Related papers: On correspondences of a K3 surface with itself. II…

Let $X$ be a K3 surface with a polarization $H$ of the degree $H^2=2rs$, $r,s\ge 1$, and the isotropic Mukai vector $v=(r,H,s)$ is primitive. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3…

Let $X$ be a K3 surface, and $H$ its primitive polarization of the degree $H^2=8$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(2,H,2)$ is again a K3 surface, $Y$. In math.AG/0206158 we gave necessary and…

Let X be a K3 surface with a polarization H of degree H^2=2rs and with a primitive Mukai vector (r,H,s). The moduli space of sheaves over X with the isotropic Mukai vector (r,H,s) is again a K3 surface Y. We prove that Y\cong X, if Picard…

Let X be a K3 surface and H a primitive polarization of degree H^2=2a^2, a>1. The moduli space of sheaves over X with the isotropic Mukai vector (a,H,a) is again a K3 surface Y which is endowed by a natural nef element h with h^2=2. We give…

Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$. Let $N(X)$ be Picard lattice of $X$. In math.AG/0309348 and math.AG/0606289, all divisors in…

Let X be a K3 surface with a polarization H with H^2=2rs. Assume that H.N(X)=Z for the Picard lattice N(X). The moduli space Y of sheaves over X with the Mukai vector (r,H,s) is again a K3 surface. We prove that Y\cong X, if there exists…

Let X be a K3 surface which is intersection of three (a net P^2) of quadrics in P^5. The curve of degenerate quadrics has degree 6 and defines a double covering of P^2 K3 surface Y ramified in this curve. This is a classical example of a…

We consider the geometry of a general polarized K3 surface $(S,h)$ of genus 16 and its Fourier-Mukai partner $(S',h')$. We prove that $S^{[2]}$ is isomorphic to the moduli space $M_{S'}(2,h',7)$ of stable sheaves with Mukai vector…

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface…

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli…

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P^2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of…

Let $X$ be a K3 surface with Picard group $\mathrm{Pic}(X)\cong\mathbb{Z} H$ such that $H^2=2n$. Let $M_{H}(\mathbf{v})$ be the moduli space of Gieseker semistable sheaves on $X$ with Mukai vector $\mathbf{v}$. We say that $\mathbf{v}$…

Let X be a K3 surface and M a smooth and projective moduli space of stable sheaves on X of Mukai vector v. A universal sheaf U over X x M induces an integral transform F from the derived category D(X) of coherent sheaves on X to that on M.…

In this paper we study deformation classes of moduli spaces of sheaves on a projective K3 surface. More precisely, let $(S1,H1)$ and $(S2,H2)$ be two polarized K3 surfaces, $m\in\mathbb{N}$, and for $i=1,2$ let $mv_{i}$ be a Mukai vector on…

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An…

We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the \'etale cohomology groups (with Q_ell…

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually…

In this paper we study equivariant moduli spaces of sheaves on a $ K3 $ surface $ X $ under a symplectic action of a finite group. We prove that under some mild conditions, equivariant moduli spaces of sheaves on $ X $ are irreducible…