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Register a new userNew divisors in the boundary of the instanton moduli space
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Let ${mathcal I}(n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on ${mathbb P}^3$. We know from several authors that ${mathcal I}(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank $2$ instanton bundle on ${mathbb P}^3$ is stable, we may regard ${mathcal I}(n)$ as an open subset of the projective Gieseker--Maruyama moduli scheme ${mathcal M}(n)$ of rank $2$ semistable torsion free sheaves $F$ on ${mathbb P}^3$ with Chern classes $c_1=c_3=0$ and $c_2=n$, and consider the closure $overline{{mathcal I}(n)}$ of ${mathcal I}(n)$ in ${mathcal M}(n)$. We construct some of the irreducible components of dimension $8n-4$ of the boundary $partial{mathcal I}(n):=overline{{mathcal I}(n)}setminus{mathcal I}(n)$. These components generically lie in the smooth locus of ${mathcal M}(n)$ and consist of rank $2$ torsion free instanton sheaves with singularities along rational curves.
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The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation.
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 order to obtain existence criteria for orthogonal instanton bundles on $mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $mathbb{P}^n$ and charge $cgeq 3$ has rank $rleq (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, the coarse moduli space of orthogonal instanton bundles with no global sections on $mathbb{P}^n$, with charge $cgeq 3$ and rank $r$, is affine, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in $mathcal{M}_{mathbb{P}^n}^{mathcal{O}}(c,r)$, whenever is non-empty.
Let $mathcal{M}(n,m;F bp^n)$ be the configuration space of $m$-tuples of pairwise distinct points in $F bp^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in $F bp^n$ with respect to the diagonal action of ${rm PU}(1,n;F)$ equipped with the quotient topology. It is an important problem in hyperbolic geometry to parameterize $mathcal{M}(n,m;F bp^n)$ and study the geometric and topological structures on the associated parameter space. In this paper, by mainly using the rotation-normalized and block-normalized algorithms, we construct the parameter spaces of both $mathcal{M}(n,m; bhq)$ and $mathcal{M}(n,m;bp(V_+))$, respectively.
We prove that the Hilbert scheme of $k$ points on $mathbb{C}^2$ (Hilb$^k[mathbb{C}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kahler and equivariant parameters as well as inverting the weight of the $mathbb{C}^times_hbar$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb$^k[mathbb{C}^2]$ is obtained via direct limit $ltoinfty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $hbar$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on $mathbb{P}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.
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