# Seminar Algebra & Geometry

Our weekly Seminar "Algebra and Geometry" takes place on

Tuesdays from 10:30 to 12:00 a.m., Mathematical Institute (Spiegelgasse 1, ground floor) with tea and coffee at 10:10 a.m. (on the 6th floor of Spiegelgasse 1).

 Sep 11-12 (Mon-Tue) Dijon-Basel Meeting in Algebraic Geometry (in Dijon) Sep 19 Anne Lonjou (Basel) - Cremona group and hyperbolic spacesThe 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. Sep 26 Hanspeter Kraft (Basel)- Small G-varietiesLet $G$ 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. Sep 28 (Thursday) Verteidigung von Christian Urech Oct 3 Yves de Cornulier (Lyon 1) -Commensurating actions of birational groupsGiven 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) Oct 10 Frederic Han (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) Oct 17 Konstantin Shramov (HSE Moscow) - Automorphisms of pointless surfacesTALK CANCELLEDI 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 that a 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. Oct 20 (Friday) EPFL-Basel Meeting in Birational Geometry (in EPFL) Oct 24 Immanuel van Santen (Basel)    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) ≃ Aut(A^n) implies the existence of an isomorphism of algebraic varieties X ≃ A^n. The following is our main result. Main Theorem. Let X be a quasi-affine irreducible variety such that Aut(X) ≃ Aut(A^n). Then X ≃ 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. Oct 31 Frank Kutzschebauch (Bern)- Holomorphic factorization of maps into the symplectic groupTALK CANCELLEDIt is (well) known that elementary symplectic matrices over a field $k$ generate the group of symplectic matrices $Sp_{2n)(k)$. For parametric versions much less is known. Here parametric means that the field k will be replaced by the ring of continuous, algebraic or holomorphic complex valued functions  on an appropriate space $X$. We will name  the  algebraic results we are aware of and give a new topological result (for all $n$) and a new holomorphic result for $n=2$. In the end  we will bother you with the strategy of proof.  This is joint work with Björn Ivarsson, Aalto University Helsinki and Erik Löw, Oslo University. Nov 7 Andrea Fanelli  (Düsseldorf) - Fibre-like Fano manifolds: a bestiaryFibre-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. Nov 14 Nov 21 Susanna Zimmermann (Angers) -Signature morphisms of the Cremona group of the planeThe 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’d like to explain some examples. Nov 27-28 BASEL-EPFL-Dijon Meeting in Birational Geometry (in Basel)27.11.2017 - Alte Universität (Rheinsprung 9) Seminarraum 20114h30 - 15h30: Rémi Bignalet-Cazalet (Dijon)  -  Inverse of an homaloïdal linear systemOne 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.16h-17h: Giulo Codogni (EPFL) - Gauss map, singularities of the theta divisor and trisecantsThe 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.28.11.2017- Rheinsprung 21 -Seminarraum 00.002 10h15-11h15 Egor Yasinsky (Moscow) - Finite groups of birational automorphismsI 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.11h30-12h30 Frédéric Deglise (Dijon) - local system in motivic homotopyThe 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.14h30-15h30 Maciek Zdanowicz (EPFL) - Around Serre-Tate theory for Calabi-Yau varieties in characteristic p>0A classical Serre-Tate theory gives a natural isomorphism between the deformation functor of an ordinary abelian variety andthe deformation functor of the associated p-divisible group.  In thecourse of his work on Tate conjecture, Nygaard's provided a similarisomorphism 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 andcompare it with a lifting arising from the Serre-Tate theory.  As anapplication, 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. Nov 30 Basel-Freiburg-Nancy-Saarbrücken-Strasbourg Seminar (in Strasbourg) Dec 5 Dec 12 Sara Durighetto (Ferrara) Dec 19

Information about former semesters can be found here.

# Location

Departement Mathematik und Informatik
Spiegelgasse 1
CH-4051 Basel