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Register a new userTwo infinite series of moduli spaces of rank 2 sheaves on $mathbb{P}^3$
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We describe new components of the Gieseker--Maruyama moduli scheme $mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $mathcal{M}(2)$ is connected, and identify a connected subscheme of $mathcal{M}(3)$ consisting of 7 irreducible components.
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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.
We study the irreducible components of the moduli space of instanton sheaves on $mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${mathcal T}(d)$ of stable sheaves on $mathbb{P}^3$ with Hilbert polynomial $P(t)=dcdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${mathcal T}(d)$ for $dle4$.
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.
The goal of this paper is to construct a compactification of the moduli space of degree $d ge 5$ surfaces in $mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $mathbb{P}^3$ and whose boundary points correspond to degenerations of such surfaces. We study a more general problem and consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = mathbb{P}^3$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.
We study the spectrum of rank $2$ torsion free sheaves on $mathbb{P}^3$ with aim of producing examples of distinct irreducible components of the moduli space with the same spetrcum answering the question presented by Rao for the case of torsion free sheaves. In order to do so, we provide a full description of the spectrum of the sheaves in the moduli space of semistable rank $2$ torsion free sheaves on $mathbb{P}^3$ with Chern classes $(c_1, c_2,c_3)$ equals to $(-1,2,0)$ and $(0,3,0)$.
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