Institut de mathématiques

Séminaire d’algèbre et de géométrie de Neuchâtel

Le séminaire a eu lieu chaque semaine dans la salle B217 du bâtiment des sciences.

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AUTOMNE 2025

18.9.2025
Anne Schnattinger (Neuchâtel)
Blow-ups of a quadric in P^4 along a curve yielding weak Fano threefolds
In this talk, we will examine the question of when the blow-up of a smooth quadric hypersurface in P^4 along a smooth, irreducible curve is weak Fano. We are able to give a complete answer depending only on some conditions on the geometry of the curve, particularly its degree and genus. We will introduce the notions, most importantly those of (weak) Fano threefolds, that are necessary in order to formulate the corresponding result and give the main ideas of its proof.

25.9.2025
Elias Kurz(Neuchâtel)
Birational Geometry of sexctic del pezzo surfaces
In this talk we look at non-rational del Pezzo surfaces of degree 6 and Picard rank 1 which are defined over a perfect field. We give a parametrization of their isomorphy classes and find a criteria to determine, which of them are birational to each other. We further study group homomorphisms from their group of birational transforms and give an example of a solid del Pezzo surface with infinite pliability. Joint work with Egor Yasinsky.

2.10.2025
Emelie Arvidsson (Neuchâtel)
Vanishing theorems in positive-characteristic birational geometry
I will discuss Kodaira vanishing and its generalization, the Kawamata–Viehweg vanishing theorem. Kodaira vanishing fails in general in positive characteristic, already for smooth surfaces. I will present special situations in which these theorems still hold for certain classes of surfaces, and explain how such cases can sometimes be leveraged to recover birational versions of vanishing for threefolds. I will also highlight why these results matter for the minimal model program. The talk will survey key themes in my research to date and aims to spark interest in these topics within the group.

9.10.2025
Stéphane Lamy (Toulouse)
Subgroups of plane polynomial automorphisms
I will discuss a classification of the subgroups of the group Aut(A^2) of polynomial automorphisms of the affine plane, over an arbitrary ground field. The main tool is an action of this group on a simplicial tree, coming from a structure of amalgamated product.

16.10.2025
Quentin Posva (Neuchâtel)
Moduli and singularities of varieties in positive characteristic
The aim of this talk is to introduce the audience to some challenging problems that I like, and hopefully to spark interest in some of them. I will start with a very brief review of the moduli theory of curves: this is by now a classical story, which provides a nice introduction to the moduli questions. I will also review the KSBA moduli theory, which is a successful theory for varieties of higher dimensions in characteristic 0. The main part of the talk will be about moduli questions in positive characteristic: I will explain the kinds of problems that appear, present some pathological examples one can play with, and sketch some ideas to overcome these difficulties.

20.10.2025 (lundi!)
Francesca Carroci (Roma)
Correlated GW invariants and DR cycles
In a joint work with T. Blomme we introduced a geometric refinement for log Gromov -Witten invariants of P^1-bundles on smooth projective varieties. I will introduce the correlated GW invariants and then try to explain how to compute them using a refinement of Pixton double-ramification cycle formula with target varieties. This is all joint work with T. Blomme.

23.10.2025
Paula Truöl (MPIM Bonn)
Non-complex cobordisms between quasipositive knots
Quasipositive knots occur in complex geometry as the transverse intersections of smooth algebraic curves in the complex affine plane with the 3-sphere. A complex cobordism is a surface that arises as the transverse intersection of such a smooth algebraic curve with the region bounded by two 4-balls of different radius with a common center. The two knots bounded by a complex cobordism are necessarily quasipositive, and every complex cobordism is necessarily optimal (defined in the talk). In 2016, Feller asked whether these two necessary conditions are also sufficient for the existence of a complex cobordism between two knots. In a joint work with Maciej Borodzik, we prove that they are not, for cobordisms of any genus. For genus 0, we extend our result to strongly quasipositive knots. In the talk, we will define the relevant terms and provide some context for our results.

6.11.2025
Anestis Tzogias (Neuchâtel)
Length-metric codes
We will talk about length-metric codes, a new variant of error-correcting code that we developed as an algebraic proxy for submodule codes used in physical-layer network coding, which were introduced by Gorla and Ravagnani. We will briefly summarise the history and importance of error-correcting codes in information theory, using perhaps the most well-known error-correcting codes, namely Hamming codes, as an example. We will then mention rank-metric codes and use the latter two examples to motivate some of the main problems of coding theory. Finally, we will introduce length-metric codes, which not only model submodule codes but generalise rank-metric codes. In particular, we will discuss code equivalence, optimal codes and local-to-global arguments in the length-metric. This is joint work with Elisa Gorla.

13.11.2025
Antoine Pinardin (Basel)
Cremona groups, quadric threefolds, and the icosahedron
The icosahedral group A(5) is the only non-cyclic finite simple group that occurs in the complex Cremona group in every dimension. Its actions carry substantial interest in birational geometry, and a book from Cheltsov and Shramov was dedicated to the study of its conjugacy classes in Cr(3,C). While describing them entirely is currently out of reach, a huge and more realistic step forward would be to describe all the non-linearizable actions of A(5) on minimal rational threefolds. We performed this work in the surprisingly open case of smooth quadrics, gave a full answer to the linearization problem for A(5)-actions on these varieties, and, in the non-linearizable case, gave explicitly all the minimal models to which they are equivariantly birational. I will present this project, carried out in collaboration with Zhijia Zhang.

20.11.2025
Claudia Stadlmayr (Neuchâtel)
Infinitesimal symmetry in algebraic geometry: group schemes and del Pezzo surfaces
Group schemes provide a refined notion of symmetry in positive characteristic: they detect infinitesimal structure invisible to the discrete automorphism group. Classical examples such as mu_p or alpha_p equip the trivial topological space with a non-trivial algebraic structure.
In this talk I will explain how this perspective can be used to classify weak and RDP del Pezzo surfaces admitting global vector fields, and how phenomena unique to small characteristic – such as non-lifting vector fields on rational double point singularities (RDPs) – can be illuminated using the group-scheme framework.
If time permits, I will outline applications and ongoing projects: towards higher-dimensional Fano varieties with infinite automorphism groups and (equivariant) compactifications of the affine plane.

27-28.11.2025
BENDZ SEMINAR (Basel-EPFL-Neuchâtel-Dijon-Zürich seminar in algebraic geometry)

4.12.2025
Simone Coccia (Basel)
Density of integral points on character varieties
Given a smooth complex algebraic variety Y and an algebraic group G, a (relative) G-character variety is a moduli space of G-local systems with specified behavior at the boundary of a compactification of Y. A well-known example of SL_2-character varieties are Markoff type cubic surfaces, and in recent years the study of their integral points has attracted much attention, starting with the work of Bourgain, Gamburd and Sarnak. In my talk I will present joint work with Daniel Litt where we prove that, for any algebraic variety Y, integral points are potentially Zariski dense in the relative character varieties parametrizing SL_2-local systems on Y with fixed algebraic integer traces along the boundary components. The proof first treats the case of Y a Riemann surface, where we construct one initial integral point and then produce a Zariski dense set of them by exploiting the dynamics of the mapping class group action on the character variety. The case of general Y is reduced to the Riemann surface case by using work of Corlette and Simpson.

11.12.2025
Aurore Boitrel (Aix-Marseille)
Automorphism groups of real rational Del Pezzo surfaces of degree 4
Del Pezzo surfaces and their automorphism groups play a key role in the classification up to conjugacy of subgroups of the Cremona group of the plane. Over an algebraically closed field, they are completely classified together with their automorphism groups. In this talk, we will focus on real rational Del Pezzo surfaces of degree 4. Unlike larger degrees, the degree 4 case involves an infinite moduli space of surfaces, already over the complex numbers. We will explain how studying the actions of automorphisms and of the Galois group on the conic bundle structures enables us to give a complete description of their automorphism groups by generators in terms of automorphisms and birational automorphisms.

18.12.2025
Ronan Terpereau (Lille)
Equivariant automorphism group and real forms of complexity-one varieties
Let G be a connected reductive algebraic group over a field of characteristic zero. I will discuss the representability of equivariant automorphism groups of certain G-varieties. More precisely, for a broad class of G-varieties known as complexity-one G-varieties, I will explain that this group is representable by a group scheme locally of finite type, and even by a linear algebraic group in the almost homogeneous case. Finally, using an exact sequence description of the equivariant automorphism group, I will show that most complexity-one G-varieties admit only finitely many real forms. This is joint work with Giancarlo Lucchini Arteche (arXiv:2507.18475).

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PRINTEMPS 2025
18.2.25
Olivier Benoist (CNRS-ENS)
On the rationality of real conic bundles
Deciding whether a given algebraic variety is rational (birational to projective space) is an important problem in algebraic geometry. Over the field of real numbers, this problem is particularly interesting for varieties that are known to be rational over the complex numbers, as it then has an arithmetic flavour. In this talk, I will review the techniques that are available, and focus on concrete examples of real conic bundles for which I will provide new positive and negative results. This is joint work with Alena Pirutka.

4.3.25
Gebhard Martin (Bonn)
Classification of non-F-split del Pezzo surfaces
Among all smooth cubic surfaces in P^3, there is a unique one without three lines forming a triangle: The Fermat cubic in characteristic p = 2. Coincidentally, this is also the unique non-F-split smooth cubic surface. By work of Hara, non-F-split del Pezzo surfaces exist only in degrees 3, 2, and 1. After describing the classification of non-F-split del Pezzo surfaces of degree 2 due to Saito, I will report on joint work with Réka Wagener in which we give a geometric characterization of non-F-split del Pezzo surfaces of degree 1.

1.4.25
Alain Valette (Neuchâtel)
Reciprocal hyperbolic elements in PSL_2(\Z)
An element A in PSL_2(\Z) is hyperbolic if |Tr(A)|>2. The maximal virtually abelian subgroup of PSL2(\Z) containing A is either infinite cyclic or infinite dihedral; say that A is reciprocal if the second case happens (A is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in PSL_2(\Z) in terms of the continued fractions of their fixed points in P^1(\R) (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke & Klein.

8.4.25
Lukas Lewark (ETHZ)
The joy of not being a PID
Two knots (circles in 3-space) are called concordant if they form the boundary of a cylinder in 4-space. Concordance classes of knots constitute an abelian group C called the concordance group, which has been one major focus of knot theory since the 1960s. In this talk, we will construct a homomorphism (based on Khovanov homology) from C to another abelian group G. This latter group G consists of chain complexes (with coefficients in a ring R) modulo a certain equivalence relation. With R a PID, G is just infinite cyclic; but with R a non-PID, G may have infinite rank, and so – oh joy – the homomorphism from C to G may be richer in information about C.

This talk will be aimed at an audience with no background knowledge in knot theory and Khovanov homology.

6.5.25 Marco Golla (Nantes)
Alexander polynomials and symplectic curves in CP^2
Libgober defined the Alexander polynomial of a (complex) plane
projective curve and showed that it detects some Zariski pairs of
curves: these are curves with the same degree and the same singularities
but with non-homeomorphic complements. He also proved that the Alexander
polynomial of a curve divides the Alexander polynomial of its link at
infinity and the product of Alexander polynomials of the links of its
singularities. We extend Libgober’s definition to the symplectic case
and prove that the divisibility relations also hold in this context.
This is joint work with Hanine Awada.

20.5.25
Sokratis Zikas (IMPA)
On Mori Dreamness of blowups along space curves
Mori Dream Spaces are a special kind of varieties introduced by Hu and Keel that exhibit an ideal behaviour from a birational point of view. While Mori Dreamness is a very desirable property, generally, it is not preserved by even the simplest birational maps: blowups. In this talk we study Mori Dreamness of blowups along space curves: we provide various sufficient criteria as well obstructions to the blowup being a Mori Dream Space. We also study how this property behaves while varying the curve in the corresponding Hilbert scheme and show that it is neither an open nor a closed condition. Furthermore we exhibit examples of Hilbert schemes whose general element does not give rise to a Mori Dream Space, while special elements do, and vice versa.
This is joint work in progress with Tiago Duarte Guerreiro

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AUTOMNE 2024

• 16 Septembre 2024 (exceptionnellement le lundi à 11h): Dhruv Ranganathan (Cambridge) – A story of
  degenerations in enumerative geometry.The basic principle of degeneration has been present in enumerative geometry and
  intersection theory since their origins in the 19th century. But with the development of Gromov-Witten theory in
  the 1990s, and more generally the systematic study of moduli, the idea grew into a robust set of techniques that
  can be used to control the geometry of moduli spaces. I’ll outline the basic ideas, and then highlight some of
  the successes of the last few years, including connections to tropical geometry, the double ramification cycle,
  the cohomological study of the moduli space of curves.
• 24 Septembre 2024: Fabio Bernasconi (Université de Neuchâtel) – Geometry of 3-dimensional del
  Pezzo fibrations in positive characteristic.A 3-dimensional del Pezzo fibration \(X \to C\) is one of the possible outcomes of the MMP, and one hopes to study
  the birational geometry of \(X\) in terms of the generic fibre \(X_{k(C)}\) and the base curve \(C\). In
  positive characteristic, an additional difficulty arises in this study: the generic fibre \(X_{k(C)}\) is defined
  over an imperfect field, where regularity is a weaker property than smoothness. In recent works with
  Fanelli-Schneider-Zimmermann and Tanaka, we study these extravagant cases and extend two classical results due
  to Enriques, Manin, and Iskovskikh: a 3-dimensional del Pezzo fibration over a curve, defined over an
  algebraically closed field, always admits a section, and the total space is rational if the base curve is
  rational and the anticanonical degree of a fibre is at least five.
• 01 Octobre 2024: Thomas Bouchet (Université Côte d’Azur) – Courbes de genre 4:
  invariants et reconstruction.Dans cet exposé, nous allons introduire des invariants algébriques pour les courbes non-hyperelliptiques de
  genre 4, qui caractérisent les classes d’isomorphismes de ces courbes sur un corps algébriquement clos de
  caractéristique nulle.
  Après avoir exposé quelques outils de théorie classique des invariants qui nous ont permis de résoudre ce
  problème, nous donnerons les idées des preuves des principaux résultats.
  Enfin, nous expliciterons un algorithme qui, donné une liste d’invariants, reconstruit une courbe
  non-hyperelliptique de genre 4 qui possède ces invariants.
  Cet algorithme généralise celui de Mestre pour les formes binaires, et il fonctionne dans un cadre plus
  général que celui de l’exposé.
• 08 Octobre 2024: Simon Machado (ETH Zurich) – Brunn-Minkowski in \(SO_n(\mathbb{R})\)Given a subset \(A\) of a group, the doubling constant is the ratio \(\frac{|A^2|}{|A|}\), where \(A^2\)
  denotes the set of products of two elements of \(A\) and \(|A|\) is a notion of size appropriate to the setting (e.g.
  number of points, volume, covering number, etc). The study of doubling, a part of additive combinatorics, has
  found applications in number theory, dynamics, probability theory, and geometric analysis.
  In this talk, I will focus on doubling in Lie groups with respect to the Haar measure. While doubling is well
  understood in Euclidean spaces, it becomes more mysterious in other cases. For example, a conjecture by
  Breuillard and Green predicting a lower bound for doubling in \(SO_n(\mathbb{R})\) remained out of reach until
  recently. We will discuss a proof of this result and its implications, including a Brunn-Minkowski type
  inequality and a stability result.
• 15 Octobre 2024: Ilia Itenberg (Sorbonne Université) – Refined invariants for real curves.The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the
  projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed
  enumeration of real algebraic curves of genus 1 and 2. These invariants admit a refinement (according
  to the quantum index) similar to the one introduced by Grigory Mikhalkin in the genus zero case. It
  turns out that two different rules of signs in the enumeration surprisingly lead to the same collection of
  refined invariants. The proof of the latter fact uses the tropical counterparts of the invariants
  under consideration.
  This is a joint work with Eugenii Shustin.
• 22 Octobre 2024: Giuseppe Melfi (Université de Neuchâtel) – On a problem of Erdös about sums of distinct powers of integers.This is a joint work with Maximilian F. Hasler. In this talk, I will present a new lower estimate of the
  counting function of the positive integers that are sums of distinct powers of 3 and 4. This problem was
  raised by Erdős in 1996. I will also present some possible generalizations for related problems.
• 29 Octobre 2024: Pas de séminaire – ColboisFest.
• 05 Novembre 2024: Marc Abboud (Université de Neuchâtel) – On the dynamics of endomorphisms of affine surfaces.Let \(K\) be an algebraically closed field, an affine surface \(S\) over \(K\) is a surface given by
  polynomial equations in \(K^n\) for some n. An endomorphism \(f\) of \(S\) is a polynomial transformation of
  \(K^n\) which preserves \(S\). One of the first dynamical invariant that one can define for such
  transformations is the dynamical degree. For polynomial maps of \(K^2\), it is defined as the n-th power of
  the degree of the formulas of f^n. Favre and Jonsson in 2007 showed that the dynamical degree of polynomial
  maps of \(\mathbb C^2\) is an algebraic integer of degree \(\leq 2\). We generalise their result to any affine
  surface. In this talk, I will give several examples of affine surfaces and explain the main tool of the proof
  of our result: we study the dynamics of \(f\) on a space of valuations associated to \(S\) and deduce results
  on the dynamics of \(f\) on \(S\).
• 12 Novembre 2024: Thomas Blomme (Université de Neuchâtel) – Correlated Gromov-Witten
  invariants (joint with Francesca Carocci).In this talk we introduce a geometric refinement of Gromov-Witten invariants for \(\mathbb P^1\)-bundles
  relative to the
  natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants.
  We’ll give certain invariance properties of the correlated invariants and provide some computations in the
  case of \(\mathbb P^1\)-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration
  formula for reduced Gromov-Witten invariants for abelian and \(K3\) surfaces.
• 19 Novembre 2024: Gabriel Dill (Université de Neuchâtel) – The modular support problem.In 1988, Erdös asked: let \(a\) and \(b\) be positive integers such that for all \(n\), the set of primes
  dividing \(a^n – 1\) is equal to the set of primes dividing \(b^n – 1\). Is \(a = b\)? In fact, work of
  Schinzel from 1960 already yields an affirmative answer to this question and even more generally that, if
  every prime dividing \(a^n – 1\) also divides \(b^n – 1\), then \(b\) is a power of \(a\).
  In joint work that was inspired by the analogy between roots of unity and singular moduli, i.e.
  \(j\)-invariants of elliptic curves with complex multiplication, Francesco Campagna and I have studied this
  so-called support problem with the Hilbert class polynomials \(H_D(T)\) instead of the polynomials \(T^n –
  1\), both over number fields and over function fields.
• 26 Novembre 2024: Jan Draisma (Université de Bern) – Subrank of bilinear maps.The subrank of a bilinear map \(f:U \times V \rightarrow W\) measures how many independent scalar multiplications can
  be performed by
  using \(f\) once. Both subrank and an approximate version called border subrank play a central role in Strassen’s work on
  complexity theory.I will discuss recent work by Derksen-Makam-Zuiddam and Pielasa-Šafránek-Shatsila, which determines the subrank of any
  sufficiently general \(f\); and joint work with Biaggi-Chang-Rupniewski in which we use a generalisation of the
  Hilbert-Mumford criterion to determine the asymptotics of the generic border subrank.
• 03 Décembre 2024: Marvin Hahn (Trinity college, Dublin) – Tropical combinatorics of the 2D Toda lattice.The study of soliton solutions of the KP hierarchy is a classical topic. In seminal work of Kodama and
  Williams a tropical geometry approach to their study was introduced. More precisely, it was proved that the
  phase structure of smooth solutions of the KP hierarchy is described by a tropical curve that naturally arises
  from combinatorial decompositions of the Grassmannian. The 2D Toda lattice is a generalisation of the KP
  hierarchy. In this talk, we show that the tropical approach introduced by Kodama and Williams generalises to
  the 2D Toda lattice answering a question posed by Kodama. This is talk is based on joint work in progress with
  Betancourt, Posch and Reda.
• 10 Décembre 2024 (à 16h au lieu de 13h, en salle E213): Emanuele Delucchi (SUPSI) – On
  the \(K(\pi,1)\)-problem for abelian arrangements.The study of arrangements of hypersurfaces is a nowadays classical field that was spurred by work of Arnol’d,
  Brieskorn and Deligne from the Seventies on arrangements of linear hyperplanes, configuration spaces and Artin
  groups of finite type. A hallmark of the classical theory is the strong interaction between geometry, algebra
  and combinatorics. This field recently broadened its scope beyond the linear case to include arrangements in
  the torus, in products of elliptic curves and, more generally, in connected Abelian Lie groups. The classical
  \(K(\pi,1)\)-problem, i.e. deciding asphericity of an arrangement’s complement in combinatorial terms, can be
  stated in general. In this talk I will review some of the classical history of this problem and present some
  recent advances in the non-linear case. The talk will report on joint work with Christin Bibby, Alessio D’Alì,
  Noriane Girard, Giovanni Paolini, Sonja Riedel.
• 17 Décembre 2024: Pas de séminaire – Conférence Géométrie des groupes et groupes de
  Cremona.
• 18 Février 2025: Olivier Benoist (École Normale Supérieure) – On the rationality of real
  conic bundles.Deciding whether a given algebraic variety is rational (birational to projective space) is an important
  problem in algebraic geometry. Over the field of real numbers, this problem is particularly interesting for
  varieties that are known to be rational over the complex numbers, as it then has an arithmetic flavour. In
  this talk, I will review the techniques that are available, and focus on concrete examples of real conic
  bundles for which I will provide new positive and negative results. This is joint work with Alena Pirutka.

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PRINTEMPS 2024

• 19 February 2024: Stefan Schroer (Dusseldorf), The inverse Galois problem for group schemes.
  Abstract:  Let G be a connected algebraic group. We show that there is some projective scheme where the connected part of the automorphism group scheme is isomorphic to G. We also discuss further results in the special case that G is twisted form of the additive group in positive characteristics, or more generally a twisted form of some vector group. This is joint work with Michel Brion. 

• 26 February 2024: Matthew Satriano (Waterloo),  Galois closures and small components of the Hilbert schemes of points

Abstract: Manjul Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal.

• 4 March 2024: Frédéric Mangolte (Marseille), Comessatti’s Theorem on Rational Surfaces and Real Fano threefolds.
  Abstract: From the classification of real rational surfaces worked out by Comessatti at the beginning of the 20th century we get the following striking characterization of real rational surfaces: a geometrically rational real surface is rational if and only if its real locus is non-empty and connected. In a work in progress with Andrea Fanelli, we explore real loci of geometrically rational Fano threefolds and study the rationality of these.

• 25 March 2024: Lisa Seccia (Neuchâtel), F-singularities of ladder determinantal varieties
  Abstract: Ladder determinantal varieties are defined by the vanishing of certain collections of minors in a matrix of indeterminates. These varieties were first introduced to investigate the singularities of Schubert varieties, and have since inspired further study due to their rich algebraic structure. In this talk, we will focus on their F-singularities, i.e., singularities defined by the Frobenius map in prime characteristic. After a brief overview of some basic concepts of F-singularity theory, we will see different algebraic techniques to prove that ladder determinantal varieties are F-pure for every p>0. This is joint work with De Stefani, Montaño, Nuñez-Betancourt, and Varbaro.

• 8 April: Matteo Ruggiero (Jussieu), On the Dynamical Manin-Mumford problem for planar polynomial endomorphisms.
  Abstract: Let X be a projective variety. The Dynamical Manin-Mumford problem consists in classifying the pairs (Y,f), where Y is a subvariety and f is a polarized endomorphism of X, such that Preper(f|_Y) is Zariski-dense in Y. In a joint work with Romain Dujardin and Charles Favre, we solve this problem when f is a regular endomorphism of P^2 coming from a polynomial endomorphism of C^2 of degree d>=2, under the additional condition that the action of f at the line at infinity doesn’t have periodic superattracting points.

• 15 April: Egor Yasinsky (Bordeaux), Birational rigidity of Fano varieties.
  Abstract:  I will discuss the notion of birational (super)rigidity, giving many examples in an equivariant setting and over non-closed fields. Then I will speak about one old question of János Kollár.  

• 22 April: Quentin Posva (Dusseldorf), Singularities of 1-foliations in positive characteristic.
  Abstract:  Over varieties in positive characteristic, 1-foliations are modules of derivations that encode factorization of the Frobenius morphism. They have been studied by several authors in the context of birational geometry, either as tools for reduction mod p arguments, or for producing pathological examples by ways of quotients. In this talk, I will report on my recent work on singularities of 1-foliations from a char p>0 point of view:

1) I will explain that quotients by reasonable classes of 1-foliations preserve MMP-singularities and F-singularities;

2) I will present a 4-dimensional quotient that is a locally stable morphism with non-weakly-normal central fiber;

3) If time permits, I will explain how methods from resolution of varieties can be applied to resolve 1-foliations.

• 29 April: Samuel Boissière (Poitiers), The Fano variety of lines on singular cyclic cubic fourfolds
  Abstract: In the framework of the compactification of the moduli spaces of prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds, a key question is to understand the geometry of limit automorphisms. I will present recent results in this direction, using symplectic resolutions of Fano varieties of lines on singular cyclic cubic fourfolds. In my talk, I will focus on the K3 surfaces whose geometric properties are at the heart of the understanding of the limit automorphisms in suitable moduli space parametrizing pairs of IHS manifolds with automorphism. These results have been obtained in collaboration with Chiara Camere, Paola Comparin, Lucas Li Bassi and Alessandra Sarti.

• 13 May: Alapan Mukhopadhyay (EPFL), Generators of bounded derived categories using the Frobenius map.
  Abstract: Since the appearance of Bondal- van den Bergh’s work on the rep- resentability of functors, proving existence of strong generators of the bounded derived category of coherent sheaves on a scheme has been a central problem. While for a quasi-excellent, separated scheme the existence of strong generators is established, explicit examples of such generators are not common. In this talk, we show that explicit generators can be produced in prime characteristics using the Frobenius pushforward functor. As a consequence, we will see that for a prime characteristic p domain R with finite Frobenius endomorphism, R1/pn – for large enough n- generates the bounded derived category of finite R-modules. This recovers Kunz’s characterization of regularity in terms of flatness of Frobenius. We will discuss examples indicating that in contrast to the affine situation, for a smooth projective scheme whether some Frobenius pushforward of the structure sheaf is a generator, depends on the geometry of the underlying scheme. Part of the talk is based on a joint work with Matthew Ballard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.

• 27 May: Massimiliano Mella (Ferrara), A special rational surface.
  Abstract: Two projective varieties are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decade Cremona equivalence has been investigated widely and we have now a complete theory for non divisorial reduced schemes. The case of irreducible divisors is completely different and not much is known beside the case of plane curves and few classes of surfaces. In particular, for plane curves it is a classical result that an irreducible  plane curve is Cremona equivalent to a line if and only if its log-Kodaira dimension is negative.  This can be interpreted as the log version of Castelnuovo rationality criterion for surfaces. One expects that a similar result for surfaces in projective space should not be true, as it is false the generalization in higher dimension of Castelnuovo’s Rationality Theorem. In this talk I will provide an example of such behaviour exhibiting a surface in the projective space with negative log-Kodaira dimension which is not Cremona equivalent to a plane, this can be thought of as  sort of log Iskovkikh-Manin, Clemens-Griffith, Artin-Mumfurd example.