Constructing rational curves on K3 surfaces

The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fibration over P with a unique section C, fibers F and λ = 2. Here λ = c1(π∗ω) is the first Chern class of the Hodge bundle π∗ω of the fibration π : S → P (see [H-M]). It is a standard result that the number of nodal fibers of an elliptic fibration are given by 12λ [H-M, p. 158]. So there are exactly 24 rational nodal curves in the linear series |F | for S general. This is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We will call a K3 surface with Picard lattice (1.1) a BL K3 surface. A BL K3 surface S lies on the boundary of the moduli space of K3 surfaces of genus g with C+gF as the corresponding primitive divisor. Every curve in the linear series |OS(C + gF )| is “totally reducible”, i.e., it consists of the −2 curve C and g elliptic “tails” attached to C. A curve D ∈ |OS(C + gF )| is the image of a stable rational map only if D = C ∪m1F1∪m2F2∪ ...∪m24F24, where F1, F2, ..., F24 are 24

Let X be a complex K3 surface, $$\textrm{Diff}(X)$$ the group of diffeomorphisms of X and $$\textrm{Diff}_0(X)$$ the identity component. We prove that the fundamental group of $$\textrm{Diff}_0(X)$$ contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg–Witten invariants. The moduli space of Einstein metrics on X is used as a key ingredient in the proof.

K3 surfaces, rational curves, and rational points

Monodromy of rational curves on K3 surfaces of low genus

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Neron–Severi group has rank 20 and is generated by divisors which are defined over ℚ. Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over ℚ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.

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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}.

Kuga and Satake associate with every polarized complex K3 surface (X, ℒ) a complex abelian variety called the Kuga-Satake abelian variety of (X, ℒ). We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized abelian varieties with level structure over ℂ. In this note we study these morphisms. We prove first that they are defined over finite extensions of ℚ. This is done by proving analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

A K3 category is by definition a Calabi–Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

1. The problem of finding the number of space cubic curves which pass through p given points and have 6 – p given lines as chords has been solved by several different methods. The similar problem, in space of four dimensions, of the number of rational quartic curves which pass through p given points and have 7 – p given trisecant planes has been solved for p > 1 by F. P. White and for p = 1 by J. A. Todd. In a recent paper I discussed a similar problem for elliptic quartic curves. My present object is to apply the method of that paper to the problems mentioned above and to obtain two other sets of numbers which I believe to be new. They are (1) the number of rational normal quartic curves which have three assigned chords, pass through p assigned points, and have 3 – p assigned trisecant planes; (2) the number of curves of intersection of three quadrics in [4] which have a suitable number of assigned points and chords. The evaluation of Schubert symbols which occur in the work is done by means of a formula due to Giambelli‖. This formula is stated in a more simple form in a note at the end of this paper (7).

In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface $X$ with at most double curves. We prove that we still have a family of smoothings of $X$ in a weak sense even when $X$ is not Kählerian or $H^{1}(X,\\mathcal{O}_{X})\\neq 0$ , in which cases the smoothability result of Friedman [Fr] is not applicable.

Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. In this paper we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices.

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic p . We prove the prime-to- p part conditionally on the finiteness of the p -primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to- p part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to- p part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of \mathscr{A}[p] or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.

We construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$--singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author.Comment: 26 pages