Bandoleros 2015 Fibrados vetoriais em geometria algébrica

One of the main ingredients to prove the popular Hartshorne's conjecture would be to prove that any vector bundle of low rank in a projective space must split as a direct sum of line bundles. We will discuss this problem, linking it to the extendibility of vector bundles. Our main inspiration will be Barth-Larsen theorem, which provides a positive answer to the extendibility of line bundles. We will introduce some generalization of Barth-Larsen theorem to other ambient spaces, which will lead to extend to those spaces our especulations about vector bundles of low rank.

We generalize the Hitchin-Kobayashi correspondence between semi stability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

On a manifold with a generalized complex structure $X$. Gualtieri define the notion of a generalized holomorphic bundle. In the case of an ordinary complex structure a generalized holomorphic bundle is equivalent to a holomorphic vector bundle $E$ on $X$ together with a "co-Higgs" field $\phi \in H^0(X,End(E)\otimes TX)$ for which $\phi \wedge \phi = 0$. Hitchin called the pair $(E,\phi)$ by co-Higgs bundle. S. Rayan showed the non-existence of stable co-Higgs bundles with non-trivial Higgs field on K3 and general-type surfaces. In this talk we will prove the following:
Let $(E,\phi)$ be a rank two co-Higgs reflexive sheaf on a Kahler compact surface $X$ with $\phi \in H^0(X,End(E)\otimes TX)$ nilpotent. If E is semi-stable, then, up to finite etale cover, is either:
i) $X$ is uniruled , if $(E,\phi)$ is stable.
ii) $X$ is a tori and $(E,\phi)$ is strictly semi-stable.
iii) $X$ is a properly elliptic surface and $(E,\phi)$ is strictly semi-stable.

Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus $g \leq 8$, the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as $g=9$. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to $g$ when $g \leq 5$. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins.

We recall the construction of the Hilberts schemes of points over $\mathbb{C}^n$ and generalize it in order to construct the scheme that parameterizes quotients of $r$ copies of $\mathbb{C}[z_0,...z_n]$ with constant Hilbert polynamial $c$.

Embeddings of algebraic curves in projective space are often studied using Brill-Noether varieties $G^r_d(X)$, which parameterize linear series on $X$. Using the theory of limit linear series, I will discuss ongoing work with Chan, Pflueger, and Teixidor, in which we use a surprising theorem on random standard Young tableaux to give a new proof of the classical formula for the genus of the Brill-Noether curve.

Given a smooth projective variety $X$ over an algebraically closed field $K$ of characteristic zero, a monad on $X$ is a complex $$ M_\bullet: 0 \longrightarrow A \stackrel{\alpha}{\longrightarrow} B \stackrel{\beta}{\longrightarrow} C \longrightarrow 0 $$ of coherent sheaves over $X$ that is exact at $A$ and $C$. The coherent sheaf ${E:= \ker\beta /\mbox{im} \: \alpha}$ is called the cohomology (sheaf) of the monad $M$.
Monads were introduced by Horrocks in the sixties and since then they have proved very useful objects for constructing vector bundles and studying their properties. In 2009 Costa and Miró-Roig defined instanton bundles on the smooth quadric in terms of monads. They furthermore proved a cohomological characterisation of sheaves over the projective space and over the smooth quadric that are the cohomology of a monad. I will report on joint work with Helena Soares and Simone Marchesi, namely on the existence of monads over an arithmetically Cohen Macaulay, projective variety, a cohomological characterisation of sheaves over the Segre variety that are the cohomology of a monad, and on the simplicity of their cohomology sheaf.

A typical phenomenon which occurs when one considers the degeneration of an object from a smooth curve to a singular one, is that the limit often depends on the chosen degeneration. To take in account this phenomenon, Laila Mainò in her Ph.D. thesis introduced the notion of enriched structure on a nodal curve. This is the datum of a line bundle for each component of the curve with certain deformation properties. She also constructed a moduli space parametrizing enriched structures on a stable curve. In this talk, we will give a toric description of a compactification of such a moduli space by means of tropical geometry. This is a joint work with Alex Abreu.

In this talk we present equivalences between some categories of vector bundles and full subcategories of representations of certain quivers. More precisely, we study cokernel bundles, Steiner bundles, syzygy bundles and monads on projective varieties and we give some decomposability criteria for such bundles. In particular, as an example, we prove that the cohomology of the monad on $$ \mathcal{O}_{\mathbb{P}^{2p}}(-1)^{2p+1} \stackrel{\alpha}{\longrightarrow} \Omega_{\mathbb{P}^{2p}}^{p}(p)^2 \stackrel{\beta}{\longrightarrow} \mathcal{O}^{2p+1}_{\mathbb{P}^{2p}}, $$ for certain maps, is indecomposable.

We will explain an attempt to compute global sections of determinant line bundles over moduli spaces of semistable sheaves over $\mathbb{P}^2$ (or rational surfaces) by means of the description of the latter in terms of monads, or, equivalently, of quiver representations. Thanks to this interpretation, the computation of global sections of determinant line bundles should be reduced to the comprehension of semi-invariant of quivers, which is a purely combinatorial problem.