Related papers: Cubic symmetroids and vector bundles on a quadric …

This is the revised version of our previous preprint. In this paper, we establish a generic smoothness result for moduli space of semistable sheaves of arbitrary rank over surfaces provided that the second Chern class of the sheaves is…

We consider irreducible logarithmic connections $(E,\,\delta)$ over compact Riemann surfaces $X$ of genus at least two. The underlying vector bundle $E$ inherits a natural parabolic structure over the singular locus of the connection…

We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we…

Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z).…

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\phi)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $\phi \colon E_{2}…

We prove the Bloch conjecture : $ c_2(E) \in H^4_\cald (X,\bbz(2))$ is torsion for holomorphic rank two vector bundles $E$ with an integrable connection over a complex projective variety $X$. We prove also the rationality of the…

Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let $\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that…

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve $C$. There is a known lower bound on the degree for the existence of such bundles; the main…

We study stable rationality of conic bundles $X$ over $\mathbb{P}^1$ defined over non-closed field $k$ via the cohomology of the Galois group of finite field extension $k'/k$ with action on the geometric Picard lattice of $X$.

In this paper, we prove that total space of every vector bundle with the base manifold on which the canonical isometric action acts freely, also carries a principal bundle structure. We also obtain another principal bundle based on the…

For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles…

In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an…

Given a vector bundle $E$ on a tree of smooth rational curves $C$, we give necessary and sufficient conditions for a vector bundle $E'$ on $\mathbb{P}^1$ to specialize to $E$ on $C$, generalizing the rank 2 case, due to Coskun.

Given a rank $r$ stable bundle over a smooth irreducible projective curve $C,$ there is an associated rank $2r$ bundle over $S^2(C),$ the second symmetric power of $C.$ In this article we study the stability of this bundle. As a consequence…

This is the story of the rediscovery of classical three-dimensional geometry, especially the geometry of quadric surfaces, while studying the semigroup $M_2(\mathbb R)$ of linear endomorphisms of a real plane. One of the surfaces that…

Let $C$ be a smooth projective curve of genus $g$ over a finite field $\mathbb{F}_q$ and let $D$ be a divisor on $C$ of degree $>2g-2$. We assume that the characteristic of $\mathbb{F}_q$ is sufficiently large. Let $n$ be an integer and let…

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of…

A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb…