Basel-Dijon-EPFL meeting (12-13 May 2022)

Location:  GC A1 416 for Thursday, and GR A3 31 for Friday ( click on the number of the room for a map)

Schedule:

12 May

14:00-15:00 Victor do Valle Pretti (Université de Bourgogne)

Determinants, even instantons and Bridgeland stability

Abstract: 

The moduli of instantons bundles over a Fano threefold have been under investigation by many authors during the last 50 years. Their relation with exceptional collections and monads is already proven to be useful in many situations, such as the ADHM equations and D.Faenzi’s work, for example. During this talk, we will see how to prove they’re stable in the sense of Bridgeland and obtain their moduli space as an open subset of the moduli of Bridgeland stable objects. This is done by finding a region where Bridgeland and quiver stability coincide, hence also obtaining general information about these moduli spaces. The general theory behind this association was proven by E.Macr\`{i}, here we provide a systematic way of describing the quiver regions for any smooth projective variety, if they exist.

15:00-15:30 coffee

15:30-16:30 Pascal Fong (Universität Basel)

Title: Algebraic subgroups of the group of birational transformations of ruled surfaces

Abstract: We classify the maximal algebraic subgroups of $\mathrm{Bir}(C\times \mathbb{P}^1)$, when $C$ is a smooth curve of positive genus.

16:30-17:00 coffee

17:00-18:00 Luca Tasin (Università di Milano)

Higher dimensional slope inequalities

Abstract: 

Consider a family of varieties f: X-> T, where T is a curve.  We prove several inequalities about the slope of f, which are generalisations of the classical Xiao and Cornalba-Harris inequalities in the case where X is a surface.  We then apply our results to the KSB moduli space of stable varieties to study the ample cone of such spaces. The talk is based on a joint work with Giulio Codogni and Filippo Viviani.

18:30 Dinner

13 May

9:00-9:15 coffee

9:15-10:15 Pierre-Alexandre Gillard (Université de Bourgogne)

Torus actions on affine varieties over characteristic zero fields

Abstract:

Torus actions on affine varieties over algebraically closed fields of characteristic zero were described by Altmann and Hausen in 2006 in terms of polyhedral divisors on a certain rational quotient for the torus action. 
Using Galois descent tools, we will explain how to extend the Altmann-Hausen description over arbitrary fields of characteristic zero. Then, we will discuss certain particular cases in more details using birational geometry tools.

 

 

10:15-10:30 coffee

10:30-11:30 Quentin Posva (EPFL)

Stable surfaces in positive characteristic and their moduli

Abstract:

The theory of KSBA stable varieties in characteristic zero is nowadays well-understood: we have a good grasp on their moduli and on their deformations. In positive characteristic, many questions are still open: in particular, there is no proof yet of the properness of the moduli space. In this talk, I will report on some recent results about stable surfaces in positive characteristic, focusing on the technical tools that are necessary to study the non-normal ones.

11:30-11:45 coffee

11:45-12:45 Jarod Alper (University of Washington, Seattle)

Coherent completeness in positive characteristic

Abstract: 

Formal GAGA is a fundamental result asserting that a coherent sheaf on a scheme proper over a complete local noetherian ring is the same as a compatible system of coherent sheaves on the thickenings of its central fiber.  We will discuss generalizations of this result to algebraic stacks and explain how such results can be used to prove local structure theorems for algebraic stacks.  After reviewing joint work with Hall and Rydh which establishes a satisfactory result in characteristic 0, we will discuss partial progress in joint work with Hall and Lim on extending this result to positive characteristic.