Title: Length in the Cremona group
Abstract: Every birational map of the plane can be decomposed into a product of linear maps and Jonquières transformations. Contrary to the case of automorphisms of the affine plane, the Cremona group of the plane does not have a structure of amalgamated product, so the length associated does not behave a priori so well and is not so clear to calculate. I will give an algorithm to compute it, show that it only depends on the homaloidal type of the map (degree and multiplicities of base-points) and that the length is semi-lower continuous for the Zariski topology. Joint work with Jean-Philippe Furter.
Title: Mori fibre spaces for 3-folds in positive characteristic
Abstract: In this talk I will discuss recent results in the Log Minimal Model Program for 3-folds in positive characteristic, with emphasis on the base point free theorem and existence of Mori fibre spaces.
Title: Geometric invariants of non-archimedean semi-algebraic sets
Abstract: Semi-algebraic sets over non-archimedean fields appear naturally in tropical and non-archimedean geometry. I will explain how one can use Hrushovski and Kazhdan's theory of motivic integration to attach geometric invariants to these semi-algebraic sets. We used this construction to propose geometric interpretations for the refined curve counting invariants of Göttsche and Shende and the corresponding refined tropical multiplicities of Block and Göttsche. This talk is based on joint work with Sam Payne and Franziska Schroeter.
Title: Birationally bi-rigid Fano threefolds of codimension 2
Abstract: A Fano variety X of Picard number one is said to be birationally rigid (resp. bi-rigid) if X is the unique Mori fiber space in the birational equivalent class of X (resp. X is birational to a Fano variety X' other than X and is not birational to any other Mori fiber space). Among the 85 families of Fano threefold weighted complete intersections of codimension 2, quasismooth members of specific 19 families are known to be birationally rigid. I will talk about the remaining families and explain that the classification of birationally bi-rigid Fano threefolds of codimension 2 is completed.
Title: Automorphisms of K3 fibrations
Abstract: If f : X → Y is an algebraic fibre space, then automorphisms of the generic fibre give birational automorphisms of X. We will explain how to apply this to some K3 fibrations obtained from blowups of Fano manifolds of co-index 2 to verify new cases of the Morrison-Kawamata conjecture. This is joint work with Izzet Coskun.
Title: On the rationality problem for the moduli space of hyperelliptic curves with an ineffective theta characteristic and the geometry of a singular del Pezzo 3-fold
Abstract: In this talk I show how the rationality problem for the moduli space of hyperelliptic curves with an ineffective theta characteristic can be studied, and solved, using the birational geometry of a del Pezzo 3-fold with a singular ODP. I will discuss also similar results.
Title: Frobenius splittings in birational geometry
Abstract: Due to the absence of the Kawamata-Viehweg vanishing theorem, the classification of algebraic varieties in positive characteristic, as of very recently, has been seen as an insurmountable task. The recent progress in the field has been inspired by the discovery of Frobenius-split varieties. In my talk, I will discuss connections between the geometry of projective varieties and properties of the Frobenius action, focusing particularly on surfaces.
Title: Birational geometry of cluster varieties
Abstract: I will talk about joint work with Sean Keel and Paul Hacking. I will explain how to view cluster varieties as natural objects to consider if one is interested in understanding log Calabi-Yau varieties. The simplest example of a log Calabi-Yau variety is a toric variety X along with its toric boundary. Cluster varieties can be viewed as the next most simple case, in which one glues together the simplest examples via a natural class of birational maps.