May 9 | Berlin / Neuchâtel |
An Algorithm for computing D-Weierstrass Places | Given a certain curve and a point P on this variety, we can look at the pole orders at P of functions defined on this curve. The set of all possible pole orders at P is surprisingly the same for all but finitely many points, namely Weierstrass points. My talk will be mainly based on an article by Florian Hess. His goal was to find an algorithm for computing Weierstrass points. More precisely, he obtained an algorithm for computing D-Weierstrass places. In my talk, I will start by introducing the concepts of algebraic geometry needed to understand the article. Then, I will define what D-Weierstrass places are and give the idea Florian Hess has for computing these places, that is using a link with linear systems. We will stop a while on this link which is a center point of his article and happens to have an error in its proof. Indeed, we have found a counter-example to one of the arguments used in his proof. We showed as well that his proof is correct if we added a new hypothesis. The proposition he states seems however correct in the general case as well, but no proof has been found up till now. Supposing the proposition is correct in the general case, I will then show how he uses this link to obtain an algorithm for computing D-Weierstrass places. |
May 30 | Paris-Saclay |
tba | tba |
June 13 | FRIAS - on leave from University of Warsaw |
Simpson's correspondence on singular varieties | I will talk about some generalizations of Simpson's correspondence to singular projective varieties, extending it to the positive characteristic case. This requires redefining Higgs sheaves on normal varieties and defining Chern classes of reflexive sheaves on normal varieties, which is another problem of an independent interest. |
June 30 (Friday) |
Brown University |
Curve classes on conic bundles threefolds and applications to rationality | In this talk I'll discuss joint work with Sarah Frei, Lena Ji, Soumya Sankar and Bianca Viray on the problem of determining when a geometrically rational variety is birational to projective space over its field of definition. Hassett--Tschinkel and Benoist--Wittenberg recently refined the classical intermediate Jacobian obstruction of Clemens--Griffiths by considering torsors under the intermediate Jacobian of a geometrically rational threefold. By work of Hassett--Tschinkel, Benoist--Wittenberg and Kuznetsov--Prokhorov, this obstruction is strong enough to characterize rationality of geometrically rational Fano threefolds of geometric Picard rank 1. Moving into higher Picard rank, we compute this obstruction for conic bundles over P^2. As a consequence of our work, when the ground field is the real numbers, we show that neither the topological obstruction nor the refined intermediate Jacobian obstruction is sufficient to determine rationality. |
Fall 2022
Date | Speaker | Title | Abstract |
---|---|---|---|
September 20 | Rennes |
Dynamical degrees of endomorphisms of complex affine surfaces are quadratic integers | Let X_{0} be a complex affine surface and let f be a dominant endomorphism of X_{0}. The dynamical degree λ_{1}
of f is defined as follows. Let X be a projective surface with an embedding ι: X_{0} ↪ X such that
ι(X_{0}) is an open dense subset of X. Take any ample divisor H on X and define λ_{1} := |
September 23 | |||
September 27 | Western Michigan University |
Actions of |
Let k be a field of characteristic zero. In their 2012 paper, Arzhantsev and Liendo write: " A regular Accordingly, we define a fundamental pair of locally nilpotent derivations of the affine k-domain B as a pair (D,U) which satisfies
Let k be an algebraically closed field of characteristic zero, and let X be a threefold such that Xₓ 𝔸_{k}^{1}\cong𝔸_{k}^{4}. If X admits a nontrivial action of |
October 11 | University College London |
Iitaka conjecture for anticanonical divisors in positive characteristic | Given a fibration f : X → Z, a natural question is how we can relate the Kodaira dimensions of X, Z, and the fibres. The Iitaka conjecture addresses this problem with an inequality. In characteristic 0, it is proven in many cases, while it is known to not hold in general in positive characteristics. Recently, a similar statement for the anticanonical divisors was proven in characteristic 0. The same results can be extended also to positive characteristics in low dimensions. |
October 18 | Bonn |
Automorphisms of del Pezzo surfaces in positive characteristic | One of the key ingredients in the classification of conjugacy classes of finite subgroups of the complex plane Cremona group is understanding the automorphism groups of conic bundles and del Pezzo surfaces. While automorphisms of del Pezzo surfaces of degree at least 5 behave uniformly in all characteristics, this is no longer true in smaller degrees. Nevertheless, a complete classification of automorphism groups of del Pezzo surfaces of degree 4 and 3 in positive characteristic has recently been obtained by Dolgachev and Duncan. In this talk, I will report on ongoing joint work with Dolgachev on the classification of automorphism groups of del Pezzo surfaces of the remaining degrees 2 and 1. |
November 1 | University of Basel |
p-elementary subgroups of the Cremona group | |
November 22 | Université Côte d'Azur |
The cone conjecture for certain Calabi-Yau manifolds obtained as fiber products | The cone conjecture is a long-lasting conjecture in birational geometry, and can be viewed as a conjectural counterpart to the cone theorem, for K-trivial varieties. It predicts for any K-trivial variety X the existence of a rational polyhedral fundamental domain for the action of the group Aut(X) on the cone Nef^{e}(X). If the conjecture is true for a certain X, it has important consequences, such as the finiteness of real structures on X, or the fact that the group Aut(X)/Aut^{0}(X) is finitely presented. In this talk, I present a construction inspired by the work of Schoen, Namikawa and Grassi-Morrison, of Calabi-Yau manifolds obtained as fiber products over the projective line, and prove the cone conjecture for these Calabi-Yau manifolds. This is joint work with Hsueh-Yung Lin and Long Wang. |
November 29 | University of Basel |
Galois Points and Cremona Transformations | We study the Galois points for any irreducible plane curve C subset from the two dimensional projective space of degree d (d greater than or equal 1) we prove that a Galois group of order at most 3 always can be extended to a group of de Jonquières Cremona transformation. We also show that there are Galois groups that can be extended to Cremona transformations but not de Jonquières maps, and there are Galois groups that cannot be extended to Cremona transformations. |
December 6 | University of Basel |
A study of dynamical degrees of polynomial automorphisms of the affine space via their action on the space of valuations | We are interested in computing the dynamical degree of polynomial automorphisms of ℂ^{n}. It seems that interesting data about the dynamics can be extracted from the action of automorphisms over the space of valuations. We will start by defining the space of valuations. More precisely, we will focus on a subspace of it with good geometric properties. One can see this space as generated by the action of tame polynomial automorphisms over the space of monomial valuations. Then, for a fixed polynomial endomorphism of ℂ^{n}, we will define a degree function as a generalisation of the standard degree, defined over the whole space of valuations. We will then focus on the dimension three and evoke our research. We will try to explain how good properties of our degree, in connection with these geometrical properties of our space could lead to get an algorithmic way to compute the dynamical degree of any (tame) polynomial endomorphism. |
December 20 | Université de Bourgogne (CNRS) |
Algebraic families of real forms | (Joint work in progress with Anna Bot) Real forms of a given real algebraic variety X, that is, real algebraic varieties whose complexifications are isomorphic to that of X, are classified after Borel-Serre in terms of the first Galois cohomology set of the group of automorphisms of the complexification of X. A result of T. Labinet asserting in particular that every complex algebraic group scheme locally of finite type has finite first Galois cohomology set implies that every real projective variety has at most countably many real forms. In contrast, some examples of affine and strictly quasi-projective smooth real varieties -with large non-algebraic automorphism groups- having uncountably many real forms have been recently constructed. These examples motivate the study of real forms from a moduli viewpoint, a typical question being whether forms of a given real variety can be organized in families depending algebraically on the real points of a suitable algebraic variety. In the talk I will propose and illustrate on examples different possible notions of "algebraic families of real forms" of a given real algebraic variety. Time permitting, I will also outline the steps of a construction for every real algebraic variety X of a natural complete family of real forms of X parametrized by a "space" whose real points are in natural bijection with the set of Galois 1-cocycles with values in the automorphism group of the complexification of X. |
Spring 2022
Date | Speaker | Title | Abstract |
---|---|---|---|
March 1 | HSE/École polytechnique |
Regularisations of pseudo-automorphisms | I will give a condition for a positive entropy pseudo-automorphism of a threefold such that if this condition is satisfied the pseudo-automorphism can not be regularized. I will recall the construction of a positive entropy pseudo-automorphism introduced by J. Blanc and prove that for a very general choice of parameters this pseudo-automorphism satisfies the condition; thus, it is non-regularizable. |
March 15 | EPFL |
Gluing theory for slc varieties | In the moduli theory of varieties, we are naturally led to consider a class of non-normal singularities, called semi-log canonical. This creates technical difficulties: for example, techniques from the MMP might fail on these non-normal varieties. Gluing theory was designed by KollÃ¡r to go back and forth between these singularities in characteristic zero and their normalization. I will report on my recent work on gluing theory in positive characteristic, and sketch some applications to birational geometry and moduli theory (e.g. abundance for surfaces, moduli of surfaces of general type in positive characteristic). |
March 29 | Saarbrücken |
Anticanonical geometry of the blow-up of ℙ^{4} in 8 points and its Fano model | Mukai realised the blow-up X of ℙ^{4} in 8 points as a moduli space of vector bundles on a degree-one del Pezzo surface S. With the same construction, Casagrande-Codogni-Fanelli associated to S a smooth Fano fourfold Y with remarkable geometric properties, and described explicitly the interplay between S, X and Y. Building on their work, we continue to explore the birational geometry of Y which is an important example of Fano fourfold. We will describe completely the base scheme of the anticanonical system of Y, and discuss the action of the Bertini involution ι_{Y} on Y induced by the Bertini involution on S. In particular, we will explain the relation between ι_{Y} and the anticanonical map of Y, and show that ι_{Y} preserves every divisor in the anticanonical system of Y. |
April 5 | Freiburg |
Counting points on projective curves | For a curve in projective space, the varieties parametrising its secant planes are among the most studied objects in classical algebraic geometry. We shall review some of their basic properties and reformulate this extrinsic geometry problem in terms of objects intrinsic to the geometry of the abstract curve, namely secant divisors to a given linear series. We consider in particular enumerative formulas counting the number of points in the intersection of two such secant varieties on a given curve and discuss their validity. We shall see that the surprising behaviour of the counting formulas arises from the complicated geometry of secant varieties, which are not always of expected dimension. |
April 12 | Orléans |
The structure of the groups of germs of diffeomorphisms and their formal completions. | In this talk I will give some properties of the group G_{n} of germs of diffeomorphisms of ℂ^{n}. For instance if H is a subgroup of G_{n}, then H is residually finite, but also Hopfian. Furthermore, if n=1, then G_{1} is Hopfian, and the automorphism group of G_{1} can be described. This is a joint work with Dominique Cerveau. |
April 26 | Bordeaux |
A question on the real locus of smooth Fano threefolds |
Smooth complex Fano threefolds have been fully classified by Iskovskikh and Mori-Mukai and their description is well understood and rather explicit. In this context it is natural to study the geometry of real smooth Fano threefolds and to investigate the connections between their algebraic properties and topological properties of their real loci. In this talk I will survey some known results and present a joint work in progress with Frédéric Mangolte, in which we investigate the connectedness of the real locus for smooth Fano threefolds. |
May 3 | Leibniz Universität Hannover |
On morphisms between connected commutative algebraic groups | Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. We construct a natural retraction from the set of pointed morphisms of algebraic varieties from (G,0) to (H,0) to the set of homomorphisms of algebraic groups from G to H (for arbitrary G and H) which commutes with the composition and addition of morphisms, unique with this property. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. This is false in positive characteristic or if K = ℂ and G and H are isomorphic only as complex-analytic varieties. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. |
May 10 | Bonn |
Symplectic birational invollutions on hyperkähler manifolds of K3 |
Hyperkähler manifolds (simply connected compact Kähler manifolds admitting a holomorphic symplectic form, HK for short) are higher dimensional analogues of K3 surfaces. These manifolds are interesting from several point of view (dynamical, arithmetical, geometric). In a joint work with Y. Dutta and Y. Prieto Montañez, we study finite order birational automorphism of HK manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. We prove that the existence of certain symplectic birational selfmaps implies that the HK manifold is birational to a moduli space of (twisted) sheaves on a K3 surface. Moreover, by means of stability conditions, we study the existence of symplectic birational involutions on such manifolds. |
May 17 | IMJ-PRG |
The Kodaira dimension of moduli spaces of irreducible holomorphic symplectic varieties | Irreducible holomorphic symplectic (IHS) varieties are one of the three building blocks of varieties with trivial canonical bundle; they can be seen as a generalisation in higher dimension of K3 surfaces. In 2007, Gritsenko, Hulek and Sankaran proved that the moduli space of K3 surfaces of degree 2d is of general type when d>61: their strategy is to reduce the question to the existence of a certain cusp form for an orthogonal modular variety. In this talk, after a short introduction to IHS varieties, we give some motivations for the study of the Kodaira dimension of moduli spaces of IHS varieties, we sketch the reduction argument by Gritsenko Hulek and Sankaran and we explain how we used this method to prove general type results for some moduli spaces of IHS varieties in higher dimension. We also explain what the challenges are when we try to imitate their approach. Finally, we provide an infinite list of unirational moduli spaces (few of them are known). This is joint work in progress with I. Barros, E. Brakkee and L. Flapan. |
May 24 | TUM |
Del Pezzo surfaces with global vector fields | If X is a del Pezzo surface (or a weak del Pezzo, or an RDP del Pezzo), then its automorphism scheme Aut_X is a, possibly non-reduced, affine group scheme of finite type. In particular, X has infinitely many automorphisms if and only if Aut_X is positive-dimensional and then X admits global vector fields (since the space of global vector fields on X is the tangent space to the automorphism scheme). The last implication is an equivalence in characteristic 0, but its converse can fail in positive characteristic. Over the complex numbers, a del Pezzo surface with rational double point singularities admits global vector fields if and only if its minimal resolution, the corresponding weak del Pezzo surface, does. In small characteristics, one implication of this equivalence breaks down due to the existence of non-lifting vector fields on rational double points. I will explain how to overcome these obstacles in order to classify weak and RDP (if p \neq 2) del Pezzo surfaces with global vector fields. Further, I will show examples displaying interesting behaviour of such surfaces in small characteristics. This is joint work with Gebhard Martin. |
May 31 | Toulouse |
Group of birational transformations of Severi-Brauer surfaces | A Severi-Brauer surface S over a field k is a surface that is isomorphic to the projective plane over the algebraic closure. It is interesting to see how its group of birational transformations differs from the Cremona group. I give the construction of a group homomorphism from Bir(S) to ℤ, implying that Bir(S) is not generated by elements of finite order. This is joint work with J. Blanc and E. Yasinski. |
June 23 |
Fall 2021
Date | Speaker | Title | Abstract |
---|---|---|---|
October 5 | (Geneva) |
Tropical Geometry and Refined Invariants | The talk is devoted to several real and tropical enumerative problems. In this talk, we introduce a refined signed count for oriented real rational curves inside toric surfaces, which pass through some points located on the toric boundary. In some situations, the result does not depend on the choice of the points provided this choice is generic, thus leading to "refined invariant". The computation of these invariants can be carried out using the tropical geometry approach. Furthermore, they provide an interpretation to some tropical refined invariants introduced by Block-Göttsche. |
October 12 | (Saarbrücken) |
Termination of flips for pseudo-effective 4-folds | The Minimal Model Program (MMP) plays a fundamental role in the classification theory of projective varieties. One of the main open problems of the MMP in higher dimensions is the termination of flips conjecture, which predicts that any sequence of flips terminates. In this talk I will sketch the proof of the termination of flips conjecture for pseudo-effective 4-folds. This is joint work with Guodu Chen. |
November 2 | (Leiden) |
The moduli space of rational elliptic surfaces of index two | Elliptic surfaces are ubiquitous in Mathematics. Examples include Enriques surfaces, Dolgachev surfaces, and all surfaces of Kodaira dimension one. In this talk we will focus on those elliptic surfaces which are rational and that have exactly one multiple fiber of multiplicity two. These are called rational elliptic surfaces of index two. Our goal will be to describe how to construct their moduli space when the choice of a bisection is part of the classification problem. This is based on work in progress joint with Rick Miranda. |
November 9 | (EPFL) |
On the boundedness of elliptic Calabi-Yau threefolds | In this talk, we will discuss the boundedness of Calabi-Yau threefold admitting an elliptic fibration. First, we will survey some ideas to address weak forms of boundedness of varieties admitting an elliptic fibration. Then, we will switch focus to Calabi-Yau varieties and discuss how the Kawamata-Morrison cone conjecture comes in the picture when studying boundedness properties for this class of varieties. To conclude, we will see how this circle of ideas applies to the case of elliptic Calabi-Yau threefolds. This talk is based on work joint with C.D. Hacon and R. Svaldi. |
November 16 | (Angers) |
Algebraic subgroups of the plane Cremona group over a perfect field | There are many algebraic groups acting birationally on an algebraic surface, notably many finite groups, and we want to classify them up to conjugation. This is very difficult in general, and I will focus on infinite algebraic groups acting on a rational surface over an arbitrary perfect field. It turns out that they are always contained in a maximal algebraic group acting birationally on the plane and that the list of the maximal ones up to conjugation is rather short (for a given value of short). In this talk I would like to explain this classification and motivate how it is proven. |
November 23 | (Neuchâtel) |
Ideals with a radical generic initial ideal | The choice of a term order allows us to associate to any ideal I a monomial ideal, called the initial ideal of I. The initial ideal of I depends not only on the choice of a term order, but also on the system of coordinates. Nevertheless, many properties of I can be inferred from those of its initial ideal(s). For a given term order and in a generic coordinate system, however, the initial ideal of I is always the same and it is then called the generic initial ideal of I. In this talk, I will introduce a family of ideals whose generic initial ideal is independent of the choice of both the term order and of the system of coordinates. These are the multigraded homogeneous ideals which have a radical generic initial ideal. Geometrically, they correspond to varieties in a product of projective spaces and the process of passing to the initial ideals is a flat deformation - often called a Groebner deformation - to a union of linear varieties. Multigraded ideals which have a radical generic initial ideal show interesting rigidity properties, which e.g. allow us to deduce information on their universal Groebner bases. In the talk, I will present examples of ideals which belong to this class and of what we can deduce about them using this machinery. The original work in the talk is joint with Aldo Conca and Emanuela De Negri. |
November 29-30 | |||
December 7 | (Bordeaux) |
Borel subgroups of the Cremona group | Following Demazure and Serre, the Cremona group is endowed with a natural topology. Borel subgroups are then defined as the maximal closed connected solvable subgroups. In this talk, we describe all Borel subgroups of the complex plane Cremona group. It turns out that their rank may be equal to 0,1 or 2 (the rank of a closed subgroup of the Cremona group being defined as the maximal dimension of a torus contained in it). In principle, this fact answers a question of Popov. We also show that all Borel subgroups of rank 1 or 2 are conjugate, but that this result no longer holds for Borel subgroups of rank 0. This is a joint work with I. Hedén. |
Spring 2021
Date | Speaker | Title | Abstract |
---|---|---|---|
May 4 | (Dijon - staying in Basel for 3 months) |
Real forms of nilpotent orbit closures in a complex semisimple Lie algebra. | Let G be a complex semisimple algebraic group acting on its Lie algebra g via the adjoint action, and let X be a nilpotent orbit closure in g . In this talk, we will study the real forms of X , i.e. the real algebraic varieties Y endowed with an action of a real algebraic group F such that F _{C} is isomorphic to G, as complex algebraic groups, and Y_{C} is isomorphic to X , as complex G-varieties. This is a joint work with Michael Bulois and Lucy Moser-Jauslin. |
May 11 | (Basel) |
Real forms on rational surfaces | A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Though for many families of complex projective algebraic varieties, it was possible to determine if the number of isomorphism classes over ℝ of real forms is finite, only recently Lesieutre [Les18] exhibited a complex projective variety with infinitely many nonisomorphic real forms. This result was then extended by Dinh, Oguiso and Yu [DOY20] to any complex projective rational varieties of dimension greater or equal to three. However, the case of a smooth complex projective rational surface still remains open. In this talk, we will present the content of [Bot21], in which for any positive integer r, we construct a smooth complex projective rational surface which has at least r real forms not isomorphic over ℝ. |
May 18 | (Basel) |
Sarkisov Links with centres space curves on smooth cubic surfaces | The Sarkisov Program is a central tool in the study of birational relations between Mori fibre spaces. It essentially decomposes birational maps into much simpler ones, called Sarkisov links.
In this talk, after explaining the basics of the Sarkisov Program, I will go over the construction and classification of a big class of Sarkisov links starting from P |
May 25 | (Basel) |
Embeddings of smooth affine varieties into algebraic groups | This is joint work with Peter Feller. In any category there are the following fundamental problems concerning embeddings from an object Z into another object X: 1. (Existence) Does there exists an embedding of Z into X? 2. (Uniqueness) Having two embeddings f, g of Z into X, does there exists an automorphism ψ of X such that g = ψ ∘ f? In this talk, we will mainly focus on the first problem in the category of affine varieties, where Z is smooth and X is an algebraic group. Amongst other things, we will discuss the following result. Theorem. For every simple algebraic group G and every smooth affine variety Z with dim G > 2dim Z + 1, there exists an embedding of Z into G. The proof is based upon parametric transversality results for flexible affine varieties due to Kaliman. We will also discuss the following result, which implies the optimality of the above existence result up to a possible improvement of the dimension bound by one. Its proof is an adaptation of a Chow-group-based argument due to Bloch, Murthy, and Szpiro for the affine space. Proposition. For every non-finite algebraic group G and every d ≤ dim G / 2, there exists an irreducible smooth affine variety of dimension d that does not admit an embedding into G. |
June 1st | (Basel) |
Algebra and geometry of conic bundles over non-closed fields | In this talk, I will focus on 2-dimensional conic bundles. Over complex numbers, their birational geometry is well understood and their "arithmetic" is basically trivial. However, conic bundles over non-closed fields (or/and fields of positive characteristic) provide a rich source of many interesting problems in algebra and geometry. I will discuss their properties in the simplest case of the field of real numbers, give some applications to Cremona groups and make connections with the classical theory of quadratic forms due to Witt and Milnor. |
June 8 Talk via ZOOM |
(Basel) |
Affine n-space is determined by its automorphisms | In recent years some research was done about the question whether an affine variety X is determined by the group of regular automorphism Aut(X). Clearly, this cannot be the case in general, because there are many examples where Aut(X) is trivial. However, if the automorphism group is large, i.e. infinite dimensional as an ind-group, then there is a certain chance. For affine n-space the problem was recently solved in a joint work of Serge Cantat, Andriy Regeta and Junyi Xie. Their solution is based on the following fundamental result. Theorem: Let X be an affine variety and let G be a closed connected and commutative subgroup of Aut(X). Then G is a union of commutative algebraic subgroups. Once we have this, the result in the title follows rather easily along some lines which appeared already in earlier work of Immanuel van Santen, Andriy Regeta and myself. I will first give the history of the problem and explain some recent results. Then I have to introduce and explain the basic concepts of ind-varieties and ind-groups, and finally I will give a proof of the theorem and the application to affine n-space. |
June 22 (in Alte Universität) |
(Dijon) |
Real Torus Actions on Real Affine Algebraic Varieties | |
June 29 (in 05.001) |
(Basel) |
Birational geometry of sextic double solids with a compound A_{n} singularity | Sextic double solids, double covers of ℙ^{3} branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are ℚ-factorial with ordinary double points, are known to be birationally rigid. In this talk, we discuss birational geometry of sextic double solids with an isolated compound A_{n} singularity. I have shown that n is at most 8, and that rigidity fails for n > 3. |
Fall 2020
Unfortunately, there were no talk, because of covid-19.Date | Speaker | Title | Abstract |
---|---|---|---|
November 20 |
Spring 2020
We only had one talk, and all others were then unfortunately cancelled, because of covid-19.Date | Speaker | Title | Abstract |
---|---|---|---|
February 25 | (Basel) |
Actions of Cremona groups on CAT(0) cube complexes. | A key tool to study the plane Cremona group is its action on a hyperbolic space. Saddly, in higher rank such an action is not available. Recently in geometric group theory, actions on CAT(0) cube complexes turned out to be a powerfull tool to study a large class of groups. 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. |
Fall 2019
Date | Speaker | Title | Abstract |
---|---|---|---|
September 17 | (ETHZ) |
Automorphisms of rational surfaces | The starting point of this talk is a finite number of distinct points lying on an irreducible cubic curve in ℙ^{2} which we blow up to obtain a rational surface S. What conditions must the points satisfy such that a fixed element of the Weyl group associated to Pic(S) is realised as an automorphism of the surface S? I will discuss the conditions I worked on in my master thesis, present the relevant notions and results, and speak about the potential directions which could be further pursued. |
September 23-27 | |||
October 8 | (EPFL) |
On continuous automorphisms of Cremona groups | Julie Déserti showed that every automorphism of the plane Cremona group is inner up to a field automorphism of the base-field. In this talk we generalize this result to Cremona groups of arbitrary rank, however, only under the additional restriction that the automorphisms are also homeomorphisms with respect to the Zariski or the Euclidean topology on the Cremona group. We will consider similar questions for groups of polynomial automorphisms and groups of birational diffeomorphisms. This is joint work with Susanna Zimmermann. |
October 15 | (Bern) |
ℚ-homology planes satisfying the Negativity Conjecture | A smooth complex normal algebraic surface S is a ℚ-homology plane if H_{i}(S,ℚ)=0 for i>0. This holds for example if S is a complement of a rational cuspidal curve in ℙ^{2}. The Negativity Conjecture of K. Palka asserts that for a smooth completion (X,D) of S, κ(K_{X}+½D)=-∞, so the minimal model of (X,½D) is a log Mori fiber space. Assume that S is of log general type, otherwise the geometry is well understood. It turns out that, as expected by tom Dieck and Petrie, all such S can be arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of lines and conics on ℙ^{2}. As a consequence, they all satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and all their automorphism groups are subgroups of S_{3}. To illustrate this surprising rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture) inductively, by iterating quadratic Cremona maps. |
October 22 | no talk |
||
October 29 | (Ioannina) |
Graded Rings and Birational Geometry | Un projection theory, initiated by Miles Reid, aims to construct and analyze complicated commutative rings in terms of simpler ones. It can also be considered as an algebraic language for birational geometry. The main purpose of the talk is to give a short introduction to the theory and describe some of its applications. |
November 5 | (Rennes) |
Birational Kleinian groups and birational structures | Let Y be a smooth complex projective surface. Let U be a connected Euclidean open set of Y. Let G be a subgroup of Bir(Y) which acts by holomorphic diffeomorphisms on U (i.e. preserves U and without indeterminacy points in U), in a free, properly discontinuous and cocompact way, so that the quotient X=U/G is a compact complex surface. Such a birational transformation group G, or more precisely such a quadruple (Y,U,G,X), will be called a birational Kleinian group. Once we have a birational Kleinian group, the quotient surface is equipped with a birational structure, i.e. an atlas of local charts with rational changes of coordinates. I will present some basic properties and subtleties of birational structures, compared to the classical geometric structures. Then I will begin by studying birational structures on a special type of non-algebraic surfaces, the Inoue surfaces, to reveal some of the general strategy. Using classification of solvable and abelian groups of the Cremona group, and by relating the foliations on Inoue surfaces with some birational dynamical systems via Ahlfors-Nevanlinna currents, I will show that the Inoue surfaces have one unique birational structure. Then I will move on to the general study of birational Kleinian groups with the additional hypothesis that the quotient surface is projective. I will explain how to use powerful results from Cremona groups, holomorphic foliations and non-abelian Hodge theory to get an almost complete classification of such birational Kleinian groups. |
November 12 | (Imperial College) |
Dual complexes of degenerations and Berkovich geometry | To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety, hence the dual complex can be embedded in the Berkovich space of the variety. In this talk I will explain how this interpretation gives an insight in the study of the dual complexes. I will focus on some degenerations of hyper-KÃ¤hler varieties and show that we are able to determine the homeomorphism type of their dual complex using techniques of Berkovich geometry. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-KÃ¤hler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown. |
November 18-20 | |||
November 26 | (Munich) |
Maximum Likelihood Estimation of Toric Fano Varieties | We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Based on joint work with Dimitra Kosta and Kaie Kubjas. |
December 3 | (Loughbourough) |
Birational models of terminal sextic double solids | It is known that quasismooth 3-fold Fano hypersurfaces with index 1 in weighted projective spaces over â„‚ are birationally rigid (not birational to any other Fano 3-folds, conic bundles or del Pezzo fibrations). But very little is known when they carry non-orbifold singularities. I consider sextic double solids, one of the simplest such 3-folds, which have an isolated cA_n singularity. I have shown that n is at most 8, and that rigidity fails for n > 3. In this talk, I will illustrate this phenomenon by giving some examples. |
December 9 (15h ) Spiegelgasse 5, 05.002 |
(Warwick) |
Prelog Chow rings and stable rationality in semistable degenerations | In this joint work with Christian Böhning and Hans-Christian von Bothmer we apply Voisin's criterion of existence of a decomposition of the diagonal to semistable degenerations. In doing so, we obtain partial results towards proving that very general cubic threefolds are stably irrational. |
Spring 2019
Date | Speaker | Title | Abstract |
---|---|---|---|
March 5 | (Université de Marseille) |
Symmetries of foliation: transverse action | We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserve F (i.e. it permutes the set of leaves). We say that the transverse action of G is finite if some finite index subgroup of G fixes each leaf of F. 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. |
March 19 | (Imperial college) |
The essential skeletons of pairs and the geometric P=W conjecture | The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems. |
April 16 | (École Polytechnique and Université de Nantes) |
Real algebraic curves in real minimal del Pezzo surfaces | The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in the real projective plane is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in the real projective space, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the real minimal surfaces. In this talk, we present some results about the classification of topological types realized by real algebraic curves in real minimal del Pezzo surfaces of degree. |
May 7 | (Steklov Mathematical Institute, Moscow) |
Birational transformations of tetragonal conic bundles | A conic bundle f: X\to S is a flat morphism of smooth varieties whose fibers are plane conics. 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 tetragonal types of conic bundles, which lie on the "boundary" between birationally rigid and non-rigid ones and are especially interesting for this reason. The second part of the talk is based on the joint work in progress with V. Shokurov. |
May 14 | (University of the Basque Country) |
Irrationality of quotient varieties | The rationality problem in algebraic geometry asks whether a given variety is birational to a projective space. We will gently introduce the problem and take a look at some recent advances, principally in the direction of negative examples constructed via cohomological obstructions. Special focus will be set on quotient varieties by linear group actions. |
May 21 | (Dept. of Biosystems Science and Engineering, ETHZ) |
Mathematical Models of Cancer Evolution | |
May 27-29 | |||
June 4-5 |
Fall 2018
Date | Speaker | Title | Abstract |
---|---|---|---|
September 18 11.00-11.30 |
(Basel) |
Vector/projective bundles over curves | Talk to present past and current work. |
September 18 11.30-12.00 |
(Basel) |
Birational Geometry of Algebraic Surfaces | Talk to present past and current work. |
September 25 10.30-11.00 |
(Basel) |
A_k-singularities of plane curves of fixed bidegree | Talk to present past and current work. |
September 25 11.00-11.30 |
(Basel) |
What transformation groups in algebraic, differential and metric geometry have in common? | Talk to present past and current work. |
September 25 11.30-12.00 |
(Basel) |
Algebraic Statistics: Gaussian Mixtures and Beyond | Talk to present past and current work. |
October 2 10.30-11.00 |
(Basel) |
Embeddings and tame automorphisms in affine geometry | Talk to present past and current work. |
October 2 11.00-11.30 |
(Basel) |
Cremona group and geometric group theory | Talk to present past and current work. |
October 2 11.30-12.00 |
(Basel) |
Birational geometry of surfaces and threefolds | Talk to present past and current work. |
October 10-11 |
(Conference in Dijon) |
||
October 16 10.30-11.15 |
Talk to present past and current work. | ||
October 16 11.15-12.00 |
(Basel) |
Talk to present past and current work. | |
October 30 | (conference in Loughbourough) |
||
November 6 | (Rennes) |
Around a big mapping class group | The plane minus a Cantor set and its mapping class group appear in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will motivate the study of this 'big mapping class groups'. I will then present the 'ray graph', which is a Gromov-hyperbolic graph on which this group acts by isometries. |
November 13 | (Bern) |
Positivity, Graphs and Unknotting | This talk will explore different notions of positivity of knots, how to encode such knots as graphs, and how to unknot them. Joint results with Baader/Liechti and Feller/Lobb will make appearances. No prerequisites in knot theory will be necessary. |
November 20 | (University of Georgia, Athens, US) |
Hyperkähler manifolds and their Betti numbers | Hyperkähler manifolds are higher-dimensional generalizations of K3 surfaces. The Beauville conjecture predicts that the number of deformation types of compact irreducible hyperkähler manifolds is finite in any dimension. In this talk I will briefly discuss some basic notions of the theory, explain why hyperkähler manifolds play a very important role in classification of complex manifolds, and then explain what are the evidences for Beauville's conjecture. |
November 26-30 |
(Conference in Rennes) |
||
December 4 | (Grenoble) |
Almost homogeneous curves and surfaces | The varieties which are homogeneous under the action of an algebraic group are very symmetric objects. More generally, we get a much wider class of objects, having a very rich geometry, by allowing the varieties to have not a unique orbit, but a dense orbit. Such varieties are said to be almost homogeneous; this includes the case of toric varities, when the group is an algebraic torus. 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. |
December 11 | (Bern) |
Equations with complex analytic coefficients | In the talk we discuss how Oka theory helps to solve systems of equations with complex analytic entries. A classical example is the fact that for every pair of complex analytic functions a, b: C^n -> C with no common zero there are complex analytic functions x, y: C^n -> C satisfying the Bézout identity ax+by=1. A more recent example is Leiterer's work, where the solvability of xax^{-1}=b for complex analytic matrix-valued maps a, b: C^n -> Mat(n x n, C) is investigated. Both examples are brought into the context of the speakers research. |
Spring 2018
Date | Speaker | Title | Abstract |
---|---|---|---|
March 6 | (Hamburg) |
non rationality of conic-bundles over IP^3 | I will review some of the history and recent developments of the rationality question for conic bundles over projective spaces. I will then explain our contribution to this question in the case of conic bundles over IP^3 (joint work with Asher Auel, Christian Boehning and Alena Pirutka). |
March 13 | (HSE Moscow) |
Automorphisms of pointless surfaces | I will speak about finite groups acting by birational automorphisms of surfaces over algebraically non-closed fields, mostly function fields. One of important observations here is thata smooth geometrically rational surface S is either birational to a product of a projective line and a conic (in particular, S is rational provided that it has a point), or finite subgroups of its birational automorphism group are bounded. We will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces. |
April 3 | (Bern) |
Some affine plane bundles over the punctured affine plane | An $\mathbb{A}^2$-fibration is a flat morphism between complex affine varieties whose fibers are isomorphic to the complex affine plane. In this talk, we study explicit families $f:\mathbb{A}^4\to\mathbb{A}^2$ of $\mathbb{A}^2$-fibrations over the affine plane. 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. |
April 17 | (Düsseldorf) |
Rational simple connectedness for Fano varieties | Even if the precise notion of rationally simply connected variety is still not clear in general, the recent works by de Jong, He and Starr produced great interest and new research directions. 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. |
April 24 | (Basel) |
Isomorphisms between complements of unicuspidal curves in the projective plane | In 2012 Costa constructed a family of unicuspidal curves of degree 9 in the projective plane that are pairwise non-equivalent but have isomorphic complements. We show that a family as the one of Costa cannot contain a unicuspidal curve C that admits a line intersecting C only in the singular point. To state the result more precisely, if D is any plane curve and there exists an isomorphism between the complements of C and D, then the two curves are projectively equivalent, even though the isomorphism is not necessarily linear. The proof works over an algebraically closed field of any characteristic and generalizes a result of Yoshihara (1984) who proved the claim over the complex numbers. We then use this result to show that any two irreducible curves of degree at most 8 have isomorphic complements if and only if they are projectively equivalent. |
May 8 | (Bern) |
Noetherianity up to conjugation | Finite-dimensional vector spaces are Noetherian, i.e. every descending chain of Zariski-closed subsets stabilizes. For infinite-dimensional spaces this is not true. However what can be true is that for some group G acting on the space every descending chain of G-stable closed subsets stablizes. We call spaces for which this holds G-Noetherian. In this talk, we will go over some known examples and non-examples of spaces that are Noetherian up to a group action and introduce some new ones. |
May 15 | (ETHZ) |
Complex plane curves, their intersection with round spheres, and knot concordance | We start by recalling that a smooth algebraic curve of degree d in CP^2 is a genus (d-1)(d-2)/2 surface (read `smooth 2-manifold'). The `Thom Conjecture', proven by Kronheimer and Mrowka, asserts that such algebraic curves have a surprising minimizing property. We derive consequences of the Thom Conjecture for transversal intersections of algebraic curves with round spheres, describe the knots one finds as such intersections following Rudolph, and give precise instances of the sentiment that these intersections constitute very special elements in the so-called smooth concordance group. In contrast, in the topological category, we prove that all knots are topological concordant to such an intersection. Based on joint work with Maciej Borodzik. No knowledge about knot theory and concordance theory---the study of 1-manifolds in the 3-dimensional sphere and surfaces in 4-dimensional ball bounding them---will be assumed. |
May 22 | (Köln) |
Automorphism groups of Danielewski surfaces (TALK CANCELLED) | In this talk we are going to discuss the following question: to which extent a so-called Danielewski surface is determined by its automorphism group seen as an abstract group or as an ind-group? |
May 29 | (Basel) |
Abelian quotients of the Cremona groups | I will describe a joint work with Stéphane Lamy and Susanna Zimmermann which provides abelian quotients of the Cremona groups in high dimension. |
Fall 2017
Date | Speaker | Title | Abstract |
---|---|---|---|
September 11-12 | |||
September 19 | (Basel) |
Cremona group and hyperbolic spaces | The Cremona group is the group of birational transformations of the projective plane. It acts on a hyperbolic space which is an infinite dimensional version of the hyperboloid model of H^n. This action is the main recent tool to study the Cremona group. After defining it, we will study its Voronoï tesselation, and describe some graphs naturally associated with this construction. Finally we will discuss which of these graphs are Gromov-hyperbolic. |
September 26 | (Basel) |
Small G-varieties | Let $G$ be a semisimple algebraic group acting on an affine variety $X$. An orbit $O \subset X$ is called {\it minimal} if it is $G$-isomorphic to the orbit of highest weight vectors in an irreducible representation of $G$. These orbits have many interesting properties. E.g. the closure of a minimal orbit in any affine $G$-variety $X$ is of the form $\bar O = O \cup \{x_0\}$ where $x_0 \in X$ is a fixed point, and they are even characterised by this property. An affine $G$-variety $X$ is called {\it small} if all non-trivial orbits in $X$ are minimal. It turns out that these varieties have many remarkable properties. The most interesting one is that the coordinate ring is a {\it graded $G$-algebra}. This allows a classification. In fact, there is an equivalence of categories of small $G$-varieties with so-called {\it fix-pointed} $k^*$-varieties, a class of well-understood objects which have been studied very carefully in different contexts. 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. |
September 28 | |||
October 3 | (Lyon 1) |
Commensurating actions of birational groups | Given a group G, a G-action on a set D commensurates a subset M if M differs from each of its G-translates by finitely many elements. Commensurating actions naturally induce actions on CAT(0) cube complexes. For every irreducible variety X, we define a set of (virtual) hypersurfaces, which contains the set of hypersurfaces of X and on which the group Bir(X) of birational self-transformations of X acts, extending its partial action on hypersurfaces. This action commensurates the set of hypersurfaces of X. This construction thus provides information about the structure of Bir(X) and its subgroups. (Joint work with Serge Cantat) |
October 10 | (Paris 7) |
On birational transformations of P3 of low degree. | After a short introduction to birational transformations of a projective space, we will focus on the case of P3. Few is known about birational transformations of P3 and the case of degree 3 is already difficult. To explain how things differ in larger degree it is natural to look at transformations of bidegree (4,4). In this talk we detail families of examples in degree 3 and bidegree (4,4) to illustrate this opposition. (Joint work with J. Déserti) |
October 20 | |||
October 24 | (Basel) |
Characterisation of varieties by their automorphisms | This is joint work with Hanspeter Kraft (University of Basel) and Andriy Regeta (University of Cologne). The main problem we address in this talk is the characterization of the affine space An by its automorphism group Aut(A^n). More precisely, we ask, whether the existence of an abstract group isomorphism Aut(X) \simeq Aut(A^n) implies the existence of an isomorphism of algebraic varieties X \simeq A^n. The following is our main result.
Main Theorem. Let X be a quasi-affine irreducible variety such that Aut(X) \simeq Aut(A^n). Then X \simeq A^n if one of the following conditions holds. (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. |
November 7 | (Düsseldorf) |
Fibre-like Fano manifolds: a bestiary | Fibre-like Fano manifolds naturally appear in the context of the minimal model program. In this talk I will discuss some examples, with special focus on: toric varieties, manifolds with high index and manifolds with high Picard rank. These have been obtained in recent joint works with Casagrande-Codogni and Codogni-Svaldi-Tasin. |
November 21 | (Angers) |
Signature morphisms of the Cremona group of the plane | The Noether-Castelnuovo theorem implies that over algebraically closed fields there is no non-trivial homomorphism from the Cremona group of the plane to a finite group. Over non-closed fields there are many, and I would like to explain some examples. |
November 27-28 | |||
November 27 | (Dijon) |
Inverse of an homaloïdal linear system | One can try to find explicitly the inverse of a birational map f=(f_0 : ... : f_n) from P^n to P^n given by n+1 homogeneous polynomials of the same degree f_0, ... , f_n in n+1 variables over an algebraically closed field of any characteristic. In 2000, F.Russo and A.Simis showed a situation in which case the inverse of f can be easily computed. Roughly, this favorable situation happen when the base ideal I=(f_0, ... , f_n) has "enough" linear syzygies. After rephrasing this previous result as a condition on P(I), the projectivization of the symmetric algebra of I, I will explain how to use P(I) to compute one representant of the inverse in a more general situation and I will apply it in a very concrete example. |
November 27 | (EPFL) |
Gauss map, singularities of the theta divisor and trisecants | The Gauss map is a finite rational dominant map naturally defined on the theta divisor of an irreducible principally polarised abelian varieties. In the first part of this talk, we study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension. This is a joint work with S. Grushevsky and E. Sernesi. In the second part of this talk, we will study the relation between the Gauss map and trisecant of the Kummer variety. Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well. This is a joint work with R. Auffarth and R. Salvati Manni. |
November 28 | (Moscow) |
Finite groups of birational automorphisms | I will speak about finite groups of birational automorphisms of algebraic varieties over real and complex numbers and, in particular, Cremona groups. Although the structure of these groups is known to be very complicated, one can study them on the level of finite subgroups. I will discuss some recent results in this direction, with special focus on the case when the base field is the field of real numbers. |
November 28 | (Dijon) |
Local system in motivic homotopy | The notion of a local system is proteiform and has been adapted in many contexts. In algebraic geometry, it has a transcendantal and an étale flavor, but a truly algebraic approach seems out of reach. As an illustration, the topological theory of Serre's fibrations has no analogue for algebraic varieties. In this talk, I will explain how a new kind of local system emerges from Voevodsky's motivic theory and I will describe its fundamental role. This first part of the talk will be mainly expository, trying to recall some main results on the so-called "motivic complexes" of Voevodsky. In the second part, I will describe and illustrate the geometric construction that gives the relation between some fibrations and these local systems. |
November 28 | (EPFL) |
Around Serre-Tate theory for Calabi-Yau varieties in characteristic p>0 | A classical Serre-Tate theory gives a natural isomorphism between the
deformation functor of an ordinary abelian variety and
the deformation functor of the associated p-divisible group.
In the course of his work on Tate conjecture, Nygaard's provided a similar
isomorphism for ordinary K3 surfaces. In the beginning of the talk, I will recall Nygaard's approach and give some evidence that the proof can be generalized for higher dimension Calabi-Yau varieties. Subsequently, I will present a construction of a Frobenius lifting on the ordinary part of the moduli space of Calabi-Yau varieties and compare it with a lifting arising from the Serre-Tate theory. As an application, I will give some results concerning hyperbolicitiy of the moduli space of Calabi-Yau varieties. This is a report on a work in progress with Piotr Achinger. |
November 30 | |||
December 12 | (Ferrara) |
Birational geometry of pairs | Let C, D be two birational subvarieties of the projective space P^{n}. I am interested in understanding when there exists a Cremona modification f: P^{n} --->P^{n} such that f(C) = D. For the sake of this talk I will restrict to a configuration of lines in P^{2}. In this special case I'll suggest a classification of configurations of lines that are contractible to a bunch of points. In doing this I will propose an unexpected configuration that seems to violate an established conjecture. |
Spring 2017
Date | Speaker | Title | Abstract |
---|---|---|---|
March 3 | (Bern) |
Orthogonal tensor decomposition from an algebraic perspective | Every real or complex matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense. Higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. Complementing this existing literature, I will present an algebro-geometric analysis of the set of orthogonally decomposable tensors. This analysis features a surprising connection between orthogonally decomposable tensors and semisimple algebras---associative in the case of ordinary or symmetric tensors, and of compact Lie type in the case of alternating tensors. (Joint work with Ada Boralevi, Emil Horobet, and Elina Robeva.) |
March 17 | |||
March 17 | (Toulouse) |
Algebraic subgroups of the real Cremona group | The first algebraic subgroups of the plane Cremona group that come to mind are the group of automorphisms of the plane and its algebraic subgroups. However, there are many more. I will explain how to find them and present the classification of the algebraic subgroups of the real Cremona group. This is joint work with M.F. Robayo. |
March 17 | (EPFL) |
Seshadri constants for curves | Seshadri constants are important local invariants of nef divisors. On surfaces they are related to the famous Nagata conjecture, while in general they touch on many topics like jet separation, sphere packings, and even have arithmetic applications. We find a natural concept of Seshadri constants for movable curves, and explore some of its properties that mirror the case of nef divisors. |
March 17 | (Basel) |
Degree sequences of birational transformations | To a birational map of a smooth projective variety one can associate the sequence of the degrees of its iterates. We will look at the question, which kind of sequences can be obtained in that way. I will first recall some results about degree sequences and dynamical degrees in the case of surfaces and then discuss some new constraints and examples in higher dimensions. We will also see that the set of all possible degree sequences is countable; this generalizes a result of Bonifant and Fornaess. |
March 24 | (Zurich) |
Hyperbolicity of moduli spaces of abelian varieties | For any positive integers g and n, let A_g(n) be the moduli space of principally polarized abelian varieties with a level-n structure (it is a smooth quasi-projective variety for n>2). Building on works of Nadel and Noguchi, Hwang and To have shown that the minimal genus of a curve contained in A_g(n) grows with n. We will explain a generalization of this result dealing with subvarieties of any dimension. In particular, we show that all subvarieties of A_g(n) are of general type when n > 6g. Similar results are true more generally for quotients of bounded symmetric domains by lattices. |
March 31 | (Roma-Strasbourg) |
Betti numbers and pseudo effective cones in 2-Fano varieties | The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) k-Fano variety and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties in analogy with the case k=1. Then, we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index ≥n-2, and also we complete the classification of weak 2-Fano varieties of Araujo and Castravet. |
April 21 | (Montpellier/Rennes) |
Geometry of the moduli of parabolic bundles on elliptic curves | We are interested in rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We will describe the moduli space associated to these objects, and state a Torelli theorem. The former moduli space is itself related to the moduli space of rank 2 parabolic bundles over a 5-punctured P^{1}. We will explain this link, recovering at the same time the beautiful geometry of del Pezzo surfaces of degree 4. |
April 28 | (Toulouse) |
From noncommutative motivic measures to subgroups of the Cremona group. | Let X be a smooth projective variety over a field k, and assume that weak factorization holds (e.g., k has characteristic zero). 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^n, we can show that the group generated by the standard Cremona transformation and PGL(n+1) is strictly contained in the group contracting rational varieties, as soon as n > 4. Another example allows to reconstruct Frumkin's genus filtration of Bir(X) in the cases where X is a uniruled threefold. |
May 5 | (Edinburgh) |
Igusa quartic and and Wiman-Edge sextics | The automorphism group of Igusa quartic is the symmetric group of degree 6. There are other quartic threefolds that admits a faithful action of this group. One of them is the famous Burkhardt quartic threefold. Together they form a pencil that contains all $\mathbb{S}_6$-symmetric quartic threefolds. 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). |
May 9-10 | |||
May 12 | |||
June 2 | (Amiens) |
Some remarks on Poonen's conjecture. | Let $c$ be a rational number and consider the polynomial map $\varphi_c(x)=x^2-c$. We are interested in cycles of $\varphi _c$ in $\Q$. More precisely, we focus on Poonen's conjecture, according to which every cycle of $\varphi _c$ in $\Q$ is of length at most $3$. In our talk, we discuss this conjecture using arithmetic, combinatorial and analytic means. In particular, we obtain an upper bound of the cardinality of the set of periodic points which we improve in the case $c \le 2$. We finish the talk by giving some properties regarding rational numbers $c$ for which $\varphi _c$ has a cycle of length $\geq 4$. |
June 9 | (Düseldorf) |
Del Pezzo fibrations in positive characteristic | Del Pezzo fibrations are possible outputs of a 3-fold MMP also in positive characterisitic. A natural question is whether geometrically non-normal del Pezzo surfaces can appear as generic fibre of such a fibration. This talk is based on a joint work with S. Schröer. |
Fall 2016
Date | Speaker | Title | Abstract |
---|---|---|---|
September 23/30 | (Basel) |
Regularization of Rational Group Actions | The aim is to give a modern proof of the famous Theorem of A. Weil showing that for every rational action of an algebraic group $G$ on a variety $X$ there is a regular action of $G$ on a variety $Y$ and a $G$-equivariant birational map $X---> Y$. As a guideline we use an approach given by D. Zaitsev (J. Lie Theory 5, 1995). |
Oct.07 | (Freiburg) |
Conjugacy classes of $n$-tuples in semi-simple Jordan algebras | Let $J$ be a (complex) semi-simple Jordan algebra, and consider the diagonal action of its automorphism group on the $n$-fold product of $J$. In this talk, geometric properties of this action are studied. In particular, a characterization of the closed orbits is given, which is similar to the description in the case where a reductive linear algebraic group acts on (the $n$-fold product of) its Lie algebra. |
Oct.14 | (Padova) |
Tropical compactifications, Mori Dream Spaces and Minkowski bases | Given a Mori Dream Space X we construct via tropicalization a model dominating all the small Q-factorial modifications. Via this construction we recover a Minkowski bases for the Newton-Okunkov bodies on X and hence the movable cone for X. This is a work in progress with Elisa Postinghel. |
Oct.21 | (MPIM Bonn) |
Moduli space of cubic surfaces and their anti canonical divisors | We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate these compactifications to the (conjectural) moduli space of logK-semistable pairs showing that any log K-stable pair is an element of our moduli and that there is a canonically defined CM line bundle that polarizes our moduli. This is a joint work with Patricio Gallardo (University of Georgia) and Cristiano Spotti (Aarhus University). |
Nov.04 | (University of Edinburgh) |
On the number of minimal models of a smooth threefold of general type | Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists, some natural question arise, such as: is the minimal model unique? If not, how many are they? After recalling all the necessary notions of the Minimal Model Program, I will explain that varieties of general type admit a finite number of minimal models. I will talk about a recent joint project with Stefan Schreieder and Luca Tasin, where we prove that in the case of threefold this number is bounded by a constant depending only on the Betti numbers. I will also show that in some cases it is possible to compute this constant explicitly. |
Nov.11 | (Fribourg) |
Algebraic aspects of hyperbolic volume | This talk starts with a survey about hyperbolic volume in three dimensions, dilogarithms and zeta values. 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. |
Dec.09 | (Bonn) |
A knot theorists approach to singularities and their deformations. | After a brief introduction to plane curve singularities, we recast some of their classical invariants in terms of modern knot theory. Then we will see connections between purely algebraic questions about deformations of singularities and purely knot theoretic questions about cobordisms between knots, and answer some of them. This talk aims to be elementary and will not assume familiarity with knot theory or singularity theory. |
Dec.16 | (EPFL) |
On projectivity of the moduli space of stable surfaces in characteristic p > 5 | Stable varieties are higher dimensional generalizations of stable curves. Their moduli space contains an open locus parametrizing varieties of general type up to birational equivalence, just as the space of stable curves contains the space of smooth curves in dimension one. Furthermore, also similarly to the one dimensional picture, it provides a compactification of the above locus, which is known in characteristic zero but it is only conjectural in positive characteristic in dimension at least two. I will present a work in progress aiming to prove the projectivity of every proper subspace of the moduli space of stable surfaces in characteristic greater than 5. |
Dec.16 | (Imperial College London) |
The space of Kähler metrics on singular varieties | The geometry and topology of the space of Kähler metrics on a compact Kähler manifold is a classical subject, first systematically studied by Calabi in relation with the existence of extremal Kähler metrics. Then, Mabuchi proposed a Riemannian structure on the space of Kähler metrics under which it (formally) becomes a non-positive curved infinite dimensional space. Chen later proved that this is a metric space of non-positive curvature in the sense of Alexandrov and its metric completion was characterized only recently by Darvas. In this talk we will talk about the extension of such a theory to the setting where the compact Kähler manifold is replaced by a compact singular normal Kähler space. As one application we give an analytical criterion for the existence of Kähler-Einstein metrics on certain mildly singular Fano varieties, an analogous to a criterion in the smooth case due to Darvas and Rubinstein. This is based on a joint work with Vincent Guedj. |
Spring 2016
Date | Speaker | Title | Abstract |
---|---|---|---|
February 5 | (Clermont-Ferrand) |
Noncommutative invariant algebras and weighted homographic actions | The natural algebraic question of extending a group action on a commutative algebra A of "functions" to a noncommutative algebra of differential (or rational differential, or pseudodifferential) "operators" with coefficients in A is closely related to the study of deformations of A. 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. |
Feb.26 | (Talca) |
Additive group actions on algebraic varieties | This is a joint work with A. Dubouloz. 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. |
Mar.03 | (Bristol) |
A new approach to the classification of singular Fano-Mori 3-folds. | I will give an overview of the geometry of Fano 3-folds after Mori theory. After discussing past approaches to the classification, I will highlight why such attempts seem hopeless. Building on recent advances in the geometry of Fanos, I introduce a new viewpoint on the classification problem. A main emphasis will be given to the unpredicted behaviour of the first examples of non-complete intersection Fanos, discovered in a joint work with Takuzo Okada. |
Mar.08 | (Singapore) |
Automorphism groups of positive entropy of normal projective varieties | We will report our recent results on solvable groups G of automorphisms of normal projective varieties X. One machinery we use is the equivariant minimal model program. One purpose is to know more geometry of X from the existence of such G acting on X. |
Mar.10 | (Moscow) |
Automorphism groups of affine algebraic surfaces preserving an A1-fibration | It is well known that A1-fibrations are of particular interest in the study of automorphism groups of affine surfaces. That is, description of automorphism groups is based on subgroups preserving A1-fibrations, except for surfaces without such fibrations. We provide a method to directly compute these subgroups in terms of a boundary divisor of an SNC-completion. We also use the concepts of arc spaces and formal neighbourhoods. This allows us to establish the following structure of a subgroup preserving a A1-fibration. Up to a finite index, it is a semidirect product of an abelian unipotent subgroup acting by translations on fibers and of a finite-dimensional subgroup that fixes a certain section. In particular, we derive an example of a surface with infinite discrete automorphism group. |
Mar.11 | (Grenoble) |
A^1-contractibility of Koras-Russell threefolds | In this talk, which is a joint work with Adrien Dubouloz, we will explain that the unstable homotopy category is unable to distinguish between the affine space of dimension 3 and the Koras-Russell threefolds of the first kind. We will first make some historical remarks and then spend some time to explain the basic features of the unstable homotopy category, in particular how to distinguish isomorphisms in some special cases. We will then proceed with the proof of the above claim. |
Mar.18 | (Roma Tre) |
Torus equivariant K-stability | We prove (using algebro-geometric methods) two results that allow to test the positivity of the Donaldson-Futaki weights of arbitrary polarised varieties via test-configurations which are equivariant with respect to a maximal torus in the automorphism group. It follows in particular that there is a purely algebro-geometric proof of the K-stability of projective spaces (or more generally of smooth toric Fanos with vanishing Futaki character, as well as of the examples of non-toric Kahler-Einstein Fano threefolds due to Ilten and Suss) and that K-stability for toric polarised manifolds can be tested via toric test-configurations. A further application is a proof of the K-stability of constant scalar curvature polarised manifolds with continuous automorphisms. Our approach is based on the method of filtrations introduced by Wytt Nystrom and Szekelyhidi. This is a joint work with J. Stoppa. |
Apr.01 | (Mulhouse) |
PSL(2,C), the exponential and some new free groups | |
Apr.08 | (Bonn) |
On a classical question about Chern numbers | Generalising a question of Hirzebruch, Kotschick asked the following: which Chern numbers are determined up to finite ambiguity by the underlying smooth manifold? Together with S. Schreieder, I treated this question in dimension higher than 3. After explaining such results, I will talk about the 3 dimensional case, where it is believed that Chern numbers are bounded. Results in this direction have been obtained in a recent preprint with P. Cascini, where tools from the Minimal Model Program have been used, combined with topology's and arithmetic's techniques. |
April 14 |