Classification of extremal elliptic <i>K</i>3 surfaces and fundamental groups of open <i>K</i>3 surfaces

Shimada, Ichio; Zhang, De‐Qi

doi:10.1017/s002776300002211x

Cited by 75 publications

(184 citation statements)

References 15 publications

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“…The choice of a root of this quadratic equation does not matter, because, in the eleven cases in Table 10.1, the orientation reversing of T X yields an isomorphic oriented lattice; that is, the GL 2 (Z)-equivalence class of T X contains only one SL 2 (Z)-equivalence class (see [33] In the following, we use the general theory of elliptic surfaces, for which we refer to [35] or [28]. Shioda and Inose [34] showed that every singular K3 surface X has a Jacobian fibration φ : X → P 1 with two singular fibers φ −1 (p) and…”

Section: Singular K3 Surfacesmentioning

confidence: 99%

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

Shimada

2015

International Mathematics Research Notices

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Abstract. 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 Néron-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. IntroductionThe automorphism group Aut(X) of an algebraic K3 surface X is an important and interesting object. Suppose that X is defined over the complex number field C, or is supersingular in odd characteristic. Then, thanks to the Torelli-type theorem due to and Ogus [21], [22], we can study Aut(X) by the Néron-Severi lattice S X of X. We denote by O(S X ) the orthogonal group of S X . Then we have a natural homomorphism ϕ X : Aut(X) → O(S X ).It is known that this homomorphism has only a finite kernel. Using the reduction theory for arithmetic subgroups of O(S X ), Sterk [36] and Lieblich and Maulik [16] proved that Aut(X) is finitely generated. The Néron-Severi lattices for which Aut(X) are finite were classified by Nikulin [18], [19] and Vinberg [40]. On the other hand, when Aut(X) is infinite, it is in general a difficult problem to give a set of generators.We also have the following related problem. Let Nef(X) denote the nef cone of X; that is, the cone of S X ⊗ R consisting of vectors x ∈ S X ⊗ R such that x, C ≥ 0 holds for any curve C on X, where , is the intersection form on S X . In order to classify various geometric objects on X (for example, smooth rational curves, Jacobian fibrations, or polarizations of a fixed degree) modulo Aut(X), it is useful to describe explicitly a fundamental domain of the action of Aut(X) on Nef(X). Several authors have studied these problems by using the idea of Borcherds [4], [5] to embed S X into an even unimodular hyperbolic lattice of rank 26. In these works, however, they required that S X should satisfy a certain strong condition (see Section 1.1 below for the details), and hence the range of applications is limited.The purpose of this paper is to present an algorithm (Algorithm 6.1) that calculates, under assumptions on S X milder than the preceding works, a finite set of generators of the image of ϕ X and a closed domain F of Nef(X) with the following properties:(i) For any v ∈ Nef(X), there exists an element g ∈ Aut(X) such that v g ∈ F .(ii) The domain F is tiled by a finite number of convex cones, which we call chambers. Each chamber is bounded by a finite number of hyperplanes and its stabilizer subgroup in Aut(X) is finite.See Remark 6.5 for the relation of F with a fundamental domain of the action of Aut(X) on Nef(X). The detailed description of the assumptions we impose on S X will be given in Section 8. The algorithm can be applied to a wide class of K3 surfaces. We give two examples. Example 1.1. As an example of a K3 surface with small Picard number and an i...

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Section: Singular K3 Surfacesmentioning

confidence: 99%

“…By the result of Shioda and Inose [34], the isomorphism class of a singular K3 surface X is determined by its oriented transcendental lattice T X , and Aut(X) is always infinite. See [33] for the standard Gram matrices a b b c of oriented transcendental lattices of singular K3 surfaces. Let disc T X denote the discriminant of T X .…”

Section: Introductionmentioning

confidence: 99%

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

Shimada

2015

International Mathematics Research Notices

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“…The double covering ramified along E ∪C produces an elliptic K3-surface (see Figure 5) of type [1,2,3,4,6,8] according to the standard notation; see [14]. This is an extremal elliptic K3-surface listed as number 34 in Shimada and Zhang's Table; see [17].…”

Section: Other Examplesmentioning

confidence: 99%

Effective invariants of braid monodromy

Bartolo¹,

Ruber²,

Agustín³

2006

Trans. Amer. Math. Soc.

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Abstract. In this paper we construct new invariants of algebraic curves based on (not necessarily generic) braid monodromies. Such invariants are effective in the sense that their computation allows for the study of Zariski pairs of plane curves. Moreover, the Zariski pairs found in this work correspond to curves having conjugate equations in a number field, and hence are not distinguishable by means of computing algebraic coverings. We prove that the embeddings of the curves in the plane are not homeomorphic. We also apply these results to the classification problem of elliptic surfaces.Our purpose in this paper is the construction of new and effective topological invariants for algebraic curves. These invariants will be derived from the braid monodromy of a triple (C, L, P ) introduced in [3], where C ⊂ P 2 is an algebraic curve, L ⊂ C is a line and P ∈ L. This invariant is derived from the classical braid monodromy; see references in [3].Several topological invariants of plane algebraic curves, obtained from the fundamental group of the complement, have been found to be effective, such as the Alexander polynomial, the Alexander module or the sequence of characteristic varieties [13]. A second group of effective invariants, described by A. Libgober in [12], depend on polynomial representations of the braid group and are invariants of the conjugation class of the monodromy group.In this paper, braid monodromy of a triple is represented by an element of B Using finite representations of braid groups we obtain new effective invariants for braid monodromies. Effectiveness is obtained by means of the free software GAP4 [10]. We combine these invariants with our main theorem in [3] which relates braid monodromy to topology.Braid monodromy may also be useful in the study of the moduli space of projective curves with prescribed degree and topological types of curve singularities. Let d be a positive integer and let T 1 , . . . , T r be topological types of curve singularities. Let us denote by Σ(T 1 , . . . , T r ; d) the space of all projective plane curves of degree d with exactly r singular points of type T 1 , . . . , T r . Let M(T 1 , . . . , T r ; d) be the

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“…He used the ideas of Urabe [30] and reduced the problem of listing up all rational double points on these complex K3 surfaces to lattice theoretic calculations via Torelli's theorem. By a similar method, the complete list of configurations of singular fibers on complex elliptic K3 surfaces has been obtained in [25] and [24]. In [9], the maximal configurations of ordinary nodes on rational surfaces in characteristic = 2 are investigated.…”

Section: Theorem 11 An Rdp -Triple (R N σ) Is Geometrically Realimentioning

confidence: 99%

“…In §2.3, we quote from Artin [4], Rudakov-Šafarevič [20], [21] and Shioda [27] some fundamental facts about the Picard lattices of supersingular K3 surfaces. These facts play, in positive characteristics, the same role as the one Torelli's theorem played for complex K3 surfaces in [24], [25], [30] and [31], [32]. The algorithms for obtaining the lists of geometrically realizable RDP -triples and of triples of extremal (quasi-)elliptic K3 surfaces are presented in §3 and §4, respectively.…”

Section: Theorem 11 An Rdp -Triple (R N σ) Is Geometrically Realimentioning

confidence: 99%

Rational double points on supersingular 𝐾3 surfaces

Shimada

2004

Math. Comp.

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Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces

Abstract: Abstract. We present a complete list of extremal elliptic K3 surfaces (Theorem 1

Cited by 75 publications

References 15 publications

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

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

Effective invariants of braid monodromy

Rational double points on supersingular 𝐾3 surfaces

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