### Basel-Dijon-EPFL meeting (3-4 May 2018)

**Location:**

3 May : GR B3 30, map

4 May: MA A3 31, map

**Schedule:**

**3 May**

14:00-15:00 **Adrien Dubouloz**

*Tilte:* Cylinders in Mori Fiber Spaces

*Abstract:* A cylinder in an algebraic variety is a Zariski open subset isomorphic to the product of a variety with the affine line. Every smooth projective variety containing such a cylinder has a birational model which is a Mori Fiber Space over a base. Cylinders in Fano varieties of dimension 3 and 4 have received quite a lot of attention recently in connection to the existence of additive group actions on certain of their affine cones. In this talk, I will focus on the question of existence of such cylinders in total spaces of certain strict Mori Fiber Spaces: del Pezzo fibrations and MFS of relative dimension 3 whose general fibers are isomorphic to the quintic del Pezzo threefold V5. (Joint work with T. Kishimoto, Saitama University).

15:00-15:30 coffee

15:30-16:30 **Nicholas Shepherd-Barron**

*Title: *del Pezzo surfaces and effective Torelli in genus three

*Abstract: *The tropes and singularities of a Kummer surface determine a curve of genus two in a straightforward way. In this talk we describe a similar picture in genus three. This is joint work with M. Fryers.

16:30-17:00 coffee

17:00-18:00 **Giulio Codogni**

*Title:* Positivity of the Chow-Mumford line bundle for families of K-stable klt Fano varieties

*Abstract:* The Chow-Mumford (CM) line bundle is a functorial line bundle defined on the base of any family of polarized varieties, in particular on the base of families of klt Fano varieties. It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable klt Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers.

In this talk, I will present a proof of the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming stability only for very general fibers. These results work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I will also present an application to the classification of Fano varieties. This is a joint work with Zs. Patakfalvi.

19:00 Dinner for the participants in the Brasserie Lausanne-Moudon

**4 May**

8:45-9:00 coffee

9:00-10:00 **Immanuel van Santen**

*Title:* Embeddings of lines and planes into algebraic groups

*Abstract:* In this talk we will study closed embeddings of affine varieties into affine algebraic groups G up to algebraic automorphisms of the underlying variety of G. After recalling some classical results concerning embeddings into the affine space, we will focus on embeddings of the affine line into arbitrary groups and embeddings of the affine plane into SL_2. Mainly we will discuss the following two results:

-If G is a group without characters of dimension different than

three, then all embeddings of the affine line are equivalent.

-There is an infinite family of pairwise non-equivalent

embeddings of the affine plane into SL_2.

This is joint work with Peter Feller and Jérémy Blanc.

10:00-10:30 coffee

10:30-11:30 **Zhiyu Tian**

*Title:* Motivic crepant resolution conjecture and Chow ring of hyperKähler varieties.

*Abstract:* I will explain a joint program with Lie Fu, where we consider a motivic version of the crepant resolution conjecture in Gromov-Witten theory. I will also discuss an application to the study of Chow rings of Hilbert schemes of points on K3 surfaces.

11:30-11:45 coffee

11:45-12:45 **Jean Fasel**

*Title:* Motivic triviality of the Koras-Russell threefolds of the first kind

*Abstract:* In this talk, we will try to axiomatize the motivic homotopy category of Morel-Voevodsky using the notion of pretriangulated category.

The advantage of this approach is to allow researchers not familiar with motives to work nevertheless in this framework.

As an illustration of the axiomatization, we will show that the Koras-Russell threefolds of the first kind are motivically trivial. This is a joint work with A. Dubouloz.”

12:45- 14:15 Lunch for the paricipants in “Le Parmentier”

14:15-15:15 **Filippo Viviani**

*Title:* On the cone of effective cycles on the symmetric products of curves

*Abstract:* I will report on a joint work with F. Bastianelli, A. Kouvidakis and A. F. Lopez in which we study the cone of (pseudo-)effective cycles on symmetric products of a curve.

We first prove that the diagonal cycles span a face of the pseudo-effective cone of cycles in any given dimension. Secondly, we look at the contractibility faces associated to the Abel-Jacobi morphism towards the Jacobian and in many cases we are able to compute their dimensions.