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We study the moduli space of coherent systems in $P^2$ using the Segre invariant. We obtain necessary conditions for the existence of $\alpha$-semistable coherent systems $(E,V)$ of type $(2, c_1, c_2, k)$, with $k \geq 2$. Afterwards, we give numerical conditions to the nonemptiness of the moduli space and compute the critical values depending of the Chern classes. Finally, we give some topological properties of the flips., Comment: 30 pages, no figures

The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;c_1,c_2+m)}$., Comment: 19 pages

Let $G$ be a group and $A$ a set equipped with a collection of finitary operations. We study cellular automata $\tau : A^G \to A^G$ that preserve the operations of $A^G$ induced componentwise from the operations of $A$. We show that $\tau$ is an endomorphism of $A^G$ if and only if its local function is a homomorphism. When $A$ is entropic (i.e. all finitary operations are homomorphisms), we establish that the set $\text{EndCA}(G;A)$, consisting of all such cellular automata, is isomorphic to the direct limit of $\text{Hom}(A^S, A)$, where $S$ runs among all finite subsets of $G$. In particular, when $A$ is an $R$-module, we show that $\text{EndCA}(G;A)$ is isomorphic to the group algebra $\text{End}(A)[G]$. Moreover, when $A$ is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over $A^G$ admitting a memory set $S$ is precisely $(k \vert S \vert)^k$, where $k$ is the number of atoms of $A$., Comment: 13 pages

Let $X$ be a non-singular irreducible complex projective curve of genus $g\geq 2$. We use $(t,\ell)$-stability to prove the existence of coherent systems over $X$ that are $\alpha$-stable for all allowed $\alpha >0$., Comment: 13 pages, Exposition was improved. No mathematical changes, typos corrected and reference [10] was added

Let $(E,V)$ be a general generated coherent system of type $(n,d,n+m)$ on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of $E$ to the semistability of the kernel of the evaluation map $V\otimes \mathcal{O}_X\to E$. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler's Conjecture in some cases. The strongest results are obtained for type $(2,d,4)$, which is the first previously unknown case., Comment: 22 pages

We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective algebraic manifolds. In particular, we show that Gieseker stability for torsion-free Higgs sheaves can be defined using only Higgs subsheaves with torsion-free quotients; and we show that a classical relation between Gieseker stability and Mumford-Takemoto stability extends naturally to Higgs sheaves. We also prove that a direct sum of two Higgs sheaves is Gieseker semistable if and only if the Higgs sheaves are both Gieseker semistable with equal normalized Hilbert polynomial and we prove that a classical property of morphisms between Gieseker semistable sheaves also holds in the Higgs case; as a consequence of this and the existing relation between Mumford-Takemoto stability and Gieseker stability, we obtain certain properties concerning the existence of Hermitian-Yang-Mills metrics, simplesness and extensions in the Higgs context. Finally, we make some comments about Jordan-H\"older and Harder-Narasimhan filtrations for Higgs sheaves., Comment: 16 pages, some minor corrections, last section has been shortened

We study the geometry of the moduli stack of vector bundles of fixed rank and degree over an algebraic curve by introducing a filtration made of open substacks build from $(k, l)$-stable vector bundles. $(k, l)$-stability was introduced by Narasimhan and Ramanan to study the geometry of the coarse moduli space of stable bundles. We will exhibit the stacky picture and analyse the geometric and cohomological properties of the moduli stacks of $(k, l)$-stable vector bundles. For particular pairs $(k, l) $ of integers we also show that these moduli stacks admit coarse moduli spaces and we discuss their interplay., Comment: Some typos corrected and minor clarifications. To appear: Journal of Geometry and Physics

Let $M(n,\xi)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,\xi)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,\xi))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove that $\dim Mor_P(\mathbb{P}^2,M(3,\xi))=8g-7$ and we give a sufficient condition for $Mor_{2ns}(\mathbb{P}^1,M(n,\xi))$ to be non-empty with $s\geq 1.$, Comment: Typos corrected and minor style changes, no mathematical changes. Final version to appear in Manuscripta Mathematica