SMGA

Séminaire Méditerranéen de Géométrie Algébrique
Seminari Mediterrani de Geometria Algebraica
Seminario Mediterráneo de Geometría Algebraica

2023 Barcelona. 16-17 novembre

Thursday

• 15h15-16h15. Room iA. Vladimiro Benedetti, 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 \(\mathbb P^7\), 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 \(\mathrm{SU}_C(2)\) 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 \(\mathrm{SU}_C(C,O(p))\) 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 \(\mathrm{SU}_C(2,O(p))\), 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.
  
  Coffee break
  
  
• 17h00-18h00. Room B7. Joaquim Roé, On the boundary of the Mori cone of general blowups of the plane
  Let \(X_n\to \mathbb P^2\) 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.
  
  
• 18h00-19h00. Room B7. Michele Bolognesi , 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 \(\mathrm{SU}_C(3)\) of rank three semistable vector bundles on C with trivial determinant a double cover of \(\mathbb P^8\) 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 \(\mathrm{SU}_C(3)\). With some work, this property allows us to deduce that \(D_{Z_10}(v)\) is isomorphic to the odd moduli space \(\mathrm{SU}_C(3,O_C(c))\) 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.
  
  

Friday

• 9h30-10h30. Room iA. Naoufel Bouchareb, 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 \(H^1(X,E)/\mathrm{Aut}(E)\). 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.
  
  
• 10h30-11h30. Room iA. Thomas Dedieu, Extensions of hyperelliptic curves
  An extension of a curve C in \(\mathbb P^N\) is a surface S in \(\mathbb P^{N+1}\) such that C is a hyperplane section of S (or, more generally, an r-dimensional variety Y in \(\mathbb P^{N+r-1}\) 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.
  
  
• 12h00-13h00. Room iA or T2. Gian Pietro Pirola, The Hessian of a cubic hypersurface
  We study the Hessian determinant variety H of a smooth cubic hypersurface \(\mathcal C\) 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 \(\mathcal C\) 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 \(\mathcal C\) general. We compute the numerical invariants of Y. These results were obtained in collaboration with D. Bricalli and F. Favale.
  
  

• Michele Bolognesi (Université de Montpellier)
• Christian Pauly (Université Côte d'Azur)
• Martí Lahoz (Universitat de Barcelona)
• Laurent Manivel (CNRS, Université Paul Sabatier)
• Frédéric Mangolte (Aix-Marseille Université)

• Carlos D'Andrea (Universitat de Barcelona)
• Martí Lahoz (Universitat de Barcelona)
• Joan Carles Naranjo (Universitat de Barcelona)
• Irene Spelta (Centre de Recerca Matemàtica)