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K3-Surfaces with Special Symmetry

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K3-Surfaces with Special Symmetry K3-surfaces with special symmetry Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften an der Fakult¨at f¨ur Mathematik der Ruhr-Universit¨at Bochum arXiv:0902.3761v1 [math.AG] 23 Feb 2009 vorgelegt von Kristina Frantzen im Oktober 2008 Contents Introduction 7 1 Finite group actions on K3-surfaces 10 1.1 Basicnotationanddefinitions . ......... 10 1.2 QuotientsofK3-surfaces . ........ 11 1.2.1 Quotients by finite groups of symplectic transformations........... 11 1.2.2 Quotients by finite groups of nonsymplectic transformations......... 13 1.3 Antisymplectic involutions on K3-surfaces . ............. 15 1.3.1 PicardlatticesofK3-surfaces . ........ 16 1.3.2 The fixed point set of an antisymplectic involution . ............ 16 1.4 Finite groups of symplectic automorphisms . ............ 17 1.4.1 Examples of K3-surfaces with symplectic symmetry . ........... 18 2 Equivariant Mori reduction 20 2.1 Theconeofcurvesandtheconetheorem . ........ 20 2.2 Surfaces with group action and the cone of invariant curves ............. 21 2.3 The contraction theorem and minimal models of surfaces . .............. 22 2.4 Equivariant contraction theorem and G-minimalmodels . 23 3 Centralizers of antisymplectic involutions 29 3.1 DelPezzosurfaces ................................ ..... 30 3.2 BranchcurvesandMorifibers. ....... 32 3.2.1 Rationalbranchcurves. ..... 34 3.2.2 Ellipticbranchcurves . ..... 37 3.3 Roughclassification.... .... ... .... .... .... ... .... ....... 38 3 4 Contents 4 Mukaigroupscentralizedbyantisymplecticinvolutions 39 4.1 The group L2(7) ....................................... 41 4.2 The group A6 ........................................ 41 4.3 The group S5 ......................................... 43 4.3.1 Double covers of Del Pezzo surfaces of degreethree . ........... 44 4.3.2 DoublecoversofDelPezzosurfacesofdegreefive . ......... 45 4.3.3 Conclusion .................................... 48 4 4.4 The group M20 = C2 ⋊ A5 ................................. 48 4 4.5 The group F384 = C2 ⋊ S4 ................................. 49 4 4.6 The group A4,4 = C2 ⋊ A3,3 ................................ 50 4.7 The groups T =(Q Q ) ⋊ S and H = C4 ⋊ D ................ 50 192 8 ∗ 8 3 192 2 12 2 4.8 The group N72 = C3 ⋊ D8 ................................. 51 2 4.9 The group M9 = C3 ⋊ Q8 ................................. 54 4.10 The group T48 = Q8 ⋊ S3 .................................. 56 5 K3-surfaces with an antisymplectic involution centralizing C3 ⋉ C7 63 5.1 BranchcurvesandMorifibers. ....... 64 5.2 Classification of the quotient surface Y .......................... 66 5.3 Fine classification - Computation of invariants . .............. 67 5.3.1 Thecase Y = Ymin = P2 .............................. 67 5.3.2 Equivariantequivalence . ...... 68 5.3.3 Thecase Y = Y ................................. 69 6 min 5.4 Klein’s quartic and the Klein-Mukai surface . ............. 69 5.5 The group L2(7) centralized by an antisymplectic involution . 71 6 The simple group of order 168 72 6.1 Finite groups containing L2(7) .............................. 72 6.2 Non-existence of K3-surfaces with an action of L (7) C .............. 73 2 × 3 6.2.1 Globalstructure............................... .... 73 6.2.2 Mori contractions and C7-fixedpoints ...................... 75 6.2.3 LiftingKlein’squartic . ...... 76 Case 1: The curve Bˆ isreducible ......................... 77 Case 2: The curve Bˆ isirreducible ........................ 80 Conclusion......................................... 82 Contents 5 7 The alternating group of degree six 83 7.1 The group A˜6 ........................................ 83 7.1.1 The centralizer G of σ in A˜6 ............................ 84 7.1.2 The group H = G/ σ ............................... 86 h i 7.2 H-minimal models of Y .................................. 86 7.3 BranchcurvesandMorifibers. ....... 89 7.3.1 Twoellipticbranchcurves. ...... 91 7.4 Rough classification of X .................................. 93 7.4.1 Theactionof H on P P ............................ 94 1 × 1 7.4.2 Invariant curves of bidegree (4,4) ........................ 95 7.4.3 Refining the classification of X .......................... 95 7.5 Summaryandoutlook ............................... .... 98 A ActionsofcertainMukaigroupsonprojectivespace 100 A.1 The action of N72 on P3 ................................... 100 A.1.1 Invariantquadricsandcubics. ....... 101 A.2 The action of M9 on P2 ................................... 102 Bibliography 104 Introduction K3-surfaces are special two-dimensional holomorphic symplectic manifolds. They come equip- ped with a symplectic form ω, which is unique up to a scalar factor, and their symmetries are naturally partitioned into symplectic and nonsymplectic transformations. An important class of K3-surfaces consists of those possessing an antisymplectic involution, i.e., a holomorphic involution σ such that σ ω = ω. ∗ − K3-surfaces with antisymplectic involution occur classically as branched double covers of the projective plane, or more generally of Del Pezzo surfaces. This construction is a prominent source of examples and plays a significant role in the classification of log Del Pezzo surfaces of index two (see the works of Alexeev and Nikulin e.g. in [AN06] and the classification by Nakayama [Nak07]). Moduli spaces of K3-surfaces with antisymplectic involution are studied by Yoshikawa in [Yos04], [Yos07], and lead to new developments in the area of automorphic forms. In this monograph we study K3-surfaces with antisymplectic involution from the point of view of symmetry. On a K3-surface X with antisymplectic involution it is natural the consider those holomorphic symmetries of X compatible with the given structure (X, ω, σ). These are symplectic automorphisms of X commuting with σ. Given a finite group G one wishes to understand if it can act in the above fashion on a K3-surface X with antisymplectic involution σ. If this is the case, i.e., if there exists a holomorphic action of G on X such that g ω = ω and g σ = σ g for all g G, then the structure of G can yield strong ∗ ◦ ◦ ∈ constraints on the geometry of X. More precisely, if the group G has rich structure or large order, it is possible to obtain a precise description of X. This can be considered the guiding classification problem of this monograph. In Chapter 3 we derive a classification of K3-surfaces with antisymplectic involution centralized by a group of symplectic automorphisms of order greater than or equal to 96. We prove (cf. Theorem 3.25): Theorem 1. Let X be a K3-surface with a symplectic action of G centralized by an antisymplectic invo- lution σ such that Fix(σ) = ∅. If G > 96, then X/σ is a Del Pezzo surface and Fix(σ) is a smooth 6 | | connected curve C with g(C) 3. ≥ By a theorem due to Mukai [Muk88] finite groups of symplectic transformations on K3-surfaces are characterized by the existence of a certain embedding into a particular Mathieu group and are subgroups of eleven specified finite groups of maximal symplectic symmetry. This result naturally limits our considerations and has led us to consider the above classification problem for a group G from this list of eleven Mukai groups. Theorem 1 above can be refined to obtain a complete classification of K3-surfaces with a sym- plectic action of a Mukai group centralized by an antisymplectic involution with fixed points (cf. Theorem 4.1). 7 8 Introduction Theorem 2. Let G be a Mukai group acting on a K3-surface X by symplectic transformations. Let σ be an antisymplectic involution on X centralizing G with Fix (σ) = ∅. Then the pair (X, G) can be found in X 6 Table 4.1. In addition to a number of examples presented by Mukai we find new examples of K3-surfaces with maximal symplectic symmetry as equivariant double covers of Del Pezzo surfaces. It should be emphasized that the description of K3-surfaces with given symmetry does however not necessary rely on the size of the group or its maximality and a classification can also be ob- tained for rather small subgroups of the Mukai groups. In order to illustrate that the approach does rather depend on the structure of the group, we prove a classification of K3-surfaces with a symplectic action of the group C3 ⋉ C7 centralized by an antisymplectic involution in Chapter 5. The surfaces with this given symmetry are characterized as double covers of P2 branched along invariant sextics in a precisely described one-dimensional family (Theorem 5.4). M Theorem 3. The K3-surfaces with a symplectic action of G = C3 ⋉ C7 centralized by an antisymplectic involution σ are parametrized by the space of equivalence classes of sextic branch curves in P . M 2 The group C3 ⋉ C7 is a subgroup of the simple group L2(7) of order 168 which is among the Mukai groups. The actions of L2(7) on K3-surfaces have been studied by Oguiso and Zhang [OZ02] in an a priori more general setup. Namely, they consider finite groups containing L2(7) as a proper subgroup and obtain lattice theoretic classification results using the Torelli theorem. Since a finite group containing L2(7) as a proper subgroup posseses, in the cases considered, an antisymplectic involution centralizing L2(7), we can apply Theorem 4.1 and improve the existing result (cf. Theorem 6.1). All classification results summarized above are proved by applying the following general strat- egy. The quotient of a K3-surface by an antisymplectic involution σ with fixed points centralized by a finite group G is a rational G-surface Y. We apply an equivariant version of the minimal model program respecting finite symmetry groups to the surface Y. Chapter
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