The Algebra and Geometry Seminar in Neuchâtel during Spring 2024 takes place each Monday, running from 1:00 PM to 2:00 PM. Here you can find an updated schedule with speakers and abstracts.

• 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.