Séminaire Méditerranéen de Géométrie Algébrique

1. Thursday 16, 15h-16h: Vladimiro Benedetti (Nice), The Coble quadric
   
   A very classical result by Coble states that the Kummer threefold of a general genus 3 curve C is singular along a unique quartic hypersurface in P^7, named thus the Coble quartic. By results of Beauville and Narasimhan-Ramanan the Coble quartic can be identified, via the theta map, with the moduli space SU_C(2) of semi-stable rank two vector bundles on C with trivial determinant. Recent results by Gruson-Sam-Weyman (GSW) showed that this quartic (and the Kummer) can be constructed from a general skew-symmetric four-form in eight variables. In fact, GSW's construction also shows that there exists a close relationship between some special representations of graded Lie algebras and moduli spaces of vector bundles on curves of small genus. In this talk we will see how the same skew-symmetric four-form also allows to explicitly construct, inside the Grassmannian G(2,8), the moduli space SU_C(C,O(p)) of semi-stable rank two vector bundles on C with determinant equal to O(p). Then we will extend Coble's result in this situation: we will see that, for generic p in C, there exists a unique quadratic section of G(2,8) which is singular exactly along SU_C(2, O(p)), and thus deserves to be coined the Coble quadric of the pointed curve (C,p). This is a joint work with Michele Bolognesi, Daniele Faenzi and Laurent Manivel.
   
   
2. Thursday 16, 16h45-17h45: Joaquim Roe (Barcelona), On the boundary of the Mori cone of general blowups of the plane
   
   Let X_n be the blowup of the plane at n points in very general position. If n>9, the shape of the Mori cone of X_n is expected to have a simple description as a consequence of the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, but relatively little has actually been proven. We will report on recent progress in this direction. This is joint work with C. Ciliberto and R. Miranda.
   
   
3. Thursday 16, 17h45-18h45: Michele Bolognesi (Montpellier), Odd determinant moduli spaces of vector bundles on a genus 2 curve
   
   This is the natural follow-up to Vladimiro's talk. Here we consider a smooth genus two curve C. The moduli space SU_C(3) of rank three semistable vector bundles on C with trivial determinant a double cover of P8 branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of C. Let V be a 9-dimensional complex vector space. Starting from a general trivector v in wedge^3(V), I will construct a Fano manifold D_{Z_10}(v) inside the Grassmannian G(3,9) as an orbital degeneracy locus. It turns out that D_{Z_10} naturally defines a family of Hecke lines in SU_C(3). With some work, this property allows us to deduce that D_{Z_10}(v) is isomorphic to the odd moduli space SU_C(3,O_C(c)) of rank three stable vector bundles on C with fixed effective determinant of degree one. As a side result, I will show that the intersection of D_{Z_10}(v) with a general translate of G(3,7) inside G(3,9) is a K3 surface of genus 19. This is joint work with V. Benedetti, D. Faenzi and L. Manivel.
   
   
4. Friday 17, 9h30-10h30, Naoufel Bouchareb (Marseille), Classification of affine bundles on the Riemann sphere
   
   First we will fix a holomorphic vector bundle E on a complex variety X and classify the affine fiber bundles A on X whose linearization is isomorphic to E. This set is in bijection with H^1(X,E)/Aut(E). We will also discuss a possible generalization to a much more general setting. We will look at the case where E is a rank two vector bundle on the Riemann sphere, and explain the link with the Jacobian ideal of a polynomial. Before tackling these points, we will provide the definitions needed for the talk. Time permitting, we will also discuss the problem using field theory.
   
   
5. Friday 17, 10h30-11h30, Thomas Dedieu (Toulouse), Extensions of hyperelliptic curves
   
   An extension of a curve C in P^N is a surface S in P^{N+1} such that C is a hyperplane section of S (or, more generally, an r-dimensional variety Y in P^{N+r-1} such that C is a linear section of Y). I will explain how extensions can be studied using ribbons over C, i.e., non-reduced schemes supported on C with the same shape as the first-infinitesimal neighbourhood of C in a surface extension. For a linearly normal hyperelliptic curve C of genus g and degree d at least 2g+3, I will give the classification of surface extensions of C, and the dimension of the projective space parametrizing ribbons over C. We will then see that every ribbon over C can indeed be realized as the first-infinitesimal neighbourhood of C in an extension if and only if d=2g+3. In this case there exists a universal extension of C, i.e., an extension Y of C of large dimension such that every surface extension of C is a linear section of Y. This is part of a more general program developed together with Ciro Ciliberto. I will concentrate on the hyperelliptic case which is fun to play around with.
   
   
6. Friday 17, 12h-13h, Gian Pietro Pirola (Pavia), The Hessian of a cubic hypersurface
   
   We study the Hessian determinant variety H of a smooth cubic hypersurface C of dimension n. The variety H has degree n+1 and it has been classically studied by many authors. We generalize some results of B. Segre and A. Adler. In particular, we generalize Beniamino Segre's result for surfaces by demonstrating that for n<5, H is normal if it is not Thom-Sebastiani (i.e. the equation of C has not separable variables). Then following Allan Adler we give a natural desingularization of H and study their singularities. Finally in the fourfold case, n=4, the singular locus Y of H is a smooth surface if C general. We compute the numerical invariants of Y. These results were obtained in collaboration with D. Bricalli and F. Favale.
   

1. Junyan Cao, Log ddbar lemma and some applications
2. Joao Pedro Dos Santos, Connexions relatives sur un schéma complexe
3. Frédéric Mangolte, Birational involutions of the real projective plane
4. Laurent Manivel, A birational involution for K3 surfaces of genus 10
5. Simone Marchesi, On stability of logarithmic sheaves and the Torelli problems
6. Eleonora Romano, C*-actions and Mori Dream Spaces

1. Arnaud Beauville, The Ceresa cycle
2. Marcello Bernardara, Fano de type K3 de dimension 4 : vers une classification ?
3. Jean-Paul Brasselet, Characteristic classes for singular algebraic varieties
4. Stéphane Druel, A global Weinstein splitting theorem for holomorphic Poisson manifolds
5. Pierre-Louis Montagard, Quelles adhérences d'orbites génériques du tore dans une variété de drapeaux sont-elles Gorenstein-Fano ?
6. Andrés Rojas, Chern degree functions

1. Thomas Dedieu, Extensions des courbes canoniques et applications gaussiennes
2. Antoine Etesse, Amplitude des fibrés vectoriels
3. Stéphane Lamy, Morphismes signatures sur le groupe de Cremona défini sur Q
4. Emmanuele Macri, Hypersurfaces cubiques de dimension quatre et une question de Hassett
5. Alan Thompson, Compactifications of the moduli space of K3 surfaces of degree 2
6. Jérémy Toulisse, Géométrie des représentations maximales en rang 2

1. Ana-Maria Castravet, Exceptional collections on moduli spaces of stable rational curves
2. Giulio Codogni, Positivity of the Chow-Mumford line bundle for families of K-stable klt Fano varieties
3. Lionel Darondeau, Hyperbolicité orbifolde
4. Gregoire Menet, Théorème de Torelli global pour les orbifoldes symplectiques irréductibles
5. Laurent Manivel, Sur les intersections complètes de Fano dans les espaces homogènes
6. Jean-Paul Mohsen, Techniques asymptotiques de Donaldson-Auroux et géométrie complexe

1. Michele Bolognesi, A variation on a theme of Faber and Fulton
2. Sylvain Brochard, La conjecture de de Smit : un nouveau critère de platitude
3. Julien Keller, J-flot et stabilité algébrique
4. Marti Lahoz, Surfaces K3 non commutatives et fourfolds cubiques
5. Federico Lo Bianco, Une application de l'intégration p-adique à la dynamique d'un automorphisme préservant une fibration
6. Laurent Manivel, Double spineurs de Calabi-Yau

1. Marian Aprodu, Ulrich bundles on projective surfaces
2. Marcello Bernardara, Mesures motiviques, décompositions semiorthogonales: applications en géométrie birationnelle
3. Francois Labourie, Métrique de pression et différentielles holomorphes
4. Boris Pasquier, Programmes des modèles minimaux pour les variétés horosphériques
5. Carlos Simpson, Systèmes locaux sur P1 privé de 5 points
6. Fabio Tanturri, Sur l'unirationalité des espaces de Hurwitz

1. Michele Bolognesi, Surfaces abéliennes et thêta caractéristiques
2. Sylvain Brochard, Dualité champêtre et applications
3. Julien Grivaux, Approximation au premier ordre des auto-intersections dérivées générales
4. Adam Parusinski, Filtration par le poids pour les variétés algébriques réelles
5. Christian Pauly, Variétés de Prym non-abéliennes et fonctions thêta généralisées
6. Xavier Roulleau, Construction de surfaces rigides avec K^2=2c_2=8 et q=p_g=2