Moduli Space Of Torsion-free Sheaves Research Articles

Virasoro constraints for moduli of sheaves and vertex algebras

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,p)$\\end{document} cohomology classes by reducing the statements to the rank 1 case.

On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces

AbstractThe 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 nonsingular 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 $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Rationality of moduli space of torsion-free sheaves over reducible curve

Let M ( 2 , w ̲ , χ ) be the moduli space of rank 2 torsion-free sheaves of fixed determinant and odd Euler characteristic over a reducible nodal curve with each irreducible component having utmost two nodal singularities. We show that in each irreducible component of M ( 2 , w ̲ , χ ) , the closure of rank 2 vector bundles is rational.

Brauer Group and Birational Type of Moduli Spaces of Torsionfree Sheaves on a Nodal Curve

Let be the moduli space of stable vector bundles of rank n and fixed determinant 𝕃 of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree d will be denoted by . We calculate the Brauer groups of . We study the question of rationality of and .

Instantons, Topological Strings, and Enumerative Geometry

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four, and two dimensions which naturally arise in the context of topological string theory on certain noncompact threefolds. We describe how the instanton counting in these gauge theories is related to the computation of the entropy of supersymmetric black holes and how these results are related to wall‐crossing properties of enumerative invariants such as Donaldson‐Thomas and Gromov‐Witten invariants. Some features of moduli spaces of torsion‐free sheaves and the computation of their Euler characteristics are also elucidated.

Seminormal Varieties, Torsionfree Sheaves, and Picard groups

We give a construction of torsionfree sheaves on a seminormal variety Y using torsionfree sheaves on the normalization X and the non-normal locus W. We use it to find a relation between Picard groups of X, Y, and W. We apply it to determine the Picard groups of the generalized Jacobian, the compactified Jacobian and some subschemes associated to the moduli spaces of torsionfree sheaves of rank 2 and odd degree on a nodal curve.

Stable sheaves on elliptic fibrations

Let X→ B be an elliptic surface and M(a,b) the moduli space of torsion-free sheaves on X which are stable of relative degree zero with respect to a polarization of type aH+ bμ, H being the section and μ the elliptic fibre ( b≫0). We characterize the open subscheme of M(a,b) which is isomorphic, via the relative Fourier–Mukai transform, with the relative compactified Simpson–Jacobian of the family of those curves D↪ X which are flat over B. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of torsion-free and semistable sheaves of rank n and degree zero on the fibres. The relative Fourier–Mukai transform induces an isomorphic between this relative moduli space and the relative nth symmetric product of the fibration. These results are relevant in the study of the conjectural duality between F-theory and the heterotic string.

Vector Bundles on Curves with Many Components

We construct the moduli spaces $M(n)$ of semistable parabolic sheaves of rank $n$ and fixed Euler characteristic on a disjoint union $X$ of integral projective curves with parabolic structures over Cartier divisors on $X$. In the case where $X$ is non-singular, $M$ is a normal projective variety. Suppose that $X$ is the desingularisation of a reducible reduced curve $Y$ with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, $M(n)$ is the normalisation of the moduli space of torsion-free sheaves of rank $n$ and fixed Euler characteristic on $Y$, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of $M(n)$ for small $n$ in some cases.

Moduli of vector bundles on projective surfaces: some basic results

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough. Actually we prove more: given a line bundle on the surface, we show that the number of moduli of sheaves which have a non-zero twisted (by the chosen line-bundle) endomorphism grows slower than the expected dimension of the moduli space (for fixed rank and increasing discriminant). The bounds we get are effective: we concentrate on the case of rank two and we give a lower bound on the discriminant guaranteeing that the moduli space is reduced of the expected dimension. We also give an effective irreducibility result for complete intersections. All of the above follows from a theorem bounding the dimension of complete subsets of the moduli space which do not intersect the boundary.