Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = −γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ℓ of a line in a smooth Lagrangian n-plane ℙn must satisfy (ℓ,ℓ) = −(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
This note deals with Lagrangian fibrations of elliptic K3 surfaces and the associated Hamiltonian monodromy. The fibration is constructed through the Weierstraß normal form of elliptic surfaces. There is given an example of K3 dynamical models with the identity monodromy matrix around 12 elementary singular loci.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.
5
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
On a general quasismooth well-formed weighted hypersurface of degree Σi=14 a i in ℙ(1, a 1, a 2, a 3, a 4), we classify all pencils whose general members are surfaces of Kodaira dimension zero.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.