Bandoleros 2016 Fibrados vetoriais em geometria algébrica

We give an introduction to instanton bundles and focus on the ones defined on the projective space $\mathbb{P}^3$. One important tool of studying these objects are the so called ADHM data (first introduced by Atiyah, Drinfeld, Hitchin and Manin). We use these data to extract information about the equivariant rank 2 instanton sheaves, with respect to a 3-Torus inherited from the actual 3-Torus acting on $\mathbb{P}^3$.

Representation theory of partially ordered set (poset) is a huge field of investigation. Such objects appears in ’moduli problems’ in algebraic geom- etry, for instance due to A.Klyachko any (stable) equivariant sheaf on toric variety gives rise to a certain family of filtrations of finite-dimensional vector spaces, and therefore to a certain (stable) representation of some poset. One of the fundamental tasks is the description up to isomorphism of indecomposables representations. But ”most” of the posets are wild, in the sense that the problem of classification of their representations is as difficult as the classification of representations of free algebras. Approaching the classification problem geometrically we consider the moduli spaces associated with given dimension vector and consisting with polystable (with respect to fixed integer weight) representations of poset. During the talk we aim to discuss when such spaces are non-empty and the canonical choice of stability for given Schurian representation of a poset. Also (if the time will permit) we will discuss the stability behaviour of certain functors on representations of the posets (developed by Yu. Drozd as the analogue of Coxeter functor of quivers).

We will generalize Floystad’s criterion for the existence of monads and we will study the properties of the sheafs defined by their cohomology. In particular we will study whenever it is locally free, simple, stable and we will describe the set of pairs of morphisms which define the monad and the moduli space of the cohomology sheaves.
This is a joint work with Pedro Macias Marques and Helena Soares.

Let $G$ be a finite group of order $n$, and $x$ an $n$-th primitive root of the unity. Let also $\mathbb{Z}$ be the set of integers. Consider the affine curve $C := Spec(\mathbb{Z}[x]\otimes R(G))$ where $R(G)$ is the Green ring of $G$, i.e., the $\mathbb{Z}$-free module generated by the irreducible characters of G where the product is taken pointwise. We study the fibers of the formal tangent sheaf of $C$ by estimating the dimension of their Zariski’s tangent space and also finding and measuring the singularities of $C$.

< Roughly speaking, mirror symmetry for hyperkahler manifolds mixes the ‘complex direction’ given by the holomorphically symplectic 2-form and the ‘metric direction’ given by the Kahler form and a B-field. One striking prediction about geometrical objects coming from physics is the Mirror Symmetry Conjecture proposed by Kontsevich in 1994. This conjecture speculates that mirror symmetry could be seen as an equivalence of the derived category of coherent sheaves and the derived Fukaya category. Another notion from physics linked to Mirror Symmetry is the notion of brane. A brane in a hyperkahler manifold is a submanifold which is either complex or lagrangian with respect to the three complex structures of the ambient manifold. Following these physical motivations, we will study how branes in $K3$ surfaces behave under the Mirror Symmetry transformation.

The geometry of the big diagonal $D_n$ in the product variety $X^n$ of a smooth quasi-projective surface $X$ is, by construction, fundamentally related to the geometry of the Hilbert scheme $X^{[n]}$ of $n$ points over $X$ and the isospectral Hilbert scheme $B^n$. We will first show how certain properties of the diagonal ideal $I_{D_n}$, namely the log-canonical thresholds of the pair $(X^n, I_{D_n})$, are related to the log-canonical thresholds of the pair $(B^n, \emptyset)$ and hence to the singularities of the isospectral Hilbert scheme $B^n$. We will then prove that powers of the ideal sheaf $I_{D_n}$ correspond, under the Bridgeland-King-Reid transform, to powers of determinants of tautological bundles over the Hilbert scheme of points $X^{[n]}$; as a consequence, by comparing cohomological properties of the two sides when $X$ is projective, we can deduce on one hand a universal formula for the Euler-Poincaré characteristic of $(det L^{[n]})^2$ for $n \geq 4$ and on the other hand an upper bound for the Casterlnuovo-Mumford regularity of the ideal sheaves $I^k_{D_n}$ and $(I_{D_n})^{S_n}$.