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Register a new userConstruction of stable rank 2 vector bundles on $mathbb{P}^3$ via symplectic bundles
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In this article we study the Gieseker-Maruyama moduli spaces $mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=ein{-1,0}, c_2=nge1$ on the projective space $mathbb{P}^3$. We construct two new infinite series $Sigma_0$ and $Sigma_1$ of irreducible components of the spaces $mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $Sigma_0$ contains components for all big enough values of $n$ (more precisely, at least for $nge146$). $Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $mathcal{B}(0,n)$ satisfying this property.
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We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $bge a$ and $c>a+b$. We show that, in the wide range when $c>2a+b-e, b>a, (e,a) e(0,0)$, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces $M(e,n)$ over all $nge1$ contains an infinite series of rational components for both $e=0$ and $e=-1$. Explicit constructions of rationality of Ein components under the above conditions on $e,a,b,c$ and, respectively, of their stable rationality in the remaining cases, are given. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for $c_1=0$ and $n$ even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition $n$ is odd, which is a usual sufficient condition for fineness.
In this paper we characterize the rank two vector bundles on $mathbb{P}^2$ which are invariant under the actions of the parabolic subgroups $G_p:=mathrm{Stab}_p(mathrm{PGL}(3))$ fixing a point in the projective plane, $G_L:=mathrm{Stab}_L(mathrm{PGL}(3))$ fixing a line, and when $pin L$, the Borel subgroup $mathbf{B} = G_p cap G_L$ of $mathrm{PGL}(3)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.
We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action.
Let $A$ be an abelian surface. We construct two complete families of stable vector bundles on the generalized Kummer variety $K_n(A)$. The first is the family of tautological bundles associated to stable bundles on $A$, and the second is the family of the wrong-way fibers of a universal family of stable bundles on the dual abelian variety $widehat{A}$ parametrized by $K_n(A)$. Each family exhibits a smooth connected component in the moduli space of stable bundles on $K_n(A)$.
We present a new family of monads whose cohomology is a stable rank two vector bundle on $PP$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. Such facts are used to prove that the moduli space of stable rank two vector bundles of zero first Chern class and second Chern class equal to 5 has exactly three irreducible components.
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