Séminaire Méditerranéen de Géométrie Algébrique 6

Montpellier, 25–26 mars 2020


In view of the international audience of the seminar, the recommended language for the talks is English

  • Hanine Awada : «New Rational Fibered Cubic Fourfolds»

    The rationality problem of smooth cubic hypersurfaces of dimension four is one of the most challenging open problem in algebraic geometry. It is expected that a very general smooth cubic fourfold should be nonrational. Until now only examples of smooth rational cubic fourfolds are known. They are "special", that is cubic fourfolds containing a surface which is not homologous to a complete intersection, and have "associated K3 surface", that means hat the Hodge structure on the non special cohomology of cubic fourfolds is essentially the Hodge structure on the primitive cohomology of a K3 surface.
    Special cubic fourfolds form a countably infinite union of irreducible families 𝒞d in the moduli space 𝒞 of cubic fourfolds. Only few divisors 𝒞d have been explicitely described in terms of particular surfaces contained in the cubic fourfolds. For example, 𝒞8 is defined as the locus of smooth cubic fourfolds containing a plane and 𝒞18 the one containing an elliptic ruled surface. To these two particular classes of cubic fourfolds, one can associate a fibration, respectively in quadrics and in del Pezzo sextics. The rationality of these fibered cubic fourfolds is strongly related to the existence of rational sections of the associated fibration. By intersecting these divisors with other ones whose elements are known to be rational, we provide explicit description of these intersections in terms of irreducible components and we exhibit new examples of rational fibered cubic four folds such that the associate fibration has no rational sections.
    In my talk, I will introduce first some definitions and some Hodge and Lattice theory related to fourfolds, crucial for studying the intersection of divisors 𝒞d in 𝒞. Then I will explicit the cases of 𝒞18 and 𝒞8 by studying its intersection with specific divisor and show the existence of some rational fibered cubic fourfolds such that the associated fibration has no section.

  • Benoît Cadorel : «Generalized algebraic Morse inequalities and jet differential equations»

    Since their introduction by Green and Griffiths in 1979, jet differential equations have become a fundamental tool for studying entire curves on complex manifolds. The existence of sufficiently many holomorphic differential equations a given manifold implies strong restrictions on the geometry of these curves: these restrictions have permitted to establish strong hyperbolicity results for many classes of manifolds, such as high degree hypersurfaces, the complements of these hypersurfaces, projective complex surfaces of general type… The main problem is then to be able to produce such differential equations on the manifolds under study.
    The most general result in this direction is due to Demailly, who proved that any complex projective manifold of general type admits "many" jet differential equations of high order. Demailly's proof is essentially analytic in nature: it relies on the use of metric methods, and in particular on his famous holomorphic Morse inequalities. We will present here an entirely algebraic proof of Demailly's theorem: the main tool in our proof will be the use of new algebraic Morse inequalities, generalizing in particular the previous versions due to Siu, Demailly and Angelini.

  • Carlos D'Andrea : «Degree and birationality of multi-graded rational maps»

    We present a new algebra called "saturated special fiber ring", based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of a rational map. This algebra turns out to be a fundamental tool to analyze the degree of a rational map, and allows us to give formulae and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties.
    This is joint work with Laurent Busé (INRIA Sophia-Antipolis) and Yairon Cid Ruiz (Max Planck Leipzig).

  • Henri Guenancia : «Quotient of bounded domains and their subvarieties»

    I will explain a recent joint work with B. Cadorel and S. Diverio where one studies the birational geometry of subvarieties of quotients X=Ω/Γ where Ω is a a bounded domain in ℂn and Γ is a discrete subgroup of Aut(X) acting properly discontinuously—one will not assume that Γ is co-compact or acts freely.

  • Thomas Baier : «Hitchin's connection from an algebraic-geometric perspective»

    Hitchin's connection, originally constructed using techniques of Kähler geometry, is a flat projective connection on bundles of non-abelian theta functions. In this talk we will discuss its construction in algebraic geometry, and notably also for ground fields of (almost any) positive characteristic. Time permitting we will also discuss applications to higher rank Prym varieties.
    This is joint work with M.Bolognesi, J.Martens and C.Pauly.

  • Jorge Pereira : «Symmetries of codimension one foliations»

    In the first part of the talk, I will review what is known about the group of birational symmetries of foliations and webs on projective surfaces. The second part will be devoted to report on a work in progress (joint with Lo Bianco, Rousseau, and Touzet) about symmetries of codimension one foliations on higher dimensional projective manifolds.


The talks take place in room 430 of the IMAG building



Thomas Baier

Carlos D'Andrea


Henri Guenancia



Benoît Cadorel

Hanine Awada


Jorge Pereira
(séminaire AGATA)




  • Carlos D'Andrea
  • Martí Lahoz


  • Erwan Rousseau
  • Antoine Etesse
  • Jorge Pereira
  • Xavier Roulleau


  • Damien Calaque
  • Hanine Awada
  • Claude Cibils
  • Lionel Darondeau
  • Michele Bolognesi
  • Pierre-Louis Montagard
  • Sylvain Brochard


  • Cécile Gachet
  • Christian Pauly
  • Andreas Höring
  • Zakaria Ouaras
  • Phan Quoc Bao Tran


  • Marcello Bernadara
  • Dominique Mattei
  • Henri Guenancia
  • Laurent Manivel
  • Thomas Dedieu


  • Thomas Baier
  • Benoît Cadorel
  • Johan Martens
  • GDR
    nous remercions pour son soutien financier
    le Groupement De Recherche n°3064 du CNRS
    Géométrie Algébrique et Géométrie Complexe
  • IMUB
    Institut de Matemàtica
    Universitat de Barcelona
  • I2M
    Institut de Mathématiques de Marseille
    Aix-Marseille Université - CNRS
  • IMAG
    Institut Montpelliérain Alexander Grothendieck
    Université de Montpellier - CNRS
  • LJD
    Laboratoire J.A. Dieudonné
    Université Côte d'Azur - CNRS
  • IMT
    Institut de Mathématiques de Toulouse
    Université de Toulouse - CNRS

Practical informations

See the map below for adresses.

The talks take place in room 430 of the IMAG building: building 9 on Campus Triolet (campus map).

Lunches will be taken at Restaurant Administratif, on Campus Triolet. We will go together by foot.

Participants supported by Univ. Montpellier will stay at Ulysse Hotel (website).

The social dinner will be on wednesday evening at Brasserie du Théâtre (website).