Workshop POITIERS 2022

Dates et lieu

Jeudi 10 et vendredi 11 mars 2022

Laboratoire de Mathématiques et Applications, Université de Poitiers, Poitiers.

 Programme 


   
 Jeudi 10 
 13h20-14h10    Andrea   A question on the real locus of smooth Fano threefolds
   14h20-15h10   Romain $K3$ surfaces with action of the Mathieu group M20 
  15h40-16h30  Thibaut   Extremal metrics on some toric fibrations 
  16h40-17h30    Liana   Combinatorial Reid's Recipe for Dimer Models  
 Vendredi 11   9h10-10h00 Sokratis   Rigid birational involutions of $\mathbb{P}^3$ and cubic $3$-folds 
  10h10-11h00  Adrien   Affine spaces and unipotent abelian groups compactifications
in Mori fibrations
  11h30-12h20  Susanna   Classifying birational involutions of the real plane 
  15h00-15h50   Boris/Enrica   Horospherical Sarkisov program 

Titres et résumés des exposés 

    
     
  • Extremal metrics on some toric fibrations (Thibaut Delcroix)

    To study the existence of extremal Kähler metrics on Hirzebruch surfaces, which are homogeneous fibers bundles in projective lines over the projective line, Calabi used the symmetries of these varieties. He restricted to metrics coming from the fiber and translated the property of extremal metrics into a simple ODE. This construction, known as Calabi's ansatz, has been widely generalized to allow both different bases and larger toric fibers. Recently, Simon Jubert proved a version of the YTD conjecture for some of these generalizations, the semisimple principal toric fibrations. Together, we have obtained easy to test sufficient conditions for uniform K-stability, and thus by Simon Jubert's results, for the existence of extremal metrics.

  •  $K3$ surfaces with action of the Mathieu group $M_{20}$ (Romain Demelle)

    It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a $K3$ surface is 960 and if such a group has order 960, then it is isomorphic to the Mathieu group $M_{20}$. In this talk, we are interested in projective $K3$ surfaces admitting a faithful symplectic action of the group M20. The aim will be to describe all of them and to understand when it is possible to construct explicit projective models for these surfaces. We will also see that it exists an infinite number of $K3$ surfaces admitting such an action of $M_{20}$.

  •  Compactifications d'espaces affines et de groupes abéliens unipotents dans les fibrations de Mori (Adrien Dubouloz)
     

    Déterminer quelles variétés de Fano lisses peuvent apparaître comme compactifications des espaces affines et classifier les paires correspondantes à isomorphisme près est un problème surprenamment difficile, dès la dimension 3. La situation se complique encore notoirement si l'on considère plus généralement des compactifications dans des espaces totaux de fibrations de Mori au-dessus d'une base de dimension positive. Une façon de "rigidifier" le problème dans l'optique de le rendre plus abordable consiste à considérer des compactifications qui sont équivariantes pour l'action transitive naturelle de l'espace vectoriel sous-jacent à l'espace affine. Le but de l'exposé est de donner dans un premier temps un petit panorama de ce qui est connu (ou pas) concernant ces compactifications équivariantes (ou pas) dans certaines classes de variétés de Fano ou de fibrations de Mori, puis de présenter quelques questions ouvertes et pistes de recherche en dimension 3 concernant plus spécifiquement les compactifications équivariantes dans des espaces totaux de fibrés projectifs au-dessus de surfaces de Hirzebruch.

  • A question on the real locus of smooth Fano threefolds (Andrea Fanelli)
     

    Smooth complex Fano threefolds have been fully classified by Iskovskikh and Mori-Mukai and their description is well understood and rather explicit. In this context it is natural to study the geometry of real smooth Fano threefolds and to investigate the connections between their algebraic properties and topological properties of their real loci. In this talk I will survey some known results and present a joint work in progress with Frédéric Mangolte, in which we investigate the connectedness of the real locus for smooth Fano threefolds.  

  •  Horospherical Sarkisov program (Enrica Floris)
     

    Let $Z$ be a uniruled variety. Then by BCHM a minimal model program on $Z$ ends with a Mori fibre space. By the work of Corti/Hacon-McKernan, two different outputs of the minimal model program on $Z$ are connected by a Sarkisov program. If $Z$ is moreover horospherical, then Pasquier proved that there is a "horospherical minimal model program" on $Z$. That is, there is a minimal model program with scaling, such that all the varieties involved are horospherical, the outcome is a horosperical variety and the MMP can be described using some combinatorial data. In this talk we will explain how two different outcomes of this "horospherical minimal model program" are connected by a "horospherical Sarkisov program" described via similar combinatorial data.

  •  Combinatorial Reid's Recipe for Dimer Models (Liana Heuberger)
     

    Reid's recipe is the equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of G-Hilb, i.e. a crepant resolution of $\mathbb{C}^3/G$ for $G\subset SL(3,\mathbb{C})$, with certain nontrivial irreducible representations of $G$. Our goal is to generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the G-Hilb case and its categorical counterpart known as Derived Reid’s Recipe. The result is a combinatorial version of the Ried's recipe algorithm via consistent dimer models, whose key ingredient is adapting Nakamura’s jigsaw transformations for G-Hilb to our context. This is joint work with Alastair Craw and Jesus Tapia Amador.

  •  Rigid birational involutions of $\mathbb{P}^3$ and cubic 3-folds (Sokratis Zikas)
     

    The Cremona group is the group of birational transformations of the projective n-space. The study of these groups dates back to the 19th century with some of the central questions still open. In the recent years new techniques, based on the Minimal Model Program, have been developed to answer some of these questions over the field of complex numbers. In this talk, using these techniques, I will explain how to construct families of birational involutions on the projective 3-space which do not fit in an elementary relation of Sarkisov links. Using these involutions, we can show that the Cremona group admits a free product structure. As corollaries, we re-prove its non-simplicity in an effective way and show that the group of automorphisms of the Cremona group is not generated by inner and field automorphisms. The same construction and counterpart results also hold for the group of birational automorphisms of a smooth cubic 3-fold.  

    •  
  •  Classifying birational involutions of the real plane (Susanna Zimmermann)
     

    Birational involutions of the complex plane had been classified up to conjugation by Bayle-Beauville 22 years ago, and they either have rational fixed curves or they fix a unique irrational curve. The first type of involutions are linearisable and for the latter type the conjugacy classes are in bijection with the isomorphism classes of the fixed curves. The classification over non-closed fields is still pending, but we already know that the classical result cannot hold. I want to explain what is known so far over the real numbers. This is work in progress with V. Cheltsov, F. Mangolte and E. Yasinsky.

Participants 

  • Samuel Boissière 
  • Thibaut Delcroix 
  • Romain Demelle 
  • Adrien Dubouloz 
  • Andrea Fanelli 
  • Enrica Floris 
  • Liana Heuberger 
  • Boris Pasquier 
  • Alessandra Sarti 
  • Ali Soidiki 
  • Ronan Terpereau 
  • Léa Villeneuve 
  • Sokratis Zikas 
  • Susanna Zimmermann 
 

Organisateurs

Enrica Floris (Université de Poitiers)   et   Boris Pasquier (Université de Poitiers)

Financement

Cette rencontre est soutenue par: