Algebraic K3 Surfaces Research Articles

K3 surfaces with two involutions and low Picard number

Let X be a complex algebraic K3 surface of degree 2d and with Picard number ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}. Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 1$$\\end{document} when d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and ρ≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\ge 2$$\\end{document} when d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. For d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}, the first example defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} with ρ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =1$$\\end{document} was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}, can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for all d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for d=2,3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2,3,4$$\\end{document}. We also show that a nodal quartic surface can be used to realise the minimum ρ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =2$$\\end{document} for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank 1≤r≤10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le r \\le 10$$\\end{document} and signature (1,r-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,r-1)$$\\end{document} there exists a K3 surface Y defined over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}$$\\end{document} such that PicYC=PicY≅N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Pic}\\,}}Y_{\\mathbb {C}}={{\\,\ extrm{Pic}\\,}}Y \\cong N$$\\end{document}.

On K3 surfaces of Picard rank 14

AbstractWe study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2‐elementary lattices. Three such lattices exist, namely, , , and . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

A K3 surface related to Leonardo Pisano’s work on congruent numbers

This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’.

On birational transformations of Hilbert schemes of points on K3 surfaces

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk\"ahler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.

On the maximal number of du Val singularities for a K3 surface

A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L \subset X to the skyscraper sheaf of a point y \in Y . We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the “Beauville–Voisin subring” in the Chow groups of Y , and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.

Symplectic topology of $K3$ surfaces via mirror symmetry

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.

High U(1) charges in type IIB models and their F-theory lift

We construct models with U(1) gauge group and matter with charges up to 6, in the context of type IIB compactifications. We show explicitly that models with charges up to 4 can be derived from corresponding models in F-theory by applying the Sen weak coupling limit. We derive which type IIB models should be the limit of charge 5 and 6 F-theory models. Explicit six dimensional type IIB models with maximal charge 5 and 6 are constructed on an algebraic K3 surface that is the double cover of ℂℙ2. By using type IIB results we are also able to rediscover the F-theory charge 4 model in a straightforward way.

On prime degree isogenies between K3 surfaces

We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodges structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai--Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.

Spherical functors on the Kummer surface

AbstractWe find two natural spherical functors associated to the Kummer surface and analyze how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface.

Spherical functors on the Kummer surface

AbstractWe find two natural spherical functors associated to the Kummer surface and analyze how their induced twists fit with Bridgeland's conjecture on the derived autoequivalence group of a complex algebraic K3 surface.

An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the N\'eron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants.

Elliptic Fibrations on K3 Surfaces

AbstractThis paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

Lattice polarized K3 surfaces and Siegel modular forms

The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H⊕E8⊕E7. Key ingredients for this classification are as follows: a normal form for these lattice polarized K3 surfaces, a coarse moduli space and an explicit description of the inverse period map in terms of Siegel modular forms. Second, we give explicit formulas for a Hodge correspondence that relates these K3 surfaces to principally polarized abelian surfaces. The Hodge correspondence in question underlies a geometric two-isogeny of K3 surfaces, the details of which are described by the authors in Clingher and Doran (2011) [7].

Donagi–Morrison’s examples on 2-elementary K3 surfaces

Let X be an algebraic K3 surface, and let L be a base point free and big line bundle on X. If X admits a map of degree 2 to the projective plane branched over a smooth sextic and L is the pullback of the line bundle \({\mathcal{O}_{\mathbb{P}^{2}}(3),}\) then the gonality of the smooth curves of the complete linear system |L| is not constant. The polarized K3 surface (X, L) consisting of the K3 surface X and the line bundle L is called Donagi–Morrison’s example. In this paper, we give a necessary and sufficient condition for the polarized K3 surface (X, L) consisting of a 2-elementary K3 surface X and an ample line bundle L to be Donagi–Morrison’s example.

Note on a Geometric Isogeny of K3 Surfaces

The paper establishes a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H ⊕ E7 ⊕ E7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces. 1 Geometric Two-Isogenies on K3 Surfaces Let X be an algebraic K3 surface defined over the field of complex numbers. A Nikulin (or symplectic) involution on X is an analytic automorphism of order two Φ: X→ X such that Φ∗(ω) = ω for any holomorphic two-form ω on X. This type of involution has many interesting properties (see [17, 18]), amongst which the most important are: (a) the fixed locus of Φ consists of precisely eight distinct points, and (b) the surface Y obtained as the minimal resolution of the quotient X/Φ is a K3 surface. Equivalently, one can construct Y as follows. Blow up the eight fixed points on X obtaining a new surface X. The Nikulin involution Φ extends to an involution Φ on X which has as fixed locus the disjoint union of the eight resulting exceptional curves. The quotient X/Φ is smooth and recovers the surface Y from above. In the context of the above construction, one has a degree-two rational map pΦ : X 99K Y with a branch locus given by eight disjoint rational curves (the even eight configuration in the sense of Mehran [16]). In addition, there is a push-forward morphism (see [12, 17]) (pΦ)∗ : H(X,Z)→ HY (1) mapping into the orthogonal complement in H(Y,Z) of the even eight curves. The metamorphosis of the surface X into Y is referred to in the literature as the Nikulin construction. The most well-known class of Nikulin involutions is given by the Shioda-Inose structures [12, 17, 18]. These consist of Nikulin involutions that satisfy two additional requirements. The first condition asks for the surface Y to be Kummer. The second requirement asserts that the morphism (1) induces a Hodge isometry between the lattices of transcendental cocycles TX(2) and TY. An effective criterion for a particular K3 surface X to admit a Shioda-Inose structure was given by Morrison [17]. In this paper, we shall work with another class of Nikulin involutions: fiber-wise translations by a section of order two in a jacobian elliptic fibration. This class of involutions was discussed by Van Geemen and Sarti [9]. Let us be precise: Definition 1.1. A Van Geemen-Sarti involution is an automorphism ΦX : X → X for which there exists a triple (φX, S1, S2) such that: (a) φX : X→ P is an elliptic fibration on X, (b) S1 and S2 are disjoint sections of φX, (c) S2 is an element of order two in the Mordell-Weil group MW(φX,S1), (d) ΦX is the involution obtained by extending the fiber-wise translations by S2 in the smooth fibers of φX using the group structure with neutral element given by S1. Under the above conditions, one says that the triple (φX, S1, S2) is compatible with the involution ΦX. Any given Van Geemen-Sarti involution is, in particular, a Nikulin involution. One can naturally regard a Van Geemen-Sarti involution ΦX as a fiber-wise two-isogeny between the original K3 surface X and the newly constructed K3 surface Y. Since ΦX acts as a translation by an element of order two in each of the smooth fibers ∗Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis MO 63108. e-mail: clinghera@umsl.edu †Department of Mathematical and Statistical Sciences, University of Alberta. Edmonton AB T6G 2G1. e-mail: doran@math.ualberta.ca

Classification of non-symplectic automorphisms of order 3 onK3 surfaces

In this paper, we study non-symplectic automorphisms of order 3 on algebraic K3 surfaces over which act trivially on the Néron-Severi lattice. In particular we shall characterize their fixed loci in terms of the invariants of 3-elementary lattices. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

SEVERI VARIETIES AND SELF-RATIONAL MAPS OF K3 SURFACES

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parameterizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such nontrivial rational maps, that is of topological degree > 1.

K3 surfaces with non-symplectic automorphisms of 2-power order

This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Néron–Severi group. Complementing a result by Vorontsov and Kondō, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on relations with mirror symmetry.

Projective models of K3 surfaces with an even set

Abstract The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a careful analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution.