Classification of Toric log Del Pezzo Surfaces having Picard Number 1 and Index ≤3
aa r X i v : . [ m a t h . AG ] S e p CLASSIFICATION OF TORIC LOG DEL PEZZO SURFACESHAVING PICARD NUMBER AND INDEX ≤ DIMITRIOS I. DAIS
To Professor Friedrich Hirzebruch on the occasion of his eightieth birthday
Abstract.
Toric log Del Pezzo surfaces with Picard number 1 have beencompletely classified whenever their index is ≤ . In this paper we extendthe classification for those having index 3 . We prove that, up to isomorphism,there are exactly 18 surfaces of this kind. Introduction
Smooth compact toric surfaces belong to the basics in the framework of toric ge-ometry. They are rational surfaces (i.e., of Kodaira dimension −∞ ) defined by2-dimensional complete fans which are composed of basic cones, and can thereforebe studied by means of handy combinatorics (see [13, Theorem 1.28, pp. 42-43]).Of course, unlike the smooth compact complex surfaces having Kodaira dimension ≥ , they do not possess uniquely determined minimal models . Nevertheless, theset of their minimal models consists of the projective plane P C together with the Hirzebruch surfaces F κ := (cid:8) ([ z : z : z ] , [ t : t ]) ∈ P C × P C (cid:12)(cid:12) z t κ = z t κ (cid:9) , κ ∈ Z ≥ , for κ = 1 (cf. [10], [8, § § elementary transformations .In contrast to this classical point of view, taking into account the fact that the anti-Kodaira dimension of smooth compact toric surfaces is 2 , and switching to theso-called antiminimal and anticanonical models (in the sense of Sakai [15, § uniquely determined up to isomorphism.However, since these models are mostly singular , in order to follow this choice weneed a more systematic study of singular compact toric surfaces.A graph-theoretic method of classifying (not necessarily smooth) compact toricsurfaces up to isomorphism (generalizing Oda’s graphs [13, pp. 44-46]) has beenproposed in [5, § wve c - graphs , for short) are isomorphic (see below Theorem 4.4).In addition, by [14, Theorem 4.3, pp. 398-399] the anticanonical models ofsmooth compact toric surfaces have to be log Del Pezzo surfaces. (A compact com-plex surface X with at worst log terminal singularities, i.e., quotient singularities,is called log Del Pezzo surface if its anticanonical divisor − K X is a Q -Cartier ampledivisor. The index of such a surface is defined to be the smallest positive integer ℓ Mathematics Subject Classification. for which ℓK X is a Cartier divisor. The family of log Del Pezzo surfaces of fixedindex ℓ is known to be bounded (see [2, Theorem 2.1, p. 332])).Consequently, it seems to be rather interesting to classify toric log Del Pezzosurfaces of given index ℓ up to isomorphism. A first attempt to understand thecombinatorial complexity of this classification problem includes naturally the in-vestigation of the case in which the Picard number ρ ( X ∆ ) := rank(Pic ( X ∆ )) ofsurfaces X ∆ of this kind (associated to complete fans ∆ in R ) equals 1 . In thiscase, X ∆ ’s turn out to be weighted projective planes or quotients thereof by a finiteabelian group. Let us first recall what is known for indices ℓ ≤ Theorem 1.1.
Up to isomorphism, there are exactly toric log del Pezzo surfaceswith Picard number and index ℓ = 1 , namely No. (i) (ii) (iii) (iv) (v) X ∆ P C P C / ( Z / Z ) P C (1 , , P C (1 , , / ( Z / Z ) P C (1 , , whose wve c -graphs are illustrated in [5, Figure 8, p. 108] . Theorem 1.2.
Up to isomorphism, there are exactly toric log del Pezzo surfaceswith Picard number and index ℓ = 2 , namely No. X ∆ No. X ∆ (i) P C (1 , ,
4) (iv) P C (1 , , / ( Z / Z )(ii) P C (1 , ,
5) (v) P C (1 , , / ( Z / Z )(iii) P C (1 , ,
8) (vi) P C (1 , , / ( Z / Z )(vii) P C (1 , , / ( Z / Z ) whose wve c -graphs are illustrated in [5, Figure 11, p. 111] . In the present paper we extend these results also for index 3 by the following:
Theorem 1.3.
Up to isomorphism, there are exactly toric log del Pezzo surfaceswith Picard number and index ℓ = 3 , namely No. X ∆ No. X ∆ (i) P C (1 , ,
3) (x) P C (1 , , P C (1 , ,
4) (xi) P C (1 , , P C (2 , ,
5) (xii) P C (1 , , / ( Z / Z )(iv) P C (1 , , / ( Z / Z ) (xiii) P C (1 , , / ( Z / Z ) × ( Z / Z )(v) P C (1 , ,
6) (xiv) P C (1 , , / ( Z / Z )(vi) P C (1 , ,
7) (xv) P C (1 , , P C (1 , , / ( Z / Z ) (xvi) P C (1 , , / ( Z / Z )(viii) P C (1 , , / ( Z / Z ) (xvii) P C (1 , , / ( Z / Z )(ix) P C / ( Z / Z ) (xviii) P C (1 , , / ( Z / Z ) whose wve c -graphs are illustrated below in Figure 3 . The paper is organized as follows: In § p = p σ and q = q σ which parametrize the2-dimensional, rational, strongly convex polyhedral cones σ, and recall how theyare involved in Hirzebruch’s minimal desingularization [11] of the 2-dimensional ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ cyclic quotient singularities orb( σ ) ∈ Spec( C [ σ ∨ ∩ Z ]) for q > . In section 3 wegive necessary and sufficient arithmetical conditions for the local indices l = l σ to be 1 or 3 . Sections 4 and 5 are devoted to a detailed description of compacttoric surfaces and of those which are log Del Pezzo surfaces. Some key-lemmas ofcombinatorial nature concerning compact toric surfaces with Picard number 1 arepresented in §
6. Based on the results of § § §
7. The proof of Theorem 1.3 (which is somewhat longer thanthat of 1.1 and 1.2) follows in four steps (in § § p i , q i ) , ≤ i ≤ , sothat the induced toric log Del Pezzo surfaces X ∆ with Picard number ρ ( X ∆ ) = 1have index ℓ = 3. A minimal set of pairwise non-isomorphic surfaces of this kind issorted out in the fourth step.We use tools only from the classical toric geometry, adopting the standard ter-minology from [7], [8], and [13] (and mostly the notation introduced in [5]).2. Two-dimensional toric singularities
Let σ = R ≥ n + R ≥ n ′ ⊂ R be a 2-dimensional, rational, strongly convex polyhe-dral cone. Without loss of generality we may assume that n = (cid:0) ab (cid:1) , n ′ = (cid:0) cd (cid:1) ∈ Z , and that both n and n ′ are primitive elements of Z , i.e., gcd( a, b ) = 1 andgcd( c, d ) = 1 . Lemma 2.1.
Consider κ, λ ∈ Z , such that κa − λb = 1 . If q := | ad − bc | , and p isthe unique integer with ≤ p < q and κc − λd ≡ p (mod q ) , then gcd( p, q ) = 1 , and there exists a primitive element n ′′ = (cid:0) eg (cid:1) ∈ Z , such that n ′ = p n + q n ′′ and { n , n ′′ } is a Z -basis of Z .Proof. We define ε := sign( ad − bc ) and write κc − λd = γq + p, γ ∈ Z . Setting g := εκ + γb and e := ελ + γa, we get gc − ed = ε ( κc − λd ) + γ ( bc − ad ) = ε ( γq + p ) + γ ( − εq ) = εp, i.e., p = ε ( gc − ed ). On the other hand,det ( a eb g ) = ag − eb = ε ( κa − λb ) = ε, which means that n ′′ is primitive, { n , n ′′ } a Z -basis of Z , and ( a cb d ) = ( a eb g ) (cid:0) p q (cid:1) ,i.e., n ′ = p n + q n ′′ , because pa + qe = ε ( gca − eda ) + ε ( ad − bc ) e = cε ( ga − be ) = c and pb + qg = ε ( gcb − edb ) + ε ( ad − bc ) g = εd ( ag − be ) = d . Since gcd( p, q ) divides both c and d, and gcd( c, d ) = 1, we obtain gcd( p, q ) = 1 . (cid:3) Lemma 2.2.
There is a linear map
Φ : R −→ R , Φ ( x ) := Ξ x , with Ξ ∈ GL ( Z ) , such that Φ ( σ ) = R ≥ (cid:0) (cid:1) + R ≥ (cid:0) pq (cid:1) . Proof.
It it enough to define as Ξ := (cid:16) ε ( d − bp ) q ε ( ap − c ) q − εb εa (cid:17) . (cid:3) D.I. DAIS
Henceforth, we call σ a ( p, q )- cone . Denoting by U σ := Spec( C [ σ ∨ ∩ Z ]) the affinetoric variety associated to σ (by means of the monoid σ ∨ ∩ Z , where σ ∨ is thedual of σ ) and by orb( σ ) the single point being fixed under the usual action of thealgebraic torus T := Hom Z ( Z , C ∗ ) on U σ , it is easy to see that U σ ∼ = C only if q = 1 . (In this case, σ is said to be a basic cone .) On the other hand, whenever q > Proposition 2.3. orb( σ ) ∈ U σ is a cyclic quotient singularity. In particular, U σ ∼ = C /G = Spec( C [ z , z ] G ) , with G ⊂ GL(2 , C ) denoting the cyclic group G of order q which is generated by diag( ζ − pq , ζ q ) ( ζ q := exp(2 π √− /q )) and acts on C = Spec( C [ z , z ]) linearly andeffectively.Proof. See [8, § (cid:3) In fact, U σ is the toric variety X ∆ σ defined by the fan∆ σ := { σ together with its faces } , and by Proposition 2.4 these two numbers p = p σ and q = q σ parametrize uniquelythe isomorphism class of the germ ( U σ , orb ( σ )), up to replacement of p by its socius b p (which corresponds just to the interchange of the coordinates). [The socius b p of p is defined to be the uniquely determined integer, so that 0 ≤ b p < q , gcd( b p, q ) = 1 , and p b p ≡ q ).] Proposition 2.4.
Let σ, τ ⊂ R be two -dimensional, rational, stronly convexpolyhedral cones. Then the following conditions are equivalent :(i) There is a T -equivariant isomorphism U σ ∼ = U τ mapping orb( σ ) onto orb( τ ) . (ii) There exists a linear map
Φ : R −→ R , Φ ( x ) := Ξ x , with Ξ ∈ GL ( Z ) , suchthat Φ ( σ ) = τ. (iii) For the numbers p σ , p τ , q σ , q τ associated to σ, τ ( by Lemma 2.1) we have q τ = q σ and either p τ = p σ or p τ = b p σ . Proof.
For the equivalence (i) ⇔ (ii) see Ewald [7, Ch. VI, Thm. 2.11, pp. 222-223].For proving (ii) ⇔ (iii) we may w.l.o.g. consider (by virtue of Lemma 2.2) the cones σ := R ≥ (cid:0) (cid:1) + R ≥ (cid:0) p σ q σ (cid:1) and τ := R ≥ (cid:0) (cid:1) + R ≥ (cid:0) p τ q τ (cid:1) instead of σ, τ. (ii) ⇒ (iii): If there is a linear map Φ : R −→ R , Φ ( x ) := Ξ x , with Ξ ∈ GL ( Z ) , such that Φ ( σ ) = τ , then eitherΦ (cid:0)(cid:0) (cid:1)(cid:1) = (cid:0) (cid:1) and Φ (cid:16)(cid:0) p σ q σ (cid:1)(cid:17) = (cid:0) p τ q τ (cid:1) or Φ (cid:0)(cid:0) (cid:1)(cid:1) = (cid:0) p τ q τ (cid:1) and Φ (cid:16)(cid:0) p σ q σ (cid:1)(cid:17) = (cid:0) (cid:1) . Thus, either Ξ = p τ − p σ q σ q τ q σ ! or Ξ = p τ − p σ p τ q σ q τ − p σ q τ q σ ! . In the first case det(Ξ) has to be equal to 1, which means that q σ = q τ and p τ − p σ ≡ q σ ), i.e., p τ = p σ (because 0 ≤ p σ , p τ ≤ q σ = q τ ). In the second ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ case, det(Ξ) = −
1; hence, q σ = q τ and 1 − p σ p τ ≡ q σ ), i.e., p τ = b p σ .(iii) ⇒ (ii): If q σ = q τ and p σ = p τ , we define Φ := id R . Otherwise, q σ = q τ and p τ = b p σ , andΦ ( x ) := p τ q σ − p σ q σ − p σ ! (cid:18) x x (cid:19) , ∀ x = (cid:18) x x (cid:19) ∈ R , is an R -vector space isomorphism with the desired property. (cid:3) To construct the minimal desingularization of U σ for a ( p, q )-cone σ = R ≥ n + R ≥ n ′ ⊂ R (with q > qq − p = [[ b , b , . . . , b s ]] := b − b −
1. . . b s − − b s , and define u := n , u := q (( q − p ) n + n ′ ) , and lattice points { u j | ≤ j ≤ s + 1 } by the formulae u j +1 := b j u j − u j − , ∀ j ∈ { , . . . , s } . It is easy to see that u s +1 = n ′ , and that the integers b j are ≥ , for all indices j ∈ { , . . . , s } . Next, we subdivide σ into s + 1 smaller basic cones by introducingnew rays passing through the points u , ..., u s . Theorem 2.5 (Toric version of Hirzebruch’s desingularization) . The refinement e ∆ σ := {{ R ≥ u j + R ≥ u j +1 | ≤ j ≤ s } together with their faces } of ∆ σ := { σ together with its faces } consists of basic cones, is the coarsest refine-ment of ∆ σ with this property, and induces the minimal T - equivariant resolution X e ∆ σ −→ X ∆ σ = U σ of the singular point orb( σ ) . Moreover, the exceptional divisoris E := P sj =1 E j , having E j := orb e ∆ σ ( R ≥ u j ) ( ∼ = P C ) , ∀ j ∈ { , . . . , s } , ( i.e., the closures of the T -orbits of the new rays w.r.t. e ∆ σ ) as its components, withself-intersection number ( E j ) = − b j . Proof.
See Hirzebruch [11, pp. 15-20] who constructs X e ∆ σ by resolving the uniquesingularity lying over ∈ C in the normalization of the hypersurface (cid:8) ( z , z , z ) ∈ C (cid:12)(cid:12) z q − z z q − p = 0 (cid:9) , and Oda [13, pp. 24-30] for a proof which uses only the tools of toric geometry. (cid:3) D.I. DAIS Local indices
Let σ ⊂ R be a ( p, q )-cone. We define the local index l = l σ of σ to be the positiveinteger l := (cid:26) , if q = 1 , min { k ∈ N | k K ( E ) is a Cartier divisor } , if q > , (3.1)where K ( E ) denotes the local canonical divisor of X e ∆ σ at orb( σ ) (in the sense of[5, p. 75]) w.r.t. the minimal resolution X e ∆ σ −→ X ∆ σ of orb( σ ) constructed inTheorem 2.5. It can be shown that l = q gcd( q, p − , (3.2)cf. [5, Note 3.19, p. 89, and Prop. 4.4, pp. 94-95], and that the self-intersectionnumber of K ( E ) equals K ( E ) = − − ( p + b p ) q + s X j =1 ( b j − , (3.3)cf. [5, Corollary 4.6, p. 96]. For the proof of Theorem 1.3 we need to know underwhich restrictions on p and q we have l ∈ { , } . Lemma 3.1. If σ ⊂ R is a ( p, q ) -cone, then l = 1 ⇐⇒ (cid:26) either p = 0 and q = 1 , or p = 1 and q ≥ , (3.4) Proof.
By (3.2), l = 1 ⇐⇒ q = gcd( q, q − p + 1) , and therefore q | p − . Since p − < p < q, p and q satisfy conditions (3.4). (cid:3) Lemma 3.2. If σ ⊂ R is a ( p, q ) -cone, then l = 3 ⇐⇒ (cid:26) either ( p, q ) ∈ A, or ( p, q ) ∈ B, (3.5) where A := { ( p, q ) ∈ N × N | q = 3( p − , p ≥ , ∤ p } , and B := (cid:26) ( p, q ) ∈ N × N | q = 32 ( p − , p odd ≥ , ∤ p (cid:27) . Moreover, if ( p, q ) ∈ A and ( p ′ , q ) ∈ B, then p ′ = b p (= the socius of p ) ⇐⇒ pp ′ ≡ q ) ⇐⇒ q ≡ . Proof. l = 3 means that q = 3 m, where m := gcd( q, p − . Write p − am. Since1 ≤ p < q, we have a ∈ { , } . Since gcd( p, q ) = 1 , in the case in which a = 1 , weget gcd(3 m, m + 1) = 1 ⇐⇒ gcd(3 , p ) = 1 ⇐⇒ ∤ p, i.e. ( p, q ) ∈ A, whereas in thecase in which a = 2 , we get gcd(3 m, m + 1) = 1 ⇐⇒ gcd(3 , p ) = 1 ⇐⇒ ∤ p, and p odd ≥ , i.e. ( p, q ) ∈ B. Hence, (3.5) is true. The last assertion can be verifiedeasily. (cid:3)
Note 3.3.
It is worthwhile to take a closer look at the sets A and B, and to thecorresponding negative-regular continued fraction expansions. ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Set A : p · · · q · · ·• First case: Whenever 9 ∤ q we have b p = p and qq − p = , if p = 2 , q = 3 , [[2 , , , if p = 5 , q = 12 , [[2 , , , ..., | {z } ( q − − ) - times , , , if p ≥ , q ≥ . • Second case: Whenever 9 | q we have b p = 2 p − qq − p = [[2 , , if p = 4 , q = 9 , [[2 , , , ..., | {z } ( q − ) - times , , if p ≥ , q ≥ . Set B : p · · · q · · ·• First case: Whenever 9 ∤ q we have b p = p and qq − p = , if p = 5 , q = 6 , [[4 , , ..., | {z } ( q − − ) - times , , if p ≥ , q ≥ . • Second case: Whenever 9 | q we have b p = ( p + 1) and qq − p = [[5 , , if p = 7 , q = 9 , [[4 , , ..., | {z } ( q − ) - times , , , if p ≥ , q ≥ . These continued fraction expansions will be useful in what follows in § Compact toric surfaces
Every compact toric surface is a 2-dimensional toric variety X ∆ associated to a complete fan ∆ in R , i.e., a fan having 2-dimensional cones as maximal cones andwhose support | ∆ | is the entire R (see [13, Theorem 1.11, p. 16]). Consider acomplete fan ∆ in R and suppose that σ i = R ≥ n i + R ≥ n i +1 , i ∈ { , . . . , ν } , (4.1) D.I. DAIS are its 2-dimensional cones (with ν ≥ n i primitive for all i ∈ { , . . . , ν } ),enumerated in such a way that n , . . . , n ν go anticlockwise around the origin exactlyonce in this order (under the usual convention: n ν +1 := n , n := n ν ). Since ∆is simplicial, the Picard number ρ ( X ∆ ) of X ∆ (i.e., the rank of its Picard groupPic( X ∆ )) equals ρ ( X ∆ ) = ν − , (4.2)(see [8, p. 65]). Now suppose that σ i is a ( p i , q i )-cone for all i ∈ { , . . . , ν } andintroduce the notation I ∆ := { i ∈ { , . . . , ν } | q i > } , J ∆ := { i ∈ { , . . . , ν } | q i = 1 } , (4.3)to separate the indices corresponding to non-basic from those corresponding tobasic cones. By [13, Theorem 1.10, p. 15] the singular locus of X ∆ equalsSing( X ∆ ) = { orb( σ i ) | i ∈ I ∆ } , and its subset { orb( σ i ) | i ∈ ˘ I ∆ } , with ˘ I ∆ := { i ∈ I ∆ | p i = 1 } , (4.4)constitutes the set of the Gorenstein singularities of X ∆ . For all i ∈ I ∆ write q i q i − p i = hh b ( i )1 , b ( i )2 , . . . , b ( i ) s i ii (4.5)and, in accordance with what is already mentioned for a single 2-dimensional non-basic cone in §
2, define u ( i )1 := n i , u ( i )1 := 1 q i (( q i − p i ) n i + n i +1 ) , and u ( i ) j +1 = b ( i ) j u ( i ) j − u ( i ) j − , ∀ j ∈ { , . . . , s i } (with u ( i ) s i +1 = n i +1 ) . By construction, the proper birational map f : X e ∆ −→ X ∆ induced by the refine-ment e ∆ := the cones { σ i | i ∈ J ∆ } and n R ≥ u ( i ) j + R ≥ u ( i ) j +1 (cid:12)(cid:12)(cid:12) i ∈ I ∆ , j ∈ { , , . . . , s i } o , together with their faces . of the fan ∆ is the minimal desingularization of X ∆ . Defining E ( i ) j := orb e ∆ ( R ≥ u ( i ) j ) , ∀ i ∈ I ∆ and ∀ j ∈ { , , . . . , s i } ,C i := orb e ∆ ( R ≥ n i ) , ∀ i ∈ { , , . . . , ν } , one observes that C i is the strict transform of C i := orb ∆ ( R ≥ n i ) w.r.t. f,E ( i ) := s i X j =1 E ( i ) j the exceptional divisor replacing orb( σ i ) via f (with ( E ( i ) j ) = − b ( i ) j , ∀ i ∈ I ∆ and ∀ j ∈ { , , . . . , s i } ), and K X e ∆ − f ∗ K X ∆ = X i ∈ I ∆ r ˘ I ∆ K ( E ( i ) ) (4.6) ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ the discrepancy divisor w.r.t. f. (By K X ∆ , K X e ∆ we denote the canonical divisorsof X ∆ and X e ∆ , respectively.) Proposition 4.1.
The Picard number of X e ∆ equals ρ ( X e ∆ ) = X i ∈ I ∆ s i + ( ν −
2) = 10 − K X ∆ − X i ∈ I ∆ r ˘ I ∆ K ( E ( i ) ) . (4.7) Proof.
The first equality follows from (4.2) and from the fact that ρ ( X e ∆ ) = ρ ( X ∆ ) + ♯ { exceptional prime divisors w.r.t. f } . (4.6) implies K X e ∆ = K X ∆ + X i ∈ I ∆ r ˘ I ∆ K ( E ( i ) ) . Substituting this expression for K X e ∆ into Noether’s formula K X e ∆ = 10 − ρ ( X e ∆ ) , we obtain the second equality of (4.7). (cid:3) Definition 4.2 (The additional characteristic numbers r i ) . For every i ∈ { , .., ν } we introduce integers r i uniquely determined by the conditions: r i n i = u ( i − s i − + u ( i )1 , if i ∈ I ′ ∆ , n i − + u ( i )1 , if i ∈ I ′′ ∆ , u ( i − s i − + n i +1 , if i ∈ J ′ ∆ , n i − + n i +1 , if i ∈ J ′′ ∆ , (4.8)where I ′ ∆ := { i ∈ I ∆ | q i − > } , I ′′ ∆ := { i ∈ I ∆ | q i − = 1 } , and J ′ ∆ := { i ∈ J ∆ | q i − > } , J ′′ ∆ := { i ∈ J ∆ | q i − = 1 } , with I ∆ , J ∆ as in (4.3).By [5, Lemma 4.3], for i ∈ { , . . . , ν } , − r i is nothing but the self-intersection number C i of the strict transform C i of C i w.r.t. f. The triples ( p i , q i , r i ) , i ∈ { , , . . . , ν } , are used to define the wve c -graph G ∆ . Definition 4.3. A circular graph is a plane graph whose vertices are points ona circle and whose edges are the corresponding arcs (on this circle, each of whichconnects two consecutive vertices). We say that a circular graph G is Z - weighted atits vertices and double Z - weighted at its edges (and call it wve c - graph , for short)if it is accompanied by two maps { Vertices of G } 7−→ Z , { Edges of G } 7−→ Z , assigning to each vertex an integer and to each edge a pair of integers, respectively.To every complete fan ∆ in R (as described above) we associate an anticlockwisedirected wve c -graph G ∆ with { Vertices of G ∆ } = { v , . . . , v ν } and { Edges of G ∆ } = { v v , . . . , v ν v } , ( v ν +1 := v ) , by defining its “weights” as follows: v i r i , v i v i +1 ( p i , q i ) , ∀ i ∈ { , . . . , ν } . The reverse graph G rev∆ of G ∆ is the directed wve c -graph which is obtained bychanging the double weight ( p i , q i ) of the edge v i v i +1 into ( b p i , q i ) and reversing theinitial anticlockwise direction of G ∆ into clockwise direction (see Figure 1). Figure 1.
Theorem 4.4 (Classification up to isomorphism) . Let ∆ , ∆ ′ be two complete fansin R . Then the following conditions are equivalent :(i)
The compact toric surfaces X ∆ and X ∆ ′ are isomorphic. (ii) Either G ∆ ′ gr. ∼ = G ∆ or G ∆ ′ gr. ∼ = G rev∆ . Here “ gr. ∼ = ” indicates graph-theoretic isomorphism (i.e., a bijection between the setsof vertices which preserves the corresponding weights). For further details and forthe proof of Theorem 4.4 the reader is referred to [5, § Toric log Del Pezzo surfaces
Let X ∆ be a compact toric surface defined by a complete fan ∆ in R having (4.1)as its 2-dimensional cones. (Throughout this section we maintain the notationintroduced in § X ∆ is a log Del Pezzo surface if and only ifthe minimal generators n , . . . , n ν of the rays of ∆ are vertices of a lattice polygon Q ∆ (cf. [5, Remark 6.7, p. 107]). Definition 5.1.
A polygon Q ⊂ R is called LDP-polygon if it contains the originin its interior, and its vertices are primitive elements of Z .In fact, there is a one-to-one correspondence isomorphism classesof toric log Del Pezzosurfaces ∋ [ X ∆ ] [ Q ∆ ] ∈ lattice-equivalenceclassesof LDP-polygons . Indeed, if X ∆ ∼ = X ∆ ′ , then by Theorem 4.4 there exists a unimodular trasformationΦ : R −→ R with Φ( Q ∆ ) = Q ∆ ′ . The inverse of the above correspondence is givenby mapping the lattice-equivalence class [ Q ] of any LDP-polygon Q onto (cid:2) X ∆ Q (cid:3) , where ∆ Q := { the cones R ≥ F together with their faces | F ∈ F ( Q ) } , ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ and F ( Q ) := { facets (edges) of Q } . (If Q is an LDP-polygon,Φ : R −→ R , Φ ( x ) := Ξ x , ∀ x ∈ R , with Ξ ∈ GL ( Z ) , and Q ′ := Φ( Q ) , then G ∆ Q ′ gr. ∼ = G ∆ Q whenever det(Ξ) = 1 , and G ∆ Q ′ gr. ∼ = G rev∆ Q whenever det(Ξ) = − . )Therefore, the classification of toric log Del Pezzo surfaces (up to isomorphism)is equivalent to the classification of LDP-polygons (up to unimodular transforma-tions). Since the number of lattice-equivalence classes of LDP-polygons Q ∆ for allthose X ∆ ’s having fixed index ℓ (with ℓ as defined in §
1) is finite , as it followsfrom results appearing in [1], [3], [9] and [12], it is reasonable (for any systematicapproach to the classification problem) to focus on ℓ. By (3.1), (3.2), (4.4) and (4.6)we obtain:
Lemma 5.2.
The index ℓ of a toric log Del Pezzo surface X ∆ equals ℓ = (cid:26) lcm { l i | i ∈ I ∆ } (= lcm { l i | i ∈ I ∆ r ˘ I ∆ } ) , if I ∆ = ∅ , , if I ∆ = ∅ , (5.1) where l i = l σ i is the local index of σ i ( cf. (3.2)). Remark 5.3.
In geometric terms, ℓ = min { k ∈ N | kQ ∗ ∆ is a lattice polygon } , where Q ∗ ∆ denotes the polar of the polygon Q ∆ . In other words, ℓ equals the least commonmultiple of the (smallest) denominators of the (rational) coordinates of the verticesof Q ∗ ∆ . Moreover, for ℓ ≥ , ν = ♯ { vertices of Q ∆ } ≤ ℓ + 1 (see [6, Lemma 3.1]). Proposition 5.4.
For any toric log Del Pezzo surface X ∆ of index ℓ ≥ thefollowing inequality holds : X i ∈ I ∆ s i ≤ − X i ∈ I ∆ r ˘ I ∆ K ( E ( i ) ) − (cid:18) ℓ (cid:19) ν. (5.2) Proof. (5.2) follows from (4.7) and K X ∆ ≥ νℓ (see [6, proof of Lemma 3.2]). (cid:3) An additional necessary condition for a compact toric surface X ∆ to be log DelPezzo is dictated by the convexity of the necessarily existing LDP-polygon Q ∆ : Proposition 5.5.
For any toric log Del Pezzo surface X ∆ of index ℓ ≥ we have X i ∈ ˘ I ∆ q i ≤ X i ∈ I ∆ r ˘ I ∆ (cid:18) − l i (cid:19) q i − ( ν − ♯ ( I ∆ )) + 8 . (5.3) Proof.
Since ♯ (int(conv( { n i , n i +1 } ) ∩ Z ) = gcd( q i , p i − − , ∀ i ∈ { , . . . , ν } , we obtain ♯ ( ∂Q ∆ ∩ Z ) = ν + ν X i =1 ♯ (int(conv( { n i , n i +1 } ) ∩ Z ) = ν X i =1 gcd( q i , p i − . (5.4)( ∂, int, and conv are used as abbreviations for boundary, interior, and convex hull,respectively.) Furthermore, sincearea( Q ∆ ) = ν X i =1 area(conv( { , n i , n i +1 } )) = 12 ν X i =1 q i ! , using Pick’s formula (cf. [8, p. 113], [13, p. 101]): ♯ ( Q ∆ ∩ Z ) = area( Q ∆ ) + 12 ♯ ( ∂Q ∆ ∩ Z ) + 1 , we get ♯ (int( Q ∆ ) ∩ Z ) = 12 ν X i =1 ( q i − gcd( q i , p i − ! + 1 . (5.5)Finally, since ℓ ≥ , Scott’s inequality [17] can be written as ♯ ( ∂Q ∆ ∩ Z ) < ♯ (int( Q ∆ ) ∩ Z ) + 7 . (5.6)By (5.4), (5.5), (5.6) and (3.2) we infer that ν X i =1 (cid:18) l i − (cid:19) q i ≤ , which can be rewritten (by keeping the involved q i ’s with non-negative coefficients)in the form (5.3). (cid:3) Compact toric surfaces with Picard number 1
By virtue of (4.2) the compact toric surfaces with Picard number 1 are defined bycomplete fans ∆ in R with exactly three 2-dimensional cones. Let ∆ be a completefan of this kind and σ = R ≥ n + R ≥ n , σ = R ≥ n + R ≥ n , σ = R ≥ n + R ≥ n , (6.1)be its 2-dimensional cones, with n i primitive and σ i a ( p i , q i )-cone for i ∈ { , , } . Lemma 6.1. X ∆ is isomorphic to the quotient space P C ( q , q , q ) /H ∆ , where H ∆ is a finite abelian group of order gcd( q , q , q ) . Proof.
Since q i = | det( n i , n i +1 ) | for i ∈ { , , } , using Cramer’s rule we obtain q n + q n + q n = . By [4, Proposition 4.7, p. 224] we have X ∆ ∼ = P C ( q , q , q ) /H ∆ , where H ∆ is agroup isomorphic to Z / ( ⊕ i =1 Z n i ) . By | H ∆ | = ♯ (cid:0) { fundamental perallelepiped of ⊕ i =1 Z n i } ∩ Z (cid:1) = det( ⊕ i =1 Z n i ) , and the fact that det( ⊕ i =1 Z n i ) = gcd( q , q , q ) , the assertion is true. (cid:3) Since we are interested in describing X ∆ up to isomorphism (cf. Lemma 2.2 andTheorem 4.4) we may henceforth assume, without loss of generality, that n = (cid:0) (cid:1) and n = (cid:0) p q (cid:1) . As all cones of ∆ are strongly convex, n belongs (as shown inFigure 2) necessarily to the set M := n(cid:0) xy (cid:1) ∈ Z (cid:12)(cid:12)(cid:12) q p x < y < o . Lemma 6.2.
We have n = (cid:0) − ( q + p q ) /q − q (cid:1) , (6.2) and therefore q | q + p q and gcd(( q + p q ) /q , q ) = 1 . Moreover, q q | b p q + p q + q , (6.3) and q q | p q + b p q + q . (6.4) ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Figure 2.
Proof.
We use Lemma 2.2. Since σ is a ( p , q )-cone and σ is a ( p , q )-cone,setting n = (cid:0) xy (cid:1) , we have | det ( x p y q ) | = q , (cid:0) xy (cid:1) ∈ M = ⇒ q x − p y = − q . (6.5)on the one hand, and (cid:12)(cid:12) det (cid:0) x y (cid:1)(cid:12)(cid:12) = q , (cid:0) xy (cid:1) ∈ M = ⇒ y = − q , on the other. Hence, (6.5) gives x = − q ( q + p q ) . Moreover, by the definition of b p there exists an integer λ such that b p p − λq = 1 . This means that b p ( − q ( q + p q )) − λ ( − q ) ≡ p (mod q ) , i.e., there is a µ ∈ Z with µq = p + q ( b p ( q + p q ) − λq q ) . Consequently, µq q = b p q + q ( b p p − λq ) + p q = b p q + p q + q ,µ ∈ N , and the divisibility condition (6.3) is true. Next, by Lemma 2.2 there is amatrix ( a bc d ) ∈ GL ( Z ) such that ( a bc d ) (cid:0) xy (cid:1) = (cid:0) (cid:1) and ( a bc d ) (cid:0) (cid:1) = (cid:0) p q (cid:1) , i.e., a = p , c = q , and q x + d y = q x − d q = 0 = ⇒ d = x,p x + b y = p x − b q = 1 x < (cid:27) = ⇒ x = b p − κq , for some κ ∈ N . By (6.5), q x − p y = q ( b p − κq ) + p q = − q = ⇒ κq q = p q + b p q + q , leading to the divisibility condition (6.4). (cid:3) The converse is also true.
Lemma 6.3.
Given a triple of pairs { ( p i , q i ) | ≤ i ≤ } of non-negative integerswith p i < q i and gcd( p i , q i ) = 1 for i ∈ { , , } , and such that q q | b p q + p q + q and q q | p q + b p q + q , the -dimensional cones σ = R ≥ (cid:0) (cid:1) + R ≥ (cid:0) p q (cid:1) , σ = R ≥ (cid:0) p q (cid:1) + R ≥ (cid:0) − ( q + p q ) /q − q (cid:1) , and σ = R ≥ (cid:0) − ( q + p q ) /q − q (cid:1) + R ≥ (cid:0) (cid:1) , ( written by means of their minimal generators ) compose, together with their faces,a complete fan in R and σ i is a ( p i , q i ) -cone, for i ∈ { , , } . Proof.
Obviously, σ is a ( p , q )-cone anddet (cid:18) p − ( q + p q ) /q q − q (cid:19) = q , det (cid:18) − ( q + p q ) /q − q (cid:19) = q . Furthermore, q q | p q + b p q + q = ⇒ q | q + p q = ⇒ (cid:0) − ( q + p q ) /q − q (cid:1) ∈ Z , and setting δ := gcd( q + p q , q q ) we obtain δ | p q + b p q + q = ⇒ δ | b p q = ⇒ δ | b p p q . Since there exists an integer γ with b p p − γq = 1 , we have δ | ( γq + 1) q δ | q q = ⇒ δ | γq q (cid:27) = ⇒ δ | q . This divisibility condition is equivalent to: gcd( q ( q + p q ) , q ) = 1 , and therefore (cid:0) − ( q + p q ) /q − q (cid:1) is primitive. On the other hand, q q | b p q + p q + q = ⇒ ∃ µ ∈ N : µq q = b p q + p q + q . Since there exists an integer λ with b p p − λq = 1 , and µq q = b p q + q ( b p p − λq )+ p q = ⇒ b p ( − q ( q + p q )) − λ ( − q ) ≡ p (mod q ) ,σ is a ( p , q )-cone. Finally, q q | p q + b p q + q = ⇒ ∃ κ ∈ N : κq q = p q + b p q + q , i.e., q ( b p − κq )+ p q = − q = q ( − q ( q + p q ))+ p q = ⇒ − q ( q + p q ) = b p − κq , giving (cid:18) p q ( p b p − − κp q b p − κq (cid:19) (cid:18) − q ( q + p q ) − q (cid:19) = (cid:18) (cid:19) , and (cid:18) p q ( p b p − − κp q b p − κq (cid:19) (cid:18) (cid:19) = (cid:18) p q (cid:19) . Hence, as it is explained in the proof of Proposition 2.4, the cone σ has to be a( p , q )-cone. (cid:3) Lemma 6.4.
Every compact toric surface X ∆ having Picard number ρ ( X ∆ ) = 1 is a log Del Pezzo surface.Proof. If X ∆ is a compact toric surface with ρ ( X ∆ ) = 1 , then the minimal genera-tors n , n , n of the tree cones (6.1) of ∆ have to be in general position because thecones are strongly convex. Hence, conv( { n , n , n } ) has to be an LDP-triangle. (cid:3) ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Note 6.5.
Compact toric surfaces X ∆ having Picard number ρ ( X ∆ ) ≥ X ∆ with ρ ( X ∆ ) = 2 are the Hirzebruch surfaces F κ , κ ≥ F ∼ = P C × P C and F (i.e., a P C blown up at one point) are DelPezzo surfaces (see [13, Proposition 2.21, p. 88] or [7, Theorem V.8.2, p. 192]).The geometric reason for that is actually very simple: Since F κ can be viewed asthe toric surface associated to the fan having (cid:0) (cid:1) , (cid:0) (cid:1) , (cid:0) − (cid:1) and (cid:0) − κ (cid:1) as minimalgenerators of its rays, setting T := conv (cid:16)n(cid:0) (cid:1) , (cid:0) − (cid:1) , (cid:0) − κ (cid:1)o(cid:17) we see that (cid:0) (cid:1) ∈ ∂ T for κ = 2 , and (cid:0) (cid:1) ∈ int( T ) for κ ≥ . Classification strategy for ρ ( X ∆ ) = 1 and ℓ = 3 Definition 7.1.
We call a triple of pairs (cid:8) ( p i , q i ) ∈ Z (cid:12)(cid:12) ≤ i ≤ (cid:9) , ≤ p i < q i , with gcd( p i , q i ) = 1 , ∀ i ∈ { , , } , (7.1) admissible whenever it satisfies both divisibility conditions q q | b p q + p q + q (7.2)and q q | p q + b p q + q (7.3)To classify all toric log Del Pezzo surfaces X ∆ having Picard number 1 and index ℓ = 3 up to isomorphism it suffices (by Lemmas 6.2, 6.3, and 6.4, and Theorem 4.4)to determine all admissible triples of pairs, and consequently the fans ∆ having σ = R ≥ n + R ≥ n , σ = R ≥ n + R ≥ n , σ = R ≥ n + R ≥ n , as 2-dimensional cones, with n = (cid:0) (cid:1) , n = (cid:0) p q (cid:1) , n = (cid:0) − ( q + p q ) /q − q (cid:1) as minimalgenerators, and Q ∆ = conv( { n , n , n } ) as their LDP-polygons, so that l i = l σ i ∈ { , } , ∀ i ∈ { , , } , and l k = 3 for at least one k ∈ { , , } , (7.4)(see (5.1)). From now on we may assume w.l.o.g that l = 3 . We also keep in mindthe two auxiliary conditions s + s + s ≤ − X i ∈ I ∆ r ˘ I ∆ K ( E ( i ) ) + 8 (7.5)(where, for our convenience, we set s i := 0 for i ∈ J ∆ , cf. (4.3)), and X i ∈ ˘ I ∆ q i ≤ X i ∈ I ∆ r ˘ I ∆ q i + ♯ ( I ∆ ) + 5 (7.6)(following from (5.2) and (5.3), respectively, for ν = ℓ = 3) which have to be satisfiedbecause of Lemma 6.4. By assumption, each pair ( p i , q i ) (belonging to a triple (7.1)which will be considered as “candidate” for being admissible) is necessarily of aspecific type . All possible types are determined by conditions (7.4), (3.4) and (3.5),and are listed in Table 1. (Since l = 3 , ( p , q ) can be of type , , , or .) Types p i b p i q i s i − K ( E ( i ) ) ξ i + 2 p i ξ i + 3 ξ i + 2 ξ i + 1 2 p i − ξ i + 1) 9 ξ i ξ i + 1 2 ξ i + 5 p i ξ i + 6 ξ i + 1 ξ i + 1 ( p i + 1) (= 3 ξ i + 1) 9 ξ i ξ i + 1 2 ≥ q i − Table 1.
Here, ξ i denotes an integer which is positive for types , and , and non-negative for type . (In particular, the entries of the last two columns are computed bythe continued fraction expansions mentioned in Note 3.3 and by the formula (3.3).)Although the pairs ( p i , q i ) of type (resp., of type , , or ) are infinitely many ,conditions (7.2), (7.3), (7.5) and (7.6) force the testable triples of pairs (7.1) to beadmissible only in finitely many cases. Note 7.2.
If orb( σ ) is a non-Gorenstein singularity, then (7.2) implies[ b p q + p q + q ] = 0 (7.7)(where [ t ] denotes the remainder in the division of a t ∈ Z by 9) because 3 | q and3 | q . Analogously, if orb( σ ) is a non-Gorenstein singularity, then (7.3) implies[ p q + b p q + q ] = 0 (7.8)These weaker, necessary conditions (7.7) and (7.8) turn out to be very useful inproving that several triples of pairs (7.1) are not admissible.The proof of Theorem 1.3 will follow in four steps: ◮ Step 1 : We determine which of the triples of pairs (7.1) corresponding to the125 (= 5 ) possible type combinations ( α , α , α ) , with α , α , α ∈ { , , , , } , are admissible, i.e., those X ∆ ’s with exactly three non-Gorenstein singularities. ◮ Step 2 : We determine which of the triples of pairs (7.1) corresponding to the100 (= 2 · (5 · α , α , α ) , with α ∈ { , , , , } and( α , α ) ∈ ( { , , , , } × { , } ) ∪ ( { , } × { , , , , } ) , are admissible, i.e., those X ∆ ’s with exactly two non-Gorenstein singularities. ◮ Step 3 : We do the same for the triples of pairs (7.1) corresponding to the 20type combinations ( α , α , α ) , with α ∈ { , , , , } and α , α ∈ { , } , i.e.,for those X ∆ ’s with exactly one non-Gorenstein singularity. ◮ Step 4 : We find out the wve c -graphs G ∆ for those X ∆ ’s determined in steps1-3 , and then, using Theorem 4.4, we pick out a suitable, minimal set of repre-sentatives of X ∆ ’s all of whose members are pairwise non-isomorphic. Finally, weidentify the chosen X ∆ ’s with weighted projective planes or quotients thereof byapplying Lemma 6.1. ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Proof of Theorem 1.3: Step 1
Lemma 8.1.
Among the possible combinations ( α , α , α ) of types of triplesof pairs (7.1), with α , α , α ∈ { , , , , } , there are only satisfying simulta-neously conditions (7.7) and (7.8); namely, ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , together with their permutations. Case [ b p ] [ q ] [ p ] [ q ] [ b p q + p q ] (7 .
7) is trueonly if( , , α ) 2 3 2 3 3 α = ( , , α ) 2 3 ∈ { , , } α = ( , , α ) 2 3 ∈ { , , } α = ( , , α ) 2 3 ∈ { , , } α ∈ { , } ( , , α ) 2 3 ∈ { , , } α = ( , , α ) ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α = ( , , α ) ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α = ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } ( , , α ) ∈ { , , } ∈ { , , } α ∈ { , } Table 2.
Case [ p ] [ q ] [ b p ] [ q ] [ p q + b p q ] (7 .
8) is trueonly if( ,α , ) 2 3 2 3 3 α = ( ,α , ) 2 3 ∈ { , , } α = ( ,α , ) 2 3 ∈ { , , } α = ( ,α , ) 2 3 ∈ { , , } α ∈ { , } ( ,α , ) 2 3 ∈ { , , } α = ( ,α , ) ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α = ( ,α , ) ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α = ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } ( ,α , ) ∈ { , , } ∈ { , , } α ∈ { , } Table 3.
Proof.
By Table 2 there are 38 combinations ( α , α , α ) of types of triples (7.1),with α , α , α ∈ { , , , , } , satisfying condition (7.7). Correspondingly, Ta-ble 3 shows that there are 38 combinations ( α , α , α ) of types of triples (7.1),with α , α , α ∈ { , , , , } , satisfying condition (7.8). Obviously, the combina-tions ( α , α , α ) of types of triples of pairs (7.1), with α , α , α ∈ { , , , , } , satisfying both (7.7) and (7.8), are the 32 combinations given in the statement ofLemma. (cid:3) Lemma 8.2.
There are no admissible triples of pairs (7.1) among those corre-sponding to the type combinations ( α , α , α ) with α , α , α ∈ { , , , , } . Sketch of proof . First, we express the triples of pairs (cid:8) ( p i , q i ) ∈ Z (cid:12)(cid:12) ≤ i ≤ (cid:9) corresponding to the 32 type combinations ( α , α , α ) found in Lemma 8.1 interms of ξ i for i ∈ { , , } as in Table 1. Setting A j := (cid:26) ( ξ , ξ , ξ ) ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) ξ + ξ + ξ ≤ , ξ j ≥ , and ξ k ≥ , ∀ k ∈ { , , } r { j } (cid:27) , for j ∈ { , , } , A j,k := (cid:26) ( ξ , ξ , ξ ) ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) ξ + ξ + ξ ≤ , ξ j = 0 , ξ k ≥ , and ξ µ ≥ , ∀ µ ∈ { , , } r { j, k } (cid:27) , for j, k ∈ { , , } , j = k, and B := (cid:8) ( ξ , ξ , ξ ) ∈ Z (cid:12)(cid:12) ξ + ξ + ξ ≤ , ξ , ξ , ξ ≥ (cid:9) , we explain what condition (7.5) means for each of these 32 cases in Table 4. Case
Condition (7.5)
Case
Condition (7.5)( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A , ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ A ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B ( , , ) ( ξ , ξ , ξ ) ∈ B Table 4.
Note that ♯ ( A j ) = X κ =2 (cid:0) κ − (cid:1) + X κ =3 (cid:0) κ − (cid:1) = 165 , ♯ ( A j,k ) = 55 , ♯ ( B ) = (cid:0) (cid:1) = 165 . ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ One can, of course, test directly the validity of (7.2) and (7.3) for all these possi-bilities. Nevertheless, there is a more economic way to proceed by using reductioad absurdum. Let us discuss it exemplarily in the case ( , , ) in which p b p q p b p q p b p q ξ + 2 3 ξ + 2 9 ξ + 3 3 ξ + 1 6 ξ + 1 9 ξ ξ + 5 6 ξ + 5 9 ξ + 6 for a 3-tuple ( ξ , ξ , ξ ) ∈ A . If (cid:8) ( p i , q i ) ∈ Z (cid:12)(cid:12) ≤ i ≤ (cid:9) were an admissible tripleof pairs, then (7.2) would give(9 ξ + 3) (9 ξ ) = 27 ξ + 81 ξ ξ | ξ + 27 ξ + 9 ξ + 54 ξ ξ + 9 , i.e.,3 (3 ξ + 1) ξ | ξ + 3 ξ + ξ + 6 ξ ξ + 1 = 3 (3 ξ + 1) ξ − ξ ξ + ξ + ξ + 1 , (8.1)meaning that 3 (3 ξ + 1) ξ | ξ ξ − ξ − ξ − . Therefore, 3 ξ ξ − ξ − ξ − ≤ ξ + 9 ξ ξ ≤ ξ ξ − ξ − ξ − , i.e.,that 10 ≤ ξ + 3 ξ + ξ + 6 ξ ξ ≤ − , a contradiction). Consequently,3 ξ ξ − ≤ ξ + ξ ≤ − ξ = ⇒ ≤ (3 ξ + 1) ξ ≤ ⇒ ( ξ , ξ ) ∈ { (1 , , (1 , , (2 , , (3 , } . (8.2)Since 0 ≤ ξ ≤ , (8.1) and (8.2) would determine the values of ξ as follows: ( ξ , ξ , ξ ) p q q b p q q q p q + b p q + q (1 , ,
1) 5 12 9 11 15 180 216(1 , ,
4) 5 12 18 29 42 504 576(2 , ,
3) 8 21 9 23 33 693 756(3 , ,
5) 11 30 9 35 51 1530 1620
Hence, these four 3-tuples ( ξ , ξ , ξ ) ∈ A would provide numbers p , q , q , b p , q which do not satisfy (7.3)! Using analogous arguments one shows that none of theremaining 31 cases leads to admissible triples of pairs. (cid:3) Proof of Theorem 1.3: Step 2
Lemma 9.1.
There are no admissible triples of pairs (7.1) among those corre-sponding to the type combinations ( α , α , α ) with α ∈ { , , , , } and ( α , α ) ∈ ( { , , , , } × { } ) ∪ ( { } × { , , , , } ) . Proof. If α , α ∈ { , , , , } and α = , then [ b p q + p q ] ∈ { , , } (cf.the sixth column of Table 2) and q = 1 , i.e., [ b p q + p q + q ] ∈ { , , } . Thus,condition (7.7) is not satisfied. Analogously, one shows that condition (7.8) is notsatisfied whenever α , α ∈ { , , , , } and α = . (cid:3) Lemma 9.2.
There exist exactly admissible triples of pairs (7.1) among thosecorresponding to the type combinations ( α , α , α ) with α ∈ { , , , , } and ( α , α ) ∈ ( { , , , , } × { } ) ∪ ( { } × { , , , , } ) . Sketch of proof . For α , α ∈ { , , , , } and α = we build Table 5. In itssecond column we tabulate [ b p q + p q ] (cf. the sixth column of Table 2). Afterhaving expressed q , q in terms of ξ , ξ (as in Table 1) we write the restrictions(inequalities) coming from (7.5) in its third column. The fourth column containsthe values of q so that both (7.5) and (7.7) are true. (In particular, in the case( , , ) the expected value q = 6 is impossible because ξ , ξ ≥ . ) Finally, thelast column informs us whether (7.8) is true for these q ’s. Case [ b p q + p q ] (7.5) is truewhenever (7.5) & (7.7) trueonly if q equals Is (7.8) truefor these q ’s?( , , ) 3 2 ≤ q ≤ , , ) 3 3 ≤ ξ + q ≤ , , ) 3 3 ≤ ξ + q ≤ , , ) 0 2 ≤ ξ + q ≤
10 9 YES( , , ) 3 3 ≤ ξ + q ≤ , , ) 3 3 ≤ ξ + q ≤ , , ) 3 4 ≤ ξ + ξ + q ≤ , , ) 3 4 ≤ ξ + ξ + q ≤ , , ) 0 3 ≤ ξ + ξ + q ≤
10 9 YES( , , ) 3 4 ≤ ξ + ξ + q ≤ , , ) 3 3 ≤ ξ + q ≤ , , ) 3 4 ≤ ξ + ξ + q ≤ , , ) 0 4 ≤ ξ + ξ + q ≤
11 9 YES( , , ) 6 3 ≤ ξ + ξ + q ≤
11 3 YES( , , ) 0 4 ≤ ξ + ξ + q ≤
11 9 YES( , , ) 0 2 ≤ ξ + q ≤
10 9 YES( , , ) 0 3 ≤ ξ + ξ + q ≤
10 9 YES( , , ) 6 3 ≤ ξ + ξ + q ≤
11 3 NO( , , ) 6 2 ≤ ξ + ξ + q ≤
12 3 or 12 YES( , , ) 6 3 ≤ ξ + ξ + q ≤
11 3 NO( , , ) 3 3 ≤ ξ + q ≤ , , ) 3 4 ≤ ξ + ξ + q ≤ , , ) 0 4 ≤ ξ + ξ + q ≤
11 9 YES( , , ) 6 3 ≤ ξ + ξ + q ≤
11 3 YES( , , ) 0 4 ≤ ξ + ξ + q ≤
11 9 YES
Table 5.
Next, we analyze in detail the 14 cases for which the answer is “yes” . • In the case ( , , ) we have q = 6 and we obtain just one admissible triple ofpairs: p q p q p q • In cases ( , , ) and ( , , ) we have ξ = 1 , q = 6 , and ξ = 1 , q = 6 , respectively, and (7.2) cannot be satisfied (because 36 ∤ • In cases ( , , ) and ( , , ) we have ξ ∈ { , } , q = 9 , and ξ ∈ { , } , q = 9 , respectively, and (7.2) cannot be satisfied for ξ = 1 , resp. for ξ = 1 (because45 ∤ p q p q p q p q p q p q • In cases ( , , ) and ( , , ) we have necessarily ξ = 1 , ξ = 0 , q = 9 , and ξ = 0 , ξ = 1 , q = 9 , respectively, and (7.2) cannot be satisfied (because 72 ∤ • In cases ( , , ) and ( , , ) we have necessarily ξ = ξ = 1 , q = 9 , and (7.2)cannot be satisfied (because 81 ∤ • In cases ( , , ) and ( , , ) we have q = 3 and ξ + ξ ∈ { , . . . , } with ξ ≥ ξ ≥ . If (7.2) were true, then in particular q | b p q + q , i.e., ξ | ξ +1 = ⇒ ( ξ , ξ ) ∈ { (1 , j ) | ≤ j ≤ }∪{ (2 , , (2 , , (2 , , (3 , , (3 , , (4 , } . ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ As for everyone of the 14 possible values of ( ξ , ξ ) at least one of the divisibilityconditions (7.2) and (7.3) is violated , there are no admissible triples of pairs. • Case ( , , ): ξ = ξ = 1 , q = 9 , and (7.2) cannot be satisfied (because81 ∤ • Case ( , , ): Here, either ξ = ξ = 0 , q = 12 , giving the admissible triple ofpairs: p q p q p q or q = 3 and ξ + ξ ∈ { , , . . . , } with ξ , ξ ≥ . If in the latter case (7.2) weretrue, then, in particular, q | b p q + q , i.e.,9 ξ + 6 | (6 ξ + 5)(9 ξ + 6) + 3 = ⇒ ξ + 2 | (3 ξ + 2)(6 ξ + 4) + 3 ξ + 3= ⇒ ξ + 2 | ξ + 3 = ⇒ ξ + 2 | ξ + 1 , i.e.,( ξ , ξ ) ∈ { (0 , , (0 , , (0 , , (0 , , (0 , , (1 , , (2 , } . As for everyone of the 7 possible values of ( ξ , ξ ) at least one of the divisibilityconditions (7.2) and (7.3) is violated , there are no further admissible triples of pairs. • Case ( , , ): ξ = ξ = 1 , q = 9 , and we obtain just one admissible triple ofpairs: p q p q p q α , α , α ) , where α , α ∈ { , , , , } and α = , we determine the admissible triples of pairs: p q p q p q , , ) , p q p q p q , , ) , p q p q p q , , ) p q p q p q , , ) , and p q p q p q , , ) . (cid:3) Proof of Theorem 1.3: Step 3
Lemma 10.1.
There exist exactly admissible triples of pairs (7.1) among thosecorresponding to the type combinations ( α , α , α ) with α ∈ { , , , , } and α , α ∈ { , } . Proof.
For every α ∈ { , , , , } we consider the combinations Case p q p = b p q ( α , , ) 1 ≥ ≥ α , , ) 1 ≥ α , , ) 0 1 1 ≥ α , , ) 0 1 0 1and examine what happens in each of the twenty cases separately. • Case ( , , ): Here, and for the next three cases, p = b p = 2 , q = 3 and s = 1 . By (7.5) and (7.6) the pair ( q , q ) has to be chosen from the 21 elements of the set (cid:8) ( q , q ) ∈ Z (cid:12)(cid:12) q ≥ , q ≥ , and q + q ≤ (cid:9) . Taking into account the divisibility conditions (7.2), (7.3), i.e., 3 q | q + q + 3and 3 q | q + q + 3 , we obtain ( q , q ) ∈ { (2 , , (5 , } . Hence, there are twoadmissible triples of pairs, namely p q p q p q p q p q p q • Case ( , , ): By (7.5) (or (7.6)) we have q ≤ . By (7.3), 3 | q − , i.e., q ∈ { , } . The value q = 7 does not satisfy (7.2): 3 q | q + 4 . Hence, there isonly one admissible triple of pairs: p q p q p q • Case ( , , ): Analogously, we find just one admissible triple of pairs: p q p q p q • Case ( , , ): In this case both divisibility conditions (7.2) and (7.3) are satisfiedautomatically and lead to the admissible triple of pairs: p q p q p q • Case ( , , ): Here, and for the next three cases, p = b p = 3 ξ + 2 , q = 9 ξ + 3and s = ξ + 2 for an integer ξ ≥ . By (7.5) we have ξ + q + q ≤ . Condition(7.2) reads as3(3 ξ + 1) q | (3 ξ + 2) q + (9 ξ + 3) + q = 3(3 ξ + 1) q − ξ q − q + (9 ξ + 3) + q , i.e.,3(3 ξ + 1) q | ξ q + q − (9 ξ + 3) − q with 6 ξ q + q − (9 ξ + 3) − q ≤ . ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Since 1 ≤ ξ ≤ , (6 ξ + 1) q ≤ ξ + 3 + q ≤ ξ + 3 + (9 − ξ − q ) = ⇒ (6 ξ + 2) q ≤ ξ + 12 ≤ , implying 8 ≤ (3 ξ + 1) q ≤ . These inequalities are satisfied if and only if( ξ , q ) ∈ { (1 , , (1 , , (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (3 , } . Since 2 ≤ q ≤ − ( ξ + q ) , the divisibility condition (7.2) is true only for ξ = 1 ,q = 3 and q = 2 . (For these values (7.3) is also true.) Hence, the only admissibletriple of pairs is the following: p q p q p q • Case ( , , ): By (7.5) we have ξ + q ≤ . Condition (7.3) gives3(3 ξ + 1) | ξ + 2 + q = ⇒ | q − ξ + 1 | q + 1 . But this means that ( ξ , q ) ∈ { (1 , , (2 , } . (2 ,
6) is not permitted because 21 ∤ ,
7) violates (7.2), so there are no admissible triples of pairs. • Case ( , , ): As in the case ( , , ) one shows that there are no admissibletriples of pairs. • Case ( , , ): Conditions (7.2) and (7.3) give q = 3( p − | p + 1 , i.e., p = 2 , but in this case p ≥ . Hence, there are no admissible triples of pairs. • Case ( , , ): Here, and for the next three cases, p = 3 ξ + 1 , b p = 6 ξ + 1 ,q = 9 ξ and s = ξ + 1 for an integer ξ ≥ . By (7.5) we have ξ + q + q ≤ . Condition (7.2) reads as9 ξ q | (6 ξ + 1) q + 9 ξ + q = 9 ξ q − ξ q + q + 9 ξ + q , i.e., 9 ξ q | ξ q − q − ξ − q with 3 ξ q − q − ξ − q ≤ . Since 1 ≤ ξ ≤ , ξ q ≤ q + q + 9 ξ ≤
11 + 8 ξ ≤
67 = ⇒ ≤ ξ q ≤ . Since 2 ≤ q ≤ − ( ξ + q ) , the divisibility conditions (7.2) and (7.3) are trueonly for ξ = 1 , q = 6 , q = 3 , or ξ = 2 , q = 4 , q = 2 , leading to two admissibletriple of pairs, namely p q p q p q p q p q p q • Case ( , , ): By (7.5) we have ξ + q ≤ . Condition (7.3) gives9 ξ | ξ + 1 + q = ⇒ ξ | ξ − q − , with 6 ξ − q − ≤ . Thus, 6 ξ ≤ q + 1 ≤ − ξ = ⇒ ξ ≤
117 = ⇒ ξ = 1 . Since 2 ≤ q ≤ , condition (7.2) (i.e., 9 q | q + 10) implies q = 5 . The corre-sponding admissible triple of pairs is the following: p q p q p q • Case ( , , ): By (7.5) we have ξ + q ≤ . Condition (7.2) gives ξ | ξ + 1 + q = ⇒ ξ | ξ − q − , with 3 ξ − q − ≤ . Thus, 3 ξ ≤ q + 1 ≤ − ξ = ⇒ ξ ≤
114 = ⇒ ξ ∈ { , } . Since 2 ≤ q ≤ , condition (7.2) implies ( ξ , q ) ∈ { (1 , , (2 , } . (2 ,
5) is notpermitted because it violates (7.3). For this reason, the only admissible triple ofpairs is the following: p q p q p q • Case ( , , ): Condition (7.3) gives q = 3( p − | p + 1 , i.e., p = 2 , but inthis case p ≥ . Hence, there are no admissible triples of pairs. • Case ( , , ): Here, and for the next three cases, p = b p = 6 ξ + 5 , q = 9 ξ + 6and s = ξ + 1 for an integer ξ ≥ . By (7.5) we have ξ + q + q ≤ . Condition(7.2) reads as(9 ξ + 6) q | (6 ξ + 5) q + 9 ξ + 6 + q = (9 ξ + 6) q − (3 ξ + 1) q + 9 ξ + 6 + q , i.e.,(9 ξ + 6) q | (3 ξ + 1) q − ξ − − q with (3 ξ + 1) q − ξ − − q ≤ . Since 1 ≤ ξ ≤ , we obtain(3 ξ + 1) q ≤ ξ + 6 + (11 − q − ξ ) = ⇒ (3 ξ + 2) q ≤
17 + 8 ξ ≤ , i.e., 4 ≤ (3 ξ + 2) q ≤ . Since 2 ≤ q ≤ − ( ξ + q ) , the divisibility conditions(7.2) and (7.3) are satisfied only for ξ ∈ { , , } . In particular , for ξ = 0 weobtain ( q , q ) ∈ { (8 , , (2 , } and the admissible triples of pairs p q p q p q p q p q p q ξ = 1 we have necessarily q = q = 5 and the admissible triple of pairs: p q p q p q
11 15 1 5 1 5Finally, for ξ = 2 we have necessarily q = q = 4 and the admissible triple ofpairs: p q p q p q
17 24 1 4 1 4 • Case ( , , ): By (7.5) we have ξ + q ≤ . Condition (7.3) gives9 ξ + 6 | ξ + 5 + q = ⇒ ξ + 6 | ξ + 1 − q , with 3 ξ + 1 − q ≤ . ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Thus, 3 ξ ≤ q − ≤ − ξ = ⇒ ξ ≤ = ⇒ ξ ∈ { , , } . Since 2 ≤ q ≤ , condition (7.3) implies ( ξ , q ) ∈ { (0 , , (1 , , (2 , } . (2 ,
7) is not permitted because it violates (7.2); therefore, the admissible triples ofpairs are p q p q p q p q p q p q
11 15 1 4 0 1 (10.12) • Case ( , , ): As in the case ( , , ) one proves that there are two admissibletriples of pairs, namely p q p q p q p q p q p q
11 15 0 1 1 4 (10.14) • Case ( , , ): Conditions (7.2) and (7.3) give q = ( p − | p + 1 , i.e., p = 5 . Thus, we find just one admissible triple of pairs: p q p q p q • Case ( , , ): Here, and for the next three cases, p = 6 ξ + 1 , b p = 3 ξ + 1 ,q = 9 ξ and s = ξ + 1 for an integer ξ ≥ . By (7.5) we have ξ + q + q ≤ . Condition (7.2) reads as9 ξ q | (3 ξ + 1) q + 9 ξ + q = 9 ξ q − ξ q + q + 9 ξ + q , i.e., 9 ξ q | ξ q − q − ξ − q with 6 ξ q − q − ξ − q ≤ . Since 1 ≤ ξ ≤ , ξ q ≤ q + q + 9 ξ ≤
11 + 8 ξ ≤
67 = ⇒ ≤ ξ q ≤ . Since 2 ≤ q ≤ − ( ξ + q ) , the divisibility conditions (7.2) and (7.3) are satisfiedonly for ξ = 1 , q = 3 , q = 6 , or ξ = 2 , q = 2 , q = 4 , leading to two admissibletriple of pairs, namely p q p q p q p q p q p q
13 18 1 2 1 4 (10.16) • Case ( , , ): By (7.5) we have ξ + q ≤ . Condition (7.3) gives9 ξ | ξ + 1 + q = ⇒ ξ | ξ − q − , with 3 ξ − q − ≤ . Thus, 3 ξ ≤ q + 1 ≤ − ξ = ⇒ ξ ≤
114 = ⇒ ξ ∈ { , } . Since 2 ≤ q ≤ , condition (7.3) implies ξ = 1 and q = 2 . The result is thefollowing admissible triple of pairs: p q p q p q • Case ( , , ): By (7.5), ξ + q ≤ . Now (7.2) reads as9 ξ | ξ + 1 + q = ⇒ ξ | ξ − q − , with 6 ξ − q − ≤ . Thus, 6 ξ ≤ q + 1 ≤ − ξ = ⇒ ξ ≤
117 = ⇒ ξ = 1 . Since 2 ≤ q ≤ , condition (7.2) implies q = 5 . The corresponding admissibletriple of pairs is the following: p q p q p q • Case ( , , ): Condition (7.3) gives q = ( p − | p + 1 , i.e., p ≤ , but inthis case p ≥ . Hence, there are no admissible triples of pairs. (cid:3)
Remark 10.2.
The majority of the admissible triples of pairs induce toric log DelPezzo surfaces admitting at least one Gorenstein singularity. This is due to the factthat the q i ’s corresponding to Gorenstein singularities can be viewed as parametersmoving freely between 2 and an upper bound dictated by conditions (7.5) and (7.6),without any further restrictions.11. Proof of Theorem 1.3: Step4
Lemma 11.1.
The toric log Del Pezzo surfaces induced by the following admissibletriples of pairs ( a ) and ( b ):( a ) (9.1) (9.4) (9.5) (10.1) (10.3) (10.5)( b ) (9.6) (9.10) (9.8) (10.2) (10.4) (10.15)( a ) (10.6) (10.8) (10.7) (10.9) (10.11) (10.12)( b ) (10.16) (10.17) (10.18) (10.10) (10.13) (10.14) are isomorphic to each other. The same is true for the four surfaces induced by thefollowing admissible triples of pairs :( a ) ( b ) ( c ) ( d )(9.2) (9.3) (9.7) (9.9)(The admissible triples of pairs are given by their reference numbers.) Proof. If X ∆ ( a ) (resp., X ∆ ( b ) ) is the toric Del Pezzo surface induced by the admissi-ble triple of pairs ( a ) (resp., ( b )) in the first list, then G ∆ ( a ) gr. ∼ = G rev∆ ( b ) . Correspond-ingly, if X ∆ ( a ) , X ∆ ( b ) , X ∆ ( c ) , X ∆ ( d ) are the four surfaces induced by the admissibletriples of pairs in the second list, then we obtain G ∆ ( a ) gr. ∼ = G rev∆ ( b ) gr. ∼ = G rev∆ ( c ) gr. ∼ = G ∆ ( d ) . It is therefore enough to apply Theorem 4.4. (cid:3)
ORIC LOG DEL PEZZOS WITH ρ = 1 AND INDEX ≤ Note 11.2.
By Lemmas 8.2, 9.1, 9.2 and 10.1 we proved that among all possibletriples of pairs there exist exactly 33 which are admissible. Lemma 11.1 informsus that, in fact, for the classification of toric Del Pezzo surfaces X ∆ having Picardnumber ρ ( X ∆ ) = 1 and index ℓ = 3 up to isomorphism , we need only 18 out ofthem. (The X ∆ ’s induced by such a choice of 18 admissible triples of pairs are obviously pairwise non-isomorphic.) End of the proof of Theorem X ∆ with ρ ( X ∆ ) = 1 and index ℓ = 3 , and we enumerate them, e.g., as in the Table 6. Thecoordinates of the third minimal generator n is computed by (6.2). The integers r i = − C i , i ∈ { , , } , are computed directly via (4.8). No. Case p q p q p q n r r r (i) ( , , ) 2 3 0 1 0 1 ( − , −
1) 0 0 − , , ) 2 3 0 1 1 4 ( − , −
4) 1 − , , ) 2 3 1 2 1 5 ( − , −
5) 1 0 1(iv) ( , , ) 2 3 2 3 1 6 ( − , −
6) 1 0 1(v) ( , , ) 5 6 0 1 0 1 ( − , −
1) 0 0 − , , ) 5 6 0 1 1 7 ( − , −
7) 1 − , , ) 5 6 1 8 1 2 ( − , −
2) 0 1 1(viii) ( , , ) 2 3 5 6 1 9 ( − , −
9) 1 0 1(ix) ( , , ) 7 9 4 9 1 9 ( − , −
9) 1 1 1(x) ( , , ) 7 9 0 1 1 5 ( − , −
5) 1 0 − , , ) 4 9 0 1 1 2 ( − , −
2) 1 0 − , , ) 4 9 1 6 1 3 ( − , −
3) 1 1 1(xiii) ( , , ) 5 6 5 6 1 12 ( − , −
12) 1 0 1(xiv) ( , , ) 5 12 1 2 1 2 ( − , −
2) 1 1 − , , ) 11 15 0 1 1 4 ( − , −
4) 1 0 − , , ) 11 15 1 5 1 5 ( − , −
5) 1 1 1(xvii) ( , , ) 7 18 1 4 1 2 ( − , −
2) 1 1 − , , ) 17 24 1 4 1 4 ( − , −
4) 1 1 0
Table 6.
The wve c -graphs G ∆ (associated to the 18 ∆’s) are depicted in Figure 3 in thisorder. (The reference to the double weight (0 ,
1) at an edge of G ∆ is always omit-ted.) Finally, we may identify the corresponding X ∆ ’s with weighted projectiveplanes or quotients thereof by a finite abelian group H ∆ via Lemma 6.1. (In thestatement of the Theorem we have w.l.o.g. rearranged the weights in ascending or-der. Computing the Smith normal form, H ∆ turns out to be cyclic for the surfaces(ix) and (xviii)). (cid:3) (2,3) (2,3) (2,3)(2,3) (5,6) (5,6)(1,4) (1,2) (1,5)(1,6)(2,3) (1,3) (1,7)
00 3 0 00 01 (cid:238) (cid:238) (cid:238) (cid:238) (cid:238) (cid:238) (5,6) (2,3) (7,9)(7,9) (5,6)(1,5) (cid:238) (cid:238) (cid:238) (4,9) (1,9) (cid:238) (cid:238)
11 0 (4,9)(1,6) (1,3) (cid:238) (cid:238) (cid:238) (4,9) (1,3) (cid:238) (1,8) (1,2) (cid:238) (cid:238) (1,9) (cid:238) (5,6) (5,12)(11,15) (1,5) (cid:238) (cid:238) (cid:238) (cid:238) (1,2) (cid:238) (11,15) (1,4) (cid:238) (cid:238) (17,24)(1,4) (1,4) (cid:238) (cid:238) (7,18)(1,4) (cid:238) (5,6) (1,12) (cid:238) (1,2) (cid:238) (1,5) (1,2) Figure 3.
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