Table Of Content

Gosset Polytopes in Picard groups 0 of del Pezzo Surfaces 1 0 2 Jae-Hyouk Lee n a January 23, 2010 J 3 2 Abstract ] G In this article, we research on the correspondences between the geometry of del Pezzo surfaces Sr and the geometry of Gosset polytopes A (r−4)21. We construct Gosset polytopes (r−4)21 in Pic Sr⊗Q whose h. vertices are lines, and we identify divisor classes in Pic Sr corresponding t to(a−1)-simplexes(a≤r),(r−1)-simplexesand(r−1)-crosspolytopes a of the polytope (r−4)21. Then we explain these classes correspond to m skew a-lines(a≤r), exceptional systems and rulings, respectively. [ As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r−4)21. And we show Gieser 2 transformation and Bertini transformation induce a symmetry of poly- v 5 topes 321 and 421, respectively. 2 0 3 . 5 0 9 0 : v i X r a 1 1 Introduction The celebratedDynkin diagramsappear as the key ingredientsin many mathe- maticalresearchareas. Inthegeometryofpolytopes,theyrepresentthedihedral angles between the hyperplanes generating the polytopes, and in the algebraic geometryofsurfaces,they arethe intersectionsbetweenthe simple rootsgener- atingarootspace. Infact,the diagramsineachofaboveresearchescorrespond to the relationships presenting symmetry groups which commonly appear in eachstudy on the objects representedby the graphs. In particular, the Dynkin diagrams of the Lie groups E 3 ≤ r ≤ 8 correspond to both the Weyl groups r W(S ) of del Pezzo surfaces S and the symmetry groups of semiregular E - r r r polytopes (r−4) , which is also known as Gosset polytopes. Therefore, there 21 arenaturalcorrespondencesbetweenthe geometryof the del Pezzo surfaceand thegeometryofthe(r−4) polytope. Thisarticleexploresthecorrespondences 21 between del Pezzo surfaces and (r−4) polytopes. 21 The delPezzosurfacesaresmoothirreduciblesurfacesS whoseanticanoni- r calclass−K isample. WecanconstructthedelPezzosurfacesbyblowingup Sr r ≤8pointsfromP2 unlessitisP1×P1. Inparticular,itisverywellknownthat there are 27 lines on a cubic surface S and the configuration of these lines is 6 acted by the Weyl groupE ([10][11][13]). The set of 27-linesin S are bijective 6 6 to the set of vertices of a Gosset 2 polytope, i.e. an E -polytope. The sim- 21 6 ilar correspondences were found for the 28-bitangents in S and the tritangent 7 planes for S . The bijection between lines in S and vertices in 2 was applied 8 6 21 tostudythegeometryof2 byCoxeter([9]). Andthe completelist(see[17])of 21 bijectionsbetweenthedivisorclassescontaininglinesandverticesiswellknown and applied in many different research fields. These divisor classes which are also called lines play key roles in this article. The lines in del Pezzo surfaces are studied in many different directions. Re- cently,LeungandZhangrelatetheconfigurationsofthelinestothegeometryof the line bundles overdel Pezzosurfacesvia the representationtheory([15][16]). Another interesting researches driven from the lines in del Pezzo surfaces and their symmetry groups can be found in [1], [12] and [18]. AlineinPicS isequivalentlyadivisorclasslwithl2 =−1andK ·l=−1. r Sr Weobservethatthe WeylgroupW(S )actsasanaffinereflectiongrouponthe r affine hyperplane given by D · K = −1. Furthermore, W(S ) acts on the Sr r set of lines in Pic S . Therefrom, we construct a Gosset polytope (r−4) in r 21 PicS ⊗QwhoseverticesareexactlythelinesinPicS . ForaGossetpolytope r r (r−4) ,subpolytopesareregularsimplexesexceptsthe facetswhichconsistof 21 (r−3)-simplexesand(r−3)-crosspolytopes. Sincethesubpolytopesin(r−4) 21 are basically configurations of vertices, we obtain natural characterization of subpolytopes in (r−4) as divisor classes in Pic S . 21 r Now,wewanttousethealgebraicgeometryofdelPezzosurfacestoidentify the divisor classes corresponding to the subpolytopes in (r − 4) . For this 21 purpose, we consider the following set of divisor classes which are called skew 01991Mathematics SubjectClassification: 51M20,14J26,22E99 2 a-lines, exceptional systems and rulings in Pic S . r The skew a-line is an extension of the definition of lines in S . We show r that eachskew a-line representsan (a−1)-simplex in an (r−4) polytope. In 21 fact, the skew a-lines also holds D2 =−a and D·K =−a. Furthermore the Sr divisors with these conditions are equivalently skew a-lines for a≤3. Theexceptional system isadivisorclassinPicS whoselinearsystemgives r a regular map from S to P2. As this regular map corresponds to a blowing up r from P2 to S , naturally we relate exceptional systems to (r−1)-simplexes in r (r−4) polytopes,whichistheoneoftwotypesoffacetsappearingin(r−4) 21 21 polytopes. We show the set E is bijective to the set of the (r−1)-simplexes in r (r−4) polytopes, for 3≤r ≤ 7. 21 The ruling is a divisor class in Pic S which gives a fibration of S over r r P1. And we show that the F is bijective to the set of (r−1)-crosspolytopes r in the (r−4) polytope. Furthermore, we explain the relationships between 21 lines and rulings according to the incidence between the vertices and (r−1)- crosspolytopes. Thisleadsusthatapairofpropercrosspolytopesinthe(r−4) 21 give the blowing down maps from S to P1× P1. r Afterpropercomparisonbetweendivisorclassesobtainedfromthegeometry ofthepolytope(r−4) andthosegivenbythegeometryofadelPezzosurface, 21 we come by the following correspondences. del Pezzo surface S E-semiregular polytopes (r−4) r 21 lines vertices skew a-lines 1≤a≤r (a−1)-simplexes 1≤a≤r exceptional systems (r−1)-simplexes (r <8) rulings (r−1)-crosspolytopes The nature of the above correspondences is macroscopic, and that we need a microscopic explanation of the correspondences to decode the local geometry of the (r−4) polytopes. Thus, we consider the monoidal transform for lines 21 ondelPezzosurfacesanddescribethelocalgeometryofthe(r−4) polytopes. 21 This blowing up procedure on lines can be applied to rulings to get a useful recursive description. This will be discussed along the corresponding geometry on the polytope (r−4) in [3][14]. 21 Asanotherapplication,weconsiderthepairsoflinesinPicS (resp. PicS ) 7 8 withintersection2(resp. 3)whicharerelatedtothe28bitangents(resp. tritan- gentplane). AndwedefineGiesertransformation(resp. Bertinitransformation) on the polytope 3 (resp. 4 ) and show that this is another symmetry. 21 21 The researches on regular and semiregular polytopes along the Coxeter- Dynkindiagramshavealonghistorywhichmaybewellknownonlyasfacts. So we begin with the preliminaries on the theories of the regular and semiregular polytopes in the next section. 3 2 Regular and Semiregular Polytopes In this article, we deal with polytopes with highly nontrivial symmetries, and their symmetry groups play key roles along the corresponding Coxeter-Dynkin diagrams. In this section, we revisit the general theory of regular and semiregular polytopes according to their symmetry groups and Coxeter-Dynkin diagrams. Especially, we consider a family of semiregular polytopes known as Gosset figures (k according to Coxeter). The combinatorial data of Gosset 21 figures along the group actions will be used everywhere in this article. For further detail of the theory, readers consult Coxeter’s papers [5][6][7][8]. Let P be a convex n-polytope in an n-dimensional Euclidean space. For n each vertex O, the midpoints of all the edges emanating from a vertex O in P n forman(n−1)-polytopeiftheylieinahyperplane. Wecallthis(n−1)-polytope the vertex figure of P at O. n A polytope P (n > 2) is said to regular if its facets are regular and there n is a regular vertex figure at each vertex. When n = 2, a polygon P is regular 2 if it is equilateral and equiangular. Naturally, the facets of regular P are all n congruent, and the vertex figures are all the same. We consider two classes of regular polytopes. (1) A regular simplex α is an n-dimensional simplex with equilateral n edges. For example, α is a line-segment, α is an equilateral triangle, and α 1 2 3 is a tetrahedron. Note α is a pyramid based on α . Thus the facets of a n n−1 regularsimplex α isaregularsimplexα ,andthe vertexfigureofα isalso n n−1 n α . Furthermore, the symmetry group of α is the Coxeter group A with n−1 n n order (n+1)!. (2) A crosspolytope β is an n-dimensional polytope whose 2n-vertices n are the intersects between an n-dimensional Cartesian coordinate frame and a sphere centered at the origin. For instance, β is a line-segment, β is a 1 2 square, and β is an octahedron. Note β is a bipyramid based on β , and 3 n n−1 the n-vertices in β form α if the choice is made as one vertex from each n n−1 Cartesian coordinate line. So the vertex figure of a crosspolytope β is also a n crosspolytope β , and the facets of β is α . And the symmetry group of n−1 n n−1 β is the Coxeter group D with order 2n−1n!. n n ApolytopeP iscalledsemiregular ifitsfacetsareregularanditsverticesare n equivalent, namely, the symmetry group of P acts transitively on the vertices n of P . n Here,weconsiderthe semiregulark polytopesdiscoveredbyGosset which 21 are(k+4)-dimensionalpolytopeswhosesymmetrygroupsaretheCoxetergroup E . Note that the vertex figure of k is (k−1) and the facets of k are k+4 21 21 21 regular simplexes α and crosspolytopes β . The list of k polytopes is k+3 k+3 21 following. 4 k E order of E k -polytopes k+4 k+4 21 −1 A ×A 12 triangular prism 1 2 0 A 5! rectified 5-cell 4 1 D 245! demipenteract 5 2 E 72×6! E -polytope 6 6 3 E 8×9! E -polytope 7 7 4 E 192×10! E -polytope 8 8 The Coxeter groups are reflection groups generated by the reflections with respect to hyperplanes (called mirrors), and the Coxeter-Dynkin diagrams of Coxeter groups are labeled graphs where their nodes are indexed mirrors and thelabelsonedgespresenttheordernofdihedralangle π betweentwomirrors. n If two mirrors are perpendicular, namely n = 2, no edge joins two nodes presenting the mirrors because there is no interaction between the mirrors. Since the dihedral angle π appears very often, we only label the edges when the cor- 3 responding order n > 3. Each Coxeter-Dynkin diagram contains at least one ringed node which represents an active mirror, i.e. there is a point off the mirror, and the constructing a polytope begins with reflecting the point through the active mirror. We call the Coxeter-Dynkin diagram of α (respectively β and k ) with n n 21 the Coxeter group A (respectively D and E ) A -type (respectively D - n n n n n and E -type), and each Coxeter-Dynkin diagram of A , D and E -type has n n n n only one ringed node and no labeled edges. For this case, the following simple procedure using the Coxeter-Dynkin diagram describes possible subpolytopes and calculates total numbers of them (see also [5][7]). The Coxeter-Dynkin diagram of each subpolytope P′ is a connected subgraph Γ containing the ringed node. And the subgraph obtained by taking off all the nodes joined with the subgraph Γ represents the isotropy group GP′ of P′. Furthermore,theindexbetweenthesymmetrygroupGoftheambientpoly- tope and isotropy group GP′, gives the total number of such subpolytope. In particular,bytakingofftheringednode,weobtainthesubgraphcorresponding totheisotropygroupofavertex,andinfacttheisotropygroupisthesymmetry group of the vertex figure. (1) Regular simplex α with the symmetry group A . n n · − · − ... − ⊙ 1 2 n Coxeter-Dynkindiagramofαn The diagram of vertex figure is A -type since it is represented by the n−1 subgraphremaining after removing the ringed node, and the facet is only α n−1 becausethesubgraphofA -typeisthebiggestsubgraphcontainingtheringed n−1 node in the graph of A -type. Furthermore, since all the possible subgraphs n containingthe ringednode are A -type, only regularsimplex α , 0≤k ≤n−1 k k appearsassubpolytopes. Andforeachα inα ,thepossibletotalnumberNαn k n αk 5 is (n+1)! n+1 Nαn =[A :A ×A ]= = . αk n k n−k−1 (k+1)!(n−k)! (cid:18) k+1 (cid:19) (2) Cross polytope β with the symmetry group D . n n 1 · p · − · − ... − ⊙ 2 3 n Coxeter-Dynkindiagramofβn The diagram of vertex figure is D -type because the subgraph remaining n−1 after removing the ringed node represents D , and the facet is only α n−1 n−1 sincesubgraphofA -typeisthebiggestsubgraphcontainingtheringednode n−1 in D -type. Only regular simplex α , k = 0,...,n−1 appears as subpolytopes n k since the possible subgraphs containing the ringed node are only A -type. And k for each α in β , the possible total number Nβn is k n αk 2n−1n! n Nβn =[D :A ×D ]= =2k+1 . αk n k n−k−1 (k+1)!2n−k−2(n−k−1)! (cid:18) k+1 (cid:19) In particular, each β contains Nβn = 2n vertices, and these vertices form n α0 n-pairs with the common center. (3) Gosset polytope k with the symmetry group E , −1≤k ≤4. 21 k+4 · p · − · − · − · − ... − ⊙ −1 0 1 k Coxeter-Dynkindiagramofk21 k6=−1 Fork 6=−1,thediagramofvertexfigureisE -typeandthe facetsarethe k+3 regular simplex α and the crosspolytope β since the subgraphs of A - k+3 k+3 k+3 type andD -type appearasthe biggestsubgraphcontainingthe ringednode k+3 inE -type. Butallthelowerdimensionalsubpolytopesareregularsimplexes. k+3 Case k = −1 is a bit different with other cases since there are two ringed nodes. ⊙ · − ⊙ −1 Coxeter-Dynkindiagramof−121 The vertex figure is an isosceles triangle instead of an equilateral triangle because the corresponding diagram is obtained by taking off a ringed node in the A -type subgraph. And the facets are the regular triangle α given by 2 2 the A -type subgraph and the square β given by the subgraph taking off the 2 2 unringed node. As above,wecancalculatethe totalnumberofsubpolytopesink byusing 21 Coxeter-Dynkin diagram. For instance, we calculate 2 . After removing the 21 6 ringednodelabelled2,weobtainasubgraphofE -type,andthereforethevertex 5 figure of 2 is 1 . Since the subgraphs of A -type and D -type are all the 21 21 5 5 possible biggestsubgraphsinthe Coxeter-Dynkindiagramof2 , there aretwo 21 types of facets in 2 , which are 5-simplexes and 5-crosspolytopes,respectively. 21 And all other subpolytopes in 2 are simplexes for the same reason. In the 21 following calculation for 2 , the nodes marked by ∗ represent deleted nodes. 21 · p · − · − · − · − ⊙ −1 0 1 2 Coxeter-Dynkindiagramof221 (a) Vertices in 2 : N221 =[E :E ]=27 21 α0 6 5 · p · − · − · − · ··· ∗ −1 0 1 2 (b) 1-simplexes(edges) in 2 : N221 =[E :A ×E ]=216 21 α1 6 1 4 · p · − · − · ··· ∗ ··· ⊙ −1 0 1 2 (c) 2-simplexes(faces) in 2 : N221 =[E :A ×E ]=720 21 α2 6 2 3 · . . . · − · ··· ∗ ··· · − ⊙ −1 0 1 2 (d) 3-simplexes(cells) in 2 : N221 =[E :A ×A ]=1080 21 α3 6 3 1 ∗ . . . · − ∗ ··· · − · − ⊙ −1 0 1 2 (e) 4-simplexes in 2 : N221 =[E :A ×A ]+[E :A ]=648 21 α4 6 4 1 6 4 ∗ · . p .. · ··· ∗ ··· · − · − ⊙ , ∗ ··· · − · − · − ⊙ −1 0 1 2 −1 0 1 2 (f) 5-simplexes in 2 : N221 =[E :A ]=72 21 α5 6 5 ∗ . . . · − · − · − · − ⊙ −1 0 1 2 7 (g) 5-crosspolytopes in 2 : N221 =[E :D ]=27 21 β5 6 5 · p ∗ ··· · − · − · − ⊙ . −1 0 1 2 As we apply the same procedure to the other E-polytopes, we get the following table. E -polytope(k ) −1 0 1 2 3 4 k+4 21 21 21 21 21 21 21 β 3 5 10 27 126 2160 k+3 vertex 6 10 16 27 56 240 α 9 30 80 216 756 6720 1 α 2 30 160 720 4032 60480 2 α 5 120 1080 10080 241920 3 α 16 648 12096 483840 4 α 72 6048 483840 5 α 576 207360 6 α 17280 7 Numbersofsubpolytopesink21 3 Del Pezzo surfaces S r A del Pezzo surface is a smooth irreducible surface whose anticanonicalclass − K isample. ItiswellknownthatadelPezzosurfaceS ,unlessitisP1×P1,can S r beobtainedfromP2 byblowingupr ≤8pointsingenericpositions,namely,no three points are on a line, no six points are on a conic, and for r =8, not all of themareonaplanecurvewhosesingularpointis oneofthem(see[10][13][17]). Notation : We do not use different notations for the divisors and the corresponding classes in Picard group unless there is confusion. We denote such a del Pezzo surface by S and the corresponding blow up r by π : S → P2. And K2 = 9 − r is called the degree of the del Pezzo r r Sr surface. Each exceptional curve and the corresponding class given by blowing up is denoted by e , and both the class of π∗(h) in S and the class of a line h i r r in P2 are referred as h. Then, we have h2 =1, h·e =0, e ·e =−δ for 1≤i,j ≤r, i i j ij and the Picard group of S is r Pic S ≃Zh⊕Ze ⊕...⊕Ze r 1 r r with the signature (1,−r). And K =−3h+ e . Sr i=1 i P 8 For any irreducible curve C on a Del Pezzo surface S , we have C ·K r Sr < 0 since −K is ample. Furthermore, if the curve C has a negative self- Sr intersection,C mustbeasmoothrationalcurvewithC2 =−1bytheadjunction formula. The ample −K on a del Pezzo surface S is very beneficial to deal with Sr r Pic S . The inner product given by the intersection on Pic S induces a neg- r r ative definite metric on (ZK )⊥ in Pic S where we can also define natural Sr r reflections. To define reflections on (ZK )⊥ in Pic S , we consider a root system Sr r R :={d∈Pic S |d2 =−2, d·K =0}, r r Sr with simple roots d =h−e −e −e ,d =e −e , 1≤i≤r−1. 0 1 2 3 i i i+1 Each element d in R defines a reflection on (ZK )⊥ in Pic S r Sr r σ (D):=D+(D·d)d for D ∈(ZK )⊥ d Sr and the corresponding Weyl group W(S ) is E where 3 ≤ r ≤ 8 with the r r Dynkin diagram d·0 p · − · − · − · − ... − · d1 d2 d4 d5 dr−1 DynkindiagramofEr r≥3 The definition of the reflection σ on (ZK )⊥ can be used to obtain a d Sr transformation both on Pic S and on Pic S ⊗Q≃Qh⊕Qe ⊕...⊕Qe via r r 1 r the linear extension of the intersections of divisors in Pic S . Here Pic S ⊗Q r r is a vector space with the signature (1,−r). Affine hyperplanes and the reflection groups Lateron,wedealwithdivisorclassesDsatisfyingD·K =α,D2 =β where Sr α and β are integers along the action of Weyl group W(S ). Here, we know r W(S ) is generated by the reflections on (ZK )⊥ given by simple roots. To r Sr extendthe actionofW(S )properly,wewanttoshowthatthesereflectionsare r defined on Pic S and preserve the above equations. Furthermore, we see that r W(S ) acts as a reflection groupon the set of divisor classes with D·K =α. r Sr We consider an affine hyperplane section in Pic S ⊗Q defined by r H˜ :={D ∈Pic S ⊗Q|−D·K =b} b r Sr where b is an arbitrary real number and an affine hyperplane section H := b H˜ ∩Pic S in Pic S . Since −K is ample, we are interested in b≥0. b r r Sr 9 By the fact that K2 =9−r >0, 3≤r≤ 8 and Hodge index theorem Sr 0=(K ·(D −D ))2 ≥K2 (D −D )2 , D ,D ∈H , Sr 1 2 Sr 1 2 1 2 b the inner product on Pic S induces a negative definite metric on H . As a r b matter of fact, the induced metric is defined on Pic S ⊗Q, and we can also r consider the induced norm by fixing a center b K in the affine hyperplane 9−r Sr section −D·K =b in Pic S ⊗Q. This norm is also negative definite. Sr r Lemma 1 (1)Let H˜ (b ≥ 0) be an affine hyperplane section in Pic S ⊗Q b r defined above and b K be a center on the affine hyperplane section. The 9−r Sr classes D in H = H˜ ∩ Pic S with a fixed self-intersection are on a sphere b b r with the center b K in H˜ . 9−r Sr b (2)ForeachrootdinR ,thecorrespondingreflectionσ definedonPicS ⊗ r d r Q is an isometry preserving K and acts as a reflection on each hyperplane Sr section H˜ with the center b K . b 9−r Sr Proof: (1) Consider b 2 2b b2 b2 D− K =D2− D·K + K2 =D2− ≤0 (cid:18) 9−r Sr(cid:19) 9−r Sr (9−r)2 Sr (9−r) and the last inequality is given by the Hodge index theorem b2 =(D·K )2 ≥D2K2 =D2(9−r). Sr Sr (2) Eachroot d in R satisfies d· K =0 and d2 =−2. Therefore, we have r Sr σ (K )=K +(d·K )d=K , d Sr Sr Sr Sr and for each D ,D ∈Pic S ⊗Q 1 2 r σ (D )·σ (D )=(D +(d·D )d)·(D +(d·D )d)=D ·D . d 1 d 2 1 1 2 2 1 2 Furthermore,foreachclassDinPicS ⊗Q,theself-intersectionD2 andD·K r Sr are invariant under σ . This implies σ acts on the hyperplane section H˜ . d d b Moreover, the hyperplane in Pic S ⊗Q preserved by the action of σ is given r d by an equation d·D = 0 for D ∈ Pic S ⊗Q, and the each center b K of r 9−r Sr H˜ is in this hyperplane. And because each class D in H˜ can be written as b b b D =D + K for some D ∈H˜ , 3 9−r Sr 3 0 we have b b b σ (D)=σ (D + K )=σ (D )+ K ∈H˜ + K =H˜ . d d 3 9−r Sr d 3 9−r Sr 0 9−r Sr b Since σ is a reflection onH˜ , we can derive a fact that the isometry σ acts as d 0 d an affine reflection on H˜ for the center b K . (cid:4) b 9−r Sr 10