#Fano #hyperkähler #birational #MMP #arithmetic_geometry #Dijon_is_beautiful
Hamid Abban — Nottingham, England
Jefferson Baudin — Lausanne, Switzerland
Marta Benozzo — Orsay, France
Fabio Bernasconi — Rome, Italy
Pietro Beri — Nancy, France
Mattia Cavicchi — Dijon, France
Francesco Denisi — Saarbrucken, Germany
Andrea Fanelli — Bordeaux, France
Pascal Fong — Hannover, Germany
Lyalya Guseva — Moscow, Russia
Daniela Paiva — Basel, Switzerland
Antoine Pinardin — Basel, Switzerland
Claudia Stadlmayr — Neuchâtel, Switzerland
Nikolaos Tsakanikas — Lausanne, Switzerland
Vanja Zuliani — Orsay, France
Aline Zanardini — Lausanne, Switzerland
Limited funding is available for early career researchers. If you plan to apply for funding, please register by March 20. You will hear back from us by April 1 (no joke).
Otherwise, the registration deadline is April 17.
Lunch 🍗 Restaurant Universitaire La Cantine (6 Avenue Alain Savay)
Social dinner👯 Tuesday, 19:30, L'Épicerie & Cie, 5 Pl. Émile Zola
Hamid Abban
A Matsushima theorem for K-polystable polarised smooth Fano threefolds
Yau-Tian-Donaldson conjecture states that a polarised manifold (X,L) admits a cscK metric in c_1(L) if and only if (X,L) is K-polystable. Matsushima proved in 1957 that existence of such cscK metric implies reductively of the automorphism group of X. In a positive direction on the YTD conjecture, we show that K-polystability implies reductively of the automorphism group of X, for smooth Fano threefolds. This is joint work with Paolo Cascini and Ivan Cheltsov.
Jefferson Baudin
An effective characterization of semi-abelian varieties
A remarkable theorem of Chen and Hacon asserts that a smooth complex projective variety X with two first plurigenera equal to 1 and first betti number equal to 2dim(X) is birational to an abelian variety. In this talk, we will explore an analoguous result for smooth quasi-projective varieties, thereby obtaining a sharp characterization of semi-abelian varieties (i.e. extensions of abelian varieties by affine tori). This is joint work with Sofia Tirabassi.
Marta Benozzo
Some hypersurface singularities over the p-adic integers
Even when trying to classify smooth varieties, it is natural to stumble upon singular varieties. Recently, there has been a lot of progress in the classification of varieties over fields of positive characteristics and over mixed characteristic DVR's like Z_p. This has been possible partly thanks to the introduction of notions of singularities related to Frobenius splittings and perfectoid methods, respectively. Given a hypersurface in a projective space over the complex numbers, we can measure how singular it is with an invariant called the ''log canonical threshold''. Similarly, in positive characteristic, we can define the ''F-pure threshold'' and in mixed characteristic the ''plus-pure threshold''. In this talk, we will explore some examples of how to compute the plus-pure threshold and how this relates to the invariants in positive characteristic and in characteristic 0. This is based on joint work with V. Jagathese, V. Pandey, P. Ramírez-Moreno, K. Schwede, P. Sridhar.
Pietro Beri
Involutions on Hilbert schemes of K3 surfaces
The Torelli theorem for hyper-Kähler varieties allows one to study the birational automorphisms of a punctual Hilbert scheme of a K3 surface by algebraic methods. In particular, these automorphisms have been classified, in joint work with Al. Cattaneo, in the case of an algebraic K3 surface with Picard rank one. This classification, however, does not shed light on their geometric realization. Finding such a description is typically a subtle and challenging problem: beyond the classical Beauville involutions, only finitely many explicit constructions are known in the literature. In this talk, I will describe a new infinite family of involutions and discuss some interesting consequences of their construction. This is a joint work with L. Manivel.
Fabio Bernasconi
Failure of Kodaira vanishing and the Brauer-Manin obstruction over global fields
For smooth varieties X defined over global fields K, studying the distribution of rational points X(K) within their adelic points is a central theme in arithmetic geometry. A key observation due to Manin is that the Brauer group cuts out a subset of the adelic points — known as the Brauer-Manin set—which contains the rational points. Since then, the study of the Brauer-Manin set has become a prominent area of research in arithmetic.
In a recent article, Valloni showed how the failure of the Bogomolov-Sommese vanishing theorem over global fields of positive characteristic (a pathology specific to characteristic p) yields an abundance of p-torsion Brauer classes, which in turn allow for a fairly precise description of the Brauer-Manin set. In joint work with Valloni, we further investigate this connection between pathologies in characteristic p and the Brauer--Manin set. We give a description of the Brauer-Manin set for varieties violating the Kodaira vanishing theorem and we apply our techniques to Raynaud surfaces, providing the first example of a rather complete description of the Brauer-Manin set for a surface of general type not contained in an abelian variety.
In the talk, we will start by recalling the basic objects involved (such as the Brauer-Manin set and differential forms in characteristic p) and outline the main ideas from Valloni's work and our joint article.
Mattia Cavicchi
The Standard Conjecture of Lefschetz type for Certain Lagrangian Fibrations
When X is a smooth complex projective variety of dimension d, the i-th iterate of cup‑product with a hyperplane section induces an isomorphism between the singular cohomology groups Hd−i(X) and Hd+i(X). The standard conjecture of Lefschetz type for X, formulated by Grothendieck in the 1960s and still largely open, predicts that the inverses of these isomorphisms should be induced by algebraic cycles on X×X. In this talk, after introducing these ideas, I will discuss joint work with Ancona, Laterveer, and Saccà, in which we prove the conjecture for certain hyperkähler varieties equipped with a Lagrangian fibration. The class of hyperkähler varieties to which our theorem applies includes notably the OG10-type varieties known as LSV (Laza-Saccà-Voisin) tenfolds.
Francesco Denisi
Some results on K-trivial varieties
The purpose of this talk is twofold. We first discuss the positivity of hypersurfaces with big normal bundle, focusing on the case where the ambient variety has numerically trivial canonical divisor. Then, we present a result about the behaviour of birational automorphism groups in families of projective irreducible holomorphic symplectic manifolds.
Andrea Fanelli
Explicit birational geometry over imperfect fields and the Cremona group
In this talk I will present joint works with Fabio Bernasconi, Julia Schneider, Stefan Schröer and Susanna Zimmermann aimed to study some aspects of the birational geometry of regular algebraic surfaces over imperfect fields. These objects naturally appear when one works with fibrations over algebraically closed fields in positive characteristics. I will also discuss applications to the Cremona group.
Pascal Fong
Automorphisms of projective bundles over a curve of general type
In this talk, we will discuss the classification of maximal connected algebraic subgroups of Bir(CxP^2), where C is a smooth projective curve of genus >2. Our main focus will be on the automorphism groups of P^2-bundles, and we will study their maximality. This is a joint work with S. Zimmermann.
Lyalya Guseva
Homological Projective Duality for Grassmannian Gr(3,6)
Homological projective duality theory, introduced by Kuznetsov, is a powerful tool for investigating the bounded derived categories of projective varieties together with their linear sections. It provides interesting semiorthogonal decompositions as well as derived equivalences. In this talk, I will give a brief introduction to this theory and describe how homological projective duality works in the case of Grassmannian Gr(3,6). This is joint work in progress with Vladimiro Benedetti.
Daniela Paiva
The generalized Gizatullin’s problem
The problem of determining which automorphisms of a smooth quartic surface S ⊂ P3 are induced by birational maps of P3 remains open. This question is known as Gizatullin’s problem. In this talk, we will discuss this problem and a generalized version, where we pose the same question for projective K3 surfaces contained in Fano threefolds. I we will provide a general overview of the theory of K3 surfaces and the birational geometry of Fano threefolds, and explain how the interaction between these two areas can be used to approach Gizatullin’s problem. The results I will present are part of several joint works with Carolina Araujo, Michela Artebani, Alice Garbagnati, Ana Quedo, and Sokratis Zikas.
Antoine Pinardin
Finite simple subgroups of the real Cremona group of rank three
Very little is known about the classification of finite subgroups of Cremona in dimension three. It is natural to start with the case of simple groups, and this step was achieved by Prokhorov in 2009 over the field of complex numbers. In the work I will present, we show that the only non-cyclic finite simple subgroups of the real Cremona group of rank three are A5 and A6. This is a joint project with I. Cheltsov and Y. Prokhorov.
Claudia Stadlmayr
G-smooth RDP del Pezzo surfaces
If f: X → Y is a quotient by a free action of a finite group G, then X is smooth if and only if Y is smooth. Working in characteristic p > 0, one can replace G by a finite, not-necessarily reduced group scheme, and then X might be singular even if Y is smooth. Turning this around, this means that there are singular varieties X with a free action of a finite group scheme G such that the quotient of X by G is smooth. Following Brion, such varieties are called G-smooth, and they can be used to construct generically non-smooth fibrations between smooth varieties. In this talk, building on the classification of weak and RDP del Pezzo surfaces with global vector fields, I will report on work in progress with DongSeon Hwang and Gebhard Martin on the classification of G-smooth RDP del Pezzo surfaces.
Nikolaos Tsakanikas
Primitive Enriques Varieties
In this talk, which will be based on joint works with F. Denisi, Á. D. Ríos Ortiz and Z. Xie, I will introduce the class of primitive Enriques varieties, whose smooth members are so-called Enriques manifolds. In particular, I will present the basic properties of these geometric objects as well as some examples of (smooth and singular) primitive Enriques varieties. I will also discuss a termination statement for Enriques pairs and various deformation-theoretic results concerning primitive Enriques varieties.
Aline Zanardini
A tale of three GIT problems
A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via Gale duality. To both nets, one can also naturally associate the same smooth plane quartic. In this talk, I will report on joint work with M. Hattori and T. Papazachariou, concerning a generalisation of this classical threefold cycle of correspondences. I will explain how, by extending these correspondences, one can obtain a complete criterion for GIT stability of the three underlying geometric objects using a birational-geometric method.
Vanja Zuliani
Toward the non commutative minimal model program
In this talk we will discuss some features of the Non commutative minimal model program proposed by Halpern-Leistner. In particular we will observe that the quantum cohomology of Grassmannians and cubic hypersurfaces in dimension 3 and 4 induce, via Bridgeland stability conditions, the standard semiorthogonal decompositions of the respective derived categories. This is joint work with T. Karube and A. Robotis.
