Coimbra Salamanca

Algebraic Geometry Seminar

I Meeting - Coimbra, 29th and 30th of January 2010

Participants:
André Gama Oliveira (UTAD); Alberto López Martín (Zürich); Ana Cristina Casimiro (FCTUNL); Ana Cristina López Martín (Salamanca); Beatriz Grana Otero (Salamanca); Carlos Rito (UTAD); Daniel Hernández Serrano (Salamanca); Darío Sánchez Gómez (Salamanca); Eduardo Dias (Warwick); Esteban Gomez Gonzalez (Salamanca); Francisco José Plaza Martín (Salamanca); Filippo Viviani (CMUC/Roma Tre); Helena Soares (ISCTE); João Nuno Mestre (Coimbra); Jorge Neves (Coimbra); Margarida Melo (Coimbra); Margarida Mendes Lopes (IST); Paula Olga Gneri (Salamanca); Pedro Macias Marques (Évora); Stavros Papadakis (IST); Susana Ferreira (IPLeiria); Ugo Bruzzo (SISSA).

Abstracts:

Ana Cristina Casimiro: Let G be a connected reductive algebraic group G defined over R. We will talk about the relationship between the notions of GIT stability and irreducibility for real and complex points of the algebraic variety of representations of a finitely generated group into G, with the action of G by conjugation. We will give some application of this relation to the case of G-Higgs bundles. This is a joint work with Carlos Florentino.

Daniel Hernández Serrano: In 1988 Hitchin introduced the concept of Higgs pairs over a compact Riemann surface, and discover a map (nowadays called Hitchin map) from the moduli space of such objects to an affine space, which turned out to be a completely integrable Hamiltonian system. He address the following question to the scientific community: can we find in some concrete way the differential equations? I will give an answer to this question using the Krichever map and the Sato Grassmannian, computing the equations in terms of residue identities. For the case when the so called spectral cover is totally ramified at a fixed point, the equations will be given in terms of the coefficients of the characteristic polynomial of Higgs field. The first part of the talk will be concerned with an introduction to the Sato Grassmannian and the Krichever construction, reviewing the well known case for smooth curves.

Darío Sánchez Gómez: We consider the problem of describing Simpson moduli spaces of semistable sheaves on a Gorenstein genus one curve. Using Fourier-Mukai transforms we prove that for a fixed Hilbert polynomial there are only a finite number of nonisomorphic moduli spaces. As a particular case, for a cycle of projective lines, we show that the connected component of the moduli space that contains vector bundles of degree 0 is isomorphic to a symmetric product of the rational curve with one node. This is a joint work with D. Hernández Ruipérez, A.C. López Martín and C. Tejero Prieto (doi:10.1093/imrn/rnp094).

Filippo Viviani: The classical Torelli map is the modular map from the moduli space of smooth projective curves of genus g into the moduli space of principally polarized abelian varieties of dimension g, sending a curve into its Jacobian. The Torelli theorem asserts that the Torelli map is injective on geometric points. We propose two extensions of the Torelli theorem: one for the compactified Torelli map and the other for the tropical Torelli map. The compactified Torelli map was constructed by Alexeev: it is a modular map from the Deligne-Mumford moduli space of stable curves to the Alexeev moduli space of stable semi-abelic pairs, sending a stable curve into its compactified Picard variety of degree g-1, endowed with its natural theta divisor and the action of the generalized Jacobian. In a joint work with L. Caporaso, we give a complete description of the fibers of the compactified Torelli map. On the other hand, in a joint work with S. Brannetti and M. Melo, we construct moduli spaces of tropical curves and tropical abelian varieties and a tropical Torelli map between them. In another joint work with L. Caporaso, we describe the fibers of the tropical Torelli map. I will report on the above two Torelli-type theorems, trying to enlight the relations between them.

Francisco José Plaza Martín: This paper is devoted to the study of the uniformization of the moduli space of pairs (X,E) consisting of an algebraic curve and a vector bundle on it. A group acting on it is found and it is shown that it acts (infinitesimally) transitively on this moduli space and its Lie algebra is computed. Finally, as a consequence, a relation between certain line bundles on this moduli space is proven.

Margarida Mendes Lopes: Two important invariants of isomorphism classes of minimal surfaces of general type are K^2 (the self-intersection of a canonical divisor) and \chi, the holomorphic characteristic. These are positive numbers that satisfy 2\chi-6\leq K^2\leq 9\chi, and it is known that for most pairs (m,n) of positive integers in this region there exists a minimal surface of general type having K^2=m and \chi=n. However the invariants of irregular surfaces (i.e. those having at least a non zero holomorphic 1-form) satisfy further restrictions. In this talk I will discuss some of these restrictions, focusing in particular on some recent joint work with R.Pardini.

Ugo Bruzzo: While there is a classical generalization of the notion of numerically effective line bundle to higher-rank vector bundles, no such concept seems to exist for principal bundles with arbitrary (reductive) structure group. In this talk I will propose a possible definition, which relies on the introduction of suitable universal principal quotient bundles. Moreover I will briefly review some properties of numerically effective and numerically flat principal bundles.

II Meeting - Salamanca, 1st and 2nd of October 2010

Participants:
Ana Cristina Casimiro (CAMGSD/FCTUNL); Luis Álvarez-Cónsul (ICMAT/CSIC, Madrid); Carlos Florentino (CAMGSD/IST, Lisbon); Esteban Gómez González (Salamanca); Beatriz Graña Otero (Salamanca); Daniel Hernández Serrano (Salamanca); Alberto López Martín (Zürich); Ana Cristina López Martín (Salamanca); Pedro Macias Marques (Évora); Margarida Melo (Coimbra); Jorge Neves (Coimbra); Alvaro Nolla (Nagoya University); André G. Oliveira (UTAD, Vila Real); Stavros Papadakis (IST); Fco. José Plaza Martin (Salamanca); Carlos Rito (UTAD, Vila Real); Darío Sánchez Gómez (Salamanca); Fernando Sancho de Salas (Salamanca); Carlos Tejero Prieto (Salamanca); Filippo Viviani (CMUC/Roma Tre).

Abstracts:

Fernando Sancho de Salas: We shall review the notions of derived category D(X), of a scheme X, and integral functor. We focus on the problem of representability of functors, that is to determine when a functor between these categories can be described as an integral functor. We solve this problem for functors between D(X) and D(Z), when X is a scheme over Z.

Luis Álvarez-Cónsul: Poset sheaves (i.e., representations of a poset in the category of coherent sheaves) and poset quiver sheaves (i.e., representations of a quiver in the category of poset sheaves) over a projective scheme respectively include parabolic sheaves and parabolic Higgs sheaves as special cases. In this talk I will provide other examples, define semistability conditions and construct their moduli spaces, embedding the corresponding moduli problems in moduli problems for poset representations in other, more linear, categories.

Carlos Tejero Prieto: We determine all the Fourier-Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by the positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier-Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Using these results we draw some conclusions about the possible birational type of the moduli spaces. We prove that for a given degree d of the vector bundle and a given dimension of the subspace of its global sections there are at most d different possible birational types for the moduli spaces.

Carlos Rito: We study surfaces of general type S with p_g=0 and K²=3 having an involution i such that the bicanonical map of S is not composed with i. It is shown that, if S/i is not rational, then S/i is birational to an Enriques surface or it has Kodaira dimension 1 and the possibilities for the ramification divisor of the covering map S -> S/i are described. We also show that these two cases do occur, providing an example. In this example S has a hyperelliptic fibration of genus 3 and the bicanonical map of S is of degree 2 onto a rational surface.

Jorge Neves: The Burniat surfaces are bi-double covers of the projective plane branched on a configuration of lines. They are of general type, their geometric genus is 0 and they have K²=6-m, where m is the number of singular points of the branch locus of a certain type. This construction produces 6 families of Burniat surfaces (two for m=2) by varying m in {0,1,2,3,4}; the terminology of tertiary Burniat surfaces refers to the family with K²=3. The family of tertiary Burniat surfaces is 1-dimensional whereas the expected dimension of the moduli space of surfaces of general type with the same invariants is 4. It is thus natural to try to describe to whole connected component to which tertiary Burniat surfaces belong. In this talk, we report on work in progress with Roberto Pignatelli, in which we give a construction of new surfaces of general type with geometric genus 0 and K²=3 that extends the 1-dimensional family of tertiary Burniat surfaces to a 4-dimensional family. We will start by giving the construction as a quotient of a surface obtained using parallel unprojection. Then, using this quotient, we shall deduce an alternative description for the surfaces constructed, as bi-double covers of the projective plane generalizing the bi-double cover data for tertiary Burniat surfaces.

Carlos Florentino: The moduli space Hom(F,G)/G of completely reducible representations of a finitely generated group F into a Lie group G, known as the G-character variety of F, appears naturally in connection with knot theory, deformation theory of Riemann surfaces and Higgs bundles on them, and with quantum field theories. We will discuss the geometry, topology and singularities of these varieties in the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, one can show that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G, so they have the same homotopy type. Moreover, if G=SL(n,C), these character varieties admit a manifold structure only when F or G are Abelian, or r+n < 6. In the non-Abelian case, when r+n=5, these moduli spaces have the homotopy type of spheres. This is joint work with S. Lawton (arxiv:0807.3317 and arXiv:0907.4720).