Generic K3 Surfaces Research Articles

Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t≠2 which admits an ample divisor D∈Div(X) with qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.

Monodromy of rational curves on K3 surfaces of low genus

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over $\mathbb{C}$ as the K3 surface varies. We prove that when the K3 surface has genus $g$, $1\leq g\leq 3$, the monodromy group is the full symmetric group.

Enhancements in F-theory models on moduli spaces of K3 surfaces with ADE rank 17

We study the moduli of elliptic K3 surfaces with a section with the $ADE$ rank 17. While the Picard number of a generic K3 surface in such moduli space is 19, the Picard number is enhanced to 20 at special points in the moduli. K3 surfaces become attractive K3 surfaces at these points. Either of the following two situations occurs at such special points: i) the Mordell-Weil rank of an elliptic K3 surface is enhanced, or ii) the gauge symmetry is enhanced. The first case i) is related to the appearance of a $U(1)$ gauge symmetry. In this note, we construct the moduli of K3 surfaces with $ADE$ types $E_7 D_{10}$ and $A_{17}$. We determine some of the special points at which K3 surfaces become attractive in the moduli of K3 surfaces with $ADE$ types $E_7 D_{10}$ and $A_{17}$. We investigate the gauge symmetries in F-theory compactifications on attractive K3 surfaces which correspond to such special points in the moduli times a K3 surface. $U(1)$ gauge symmetry arises in some F-theory compactifications.

Stable maps and Chow groups

According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^2(X). We discuss this conjecture for symplectic involutions and prove it in one third of all cases. The main point is to use special elliptic K3 surfaces and stable maps to produce covering families of elliptic curves on the generic K3 surface that are invariant under the involution.

On the Brill-Noether theory for K3 surfaces

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.

On Cox rings of K3 surfaces

AbstractWe study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.

Degeneration of linear systems through fat points on $K3$ surfaces

In this paper we introduce a technique to degenerate K3 surfaces and linear systems through fat points in general position on K3 surfaces. Using this degeneration we show that on generic K3 surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through n = 4 u 9 w fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through n = 4 u 9 w fat points in general position on a general quartic surface in P 3 .