Related papers: On the moduli space of singular principal G-bundle…

We find locally free resolutions of length one for all semi-stable sheaves supported on curves of multiplicity five in the complex projective plane. In some cases we also find geometric descriptions of these sheaves by means of extensions.…

Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex…

Let $X$ be a smooth complex projective curve of genus $g\geq 3$. Let $\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with trivial determinant over $X$. We construct a desingularization $\mathbf{S}$ of $\mathbf{M}_2$…

Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic…

We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension Z/2 by the fundamental group. By comparison with the space of…

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal…

We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.

Let $G$ be a reductive group acting on an affine scheme $V$. We study the set of principal $G$-bundles on a smooth projective curve $\mathcal C$ such that the associated $V$-bundle admits a section sending the generic point of $\mathcal C$…

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ has…

We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of…

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected.…

Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the…

Let $\mhu$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u=(0,L,\chi(u)=0),$ i.e. sheaves in $u$ are of dimension $1$. There is a natural morphism $\pi$ from the moduli…

We show that the CM line bundle on a proper family parametrizing specially K-stable varieties with maximal variation is ample. As an application, we show projectivity of any proper subspace of the coarse moduli space of uniformly…

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $\mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d \in \pi_1(G)$, where G is…

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed…

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable…

In a previous paper, arXiv:1301.4409, we showed that the moduli space of curves C with a G-symmetry (that is, with a faithful action of a finite group G), having a fixed generalized homological invariant, is irreducible if the genus g' of…