Seminar Algebra and Geometry - Fall 2023

It happens usually on Tuesdays from 10:30 to 12:00 in Spiegelgasse 5, Seminarraum 05.001. Some tea takes place before the talk, at 10:00 in Spiegelgasse 1, 6th floor.
September 19 Elisa Lorenzo García
University of Neuchâtel
Reduction of plane quartics and Cayley octads In this talk I will give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst either the 42 possible types in the general case or the 32 possible types when the reduction is hyperelliptic. These criteria are in the vein of the new machinery of "cluster pictures" for hyperelliptic curves. We will also construct explicit families of quartic curves that realise all possible stable types, against which we have tested these criteria. We will give many numerical examples that illustrate how to use these criteria in practice.
This is a joint work with Raymond Van Bommel, Jordan Docking, Vladimir Dockitser and Reynarld Lercier
September 26 Pascale Voegtli
UCL (London)
Flop connections between minimal models for corank 1 foliations over threefolds In recent years the understanding of the fundamental birational geometry of foliations, especially on 3-folds, has been promoted by several groundbreaking works. Cascini and Spicer have extended most parts of the classical MMP to threefold pairs equipped with a mildly singular corank 1 foliation. In particular, the existence of log-flips has been demonstrated. The successful establishment of a foliated analogue of the classical MMP in low dimensions naturally raises the question whether classical results being closely related to the MMP do find their natural generalizations to foliated pairs. One such classical result one might strive to convey to foliations is the well-known theorem of Kawamata from 2007, stating that two minimal models with terminal singularities are related by a sequence of flops. In the talk, after having introduced the relevant notions, we will sketch a proof of an analogue of Kawamata's theorem for foliated 3-folds. This is recent joint work with D. Jiao.
October 3 no talk
October 10 Fabio Bernasconi
University of Basel
Bounding del Pezzo fibrations in positive characteristic Fano varieties form one of the basic building blocks of algebraic varieties predicted by the Minimal Model Program, as they appear as the generic fibres of Mori fibre spaces. In this talk, I will focus on the study of generic fibres of 3-dimensional Mori fibre spaces of relative dimension 2 in positive characteristic. These are geometrically integral del Pezzo surfaces defined over an imperfect field. Due to inseparability phenomena, these surfaces reveal to be more difficult to understand both geometrically and cohomologically. In this talk, I will report on a recent work with G.Martin where we obtain a classification of canonical del Pezzo surfaces and we prove the BAB conjecture for ε-klt ones.
October 17 Cinzia Casagrande
University of Torino
Fano 4-folds with large Picard number Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We will discuss the following theorem: if rho(X)>12, then X is a product of del Pezzo surfaces. This implies, in particular, that the maximal Picard number of a Fano 4-fold is 18. After an introduction and a discussion of examples, we will introduce the Lefschetz defect, an integral invariant of X, and see how its properties are related to this theorem. In the second part of the talk we will discuss the strategy of proof of the theorem in one particular case: when X has a rational contraction X-->Y where Y has a dimension 3. A rational contraction is a rational map that factors as a sequence of flips followed by a surjective morphism with connected fibers; we will see an explicit example of such setting.
October 24 no talk
October 31 Christian Urech
University of Basel
Finitely generated subgroups of algebraic elements of plane Cremona groups Let S be a variety. A birational transformation of S is called algebraic if it is contained in an algebraic subgroup of the group of birational transformations Bir(S) of S. In this talk, I will explain, why a finitely generated subgroup of Bir(S) is itself contained in an algebraic subgroup of Bir(S). This answers a question of Charles Favre. We will apply this result to describe the degree growth of finitely generated subgroups of plane Cremona groups. This is joint work with Anne Lonjou and Piotr Przytycki.
November 7 Aurore Boitrel
University of Angers
Automorphism groups of del Pezzo surfaces Del Pezzo surfaces and their automorphism groups play a key role in the study of algebraic subgroups of the Cremona group of the plane. Over an algebraically closed field, it is classically known that a del Pezzo surface is either isomorphic to 1 ×1 or to the blow-up of 2 in at most 8 points in general position, and in this case, automorphisms of del Pezzo surfaces are known and have been described. In particular, there is only one isomorphism class of del Pezzo surfaces of degree 5 over an algebraically closed field. In this talk, we will focus on del Pezzo surfaces of degree 5 defined over a perfect field k. In this case, there are many more extra surfaces (as we can already see for rational real forms of del Pezzo surfaces of large degrees), and the classification as well as the description of the automorphism groups of these surfaces over k is reduced to understanding the actions of the Galois group Gal(K/k) on the graph of (-1)-curves, where K is the algebraic closure of k.
November 13-14 Basel-Dijon-EPFL meeting in Basel    
November 21 Elena Sammarco
Università Roma Tre
Divisors in the moduli space of cubic fourfolds In his Ph.D. thesis, Brendan Hassett introduced the definition of special cubic fourfolds, the ones that contain a surface not homologous to a complete intersection. They have rich geometric properties that in many cases involve K3 surfaces. Also, they form a countably infinite union of divisors Cd in the moduli space C of cubic fourfolds. In this context, in which we find some conjectures on the rationality of the cubic fourfold, it is interesting to know what happens outside these divisors. I'll show a very explicit method to construct some non-special divisors in C.
November 28 no talk
December 5 Peter Feller
Gauss composition and distinct surfaces in the 4-ball In Disquisitiones Arithmeticae, Gauss described a composition law that turns primitive integral binary quadratic forms (IBQFs for short, expressions of the form ax^2+bxy+cy^2 considered up to linear coordinate changes in the variables x and y) of a fixed discriminant into a group known as the class group. In the first part of the talk, we discuss this group via Bhargava's celebrated cube law approach and provide a new geometric description of Gauss composition. The latter is phrased in terms of a correspondence between planes in the space of two-by-two matrices and pairs of IBQFs discovered by Aka, Wieser, and Einsiedler.

In a second, more topological part of the talk, we discuss knotting phenomena of surfaces in dimension 4. After some context, we will discuss a 40 year old question of Livingston concerning the existence of certain distinct surfaces in the 4-ball. Our contribution is the systematic construction of such examples by relating the problem to the existence of certain symplectic planes in the space of two-by-two matrices. The key step in the proof uses the geometric description of Gauss composition from above.

This talk will aim to stay light with a focus on providing context. In particular, no knowledge concerning algebraic number theory or knot theory will be presupposed. Based on joint work with M. Aka, A. Miller, and A. Wieser.
December 12 Stefan Kebekus
University of Freiburg
Extension Theorems for differential forms and applications We discuss various notions of differential forms on singular complex spaces, present a Hartogs-type extension theorem for differential forms on a resultion of singularities and explain their use in the study of minimal varieties. We survey a number of applications, pertaining to classification and characterisation of special varieties, non-Abelian Hodge Theory in the singular setting, and quasi-étale uniformization.
December 18-20 Algebraic geometry conference in Basel