K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
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1 2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ ) λ 0 Λ, λ Λ T λ0 T λ, MWL(f λ0 ), {f λ } λ Λ 1
2 K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X P 1 2 f : X P 1 (1) (5), 4 2K 2 X, : (1) K 2 X = 4 : (2) K 2 X = 3 : 3 1 (3) K 2 X = 2 f ( 1) : (4) K 2 X = 2 f ( 1) : (5) K 2 X = 1 :,, 2 ( k), k 1 (1) [13] [5], [12], MWG(f), 2
3 2 ([8]) X K 2 X = 4, f : X P 1 2 MWG(f), f 1, ,, 1 (1), 1, 1, MWG(f) f 2, [1], X 0, B, f : X B, F 2 X 0, NS(X) H 2 (X, Z) {DF Z D H 2 (X, Z)} Z, n 1 nz H 2 (X, Z) Z D (1/n)(DF ), D H 2 (X, Z) DF = (1/n)(DF ) F H 2 (X, Z) F nf n 2 F 2 = F 2 = 0 F 2 = 0 nf K X = FK X +F 2 = 2, F K X = F (K X +F ) n = 1 D 0 F = 1 D 0 H 2 (X, Z), B L, E D 0 + f L E, CF = 1 C 3
4 X C, f : X P f, f F 0 O X (K X ) O X (K X +F ) O F (K F ) 0 H 0 (X, K X + F ) H 0 (F, K F ) K X + F, Z f 2 [3],,, f Φ KX +F Z : X W := P 1 P 1 P(f O X (K X/P 1)) B W Γ, 0 B, (k 1), k, B k, W, B 4 k 1 B, W, : (0) (I k ) B Γ, B Γ, (2k 1) (2k) (II k ) B Γ, B, (2k) (2k + 1) (III k ) B Γ, B Γ Γ (4k 2) (4k 1) (IV k ) B Γ, B Γ (4k) (4k + 1) (V) B Γ, B Γ Γ B Γ, Γ f f Φ KX +F Z Γ B 4, f (I k ), (II k ), (III k ), (IV k ), (V), f,, (0) 4
5 , f Γ, Γ B, B, (0) 5 ( ) #{ } : B (ε + K X 2 + 6)Γ, ε = k (#{I k } + #{III k }) + #{V} K X =#{V} + k ((2k 1)(#{I k } + #{III k }) + 2k(#{II k } + #{IV k })) (21) (21) H f := K 2 X + 4 ( ), ( ) 1 : (V) (I k ) (III k ) (II k ) (IV k ) 4 2k 2k + 1 2k + 1 2k + 2 ρ(x) (10 K 2 X ),, rkmwl(f) = ρ(x) 2 (f 1 (t) 1) t P H f π : X W B π : X W B 2H f σ : W W, B W X, σ : W W σ : W W σ Φ Γ σ π : X P 1, H f ( 1) ẽ l, 2 f : X P 1 ẽ l ϕ : X X B (I k ), (III k ) (V) ε W σ, π, X ε ( 1) ε ( 1) E (σ π) ( K W ) ϕ (2K X (K 2 X)F ), (σ π) 0 ϕ (K X + F Z) E 5
6 (σ π) Γ ϕ F,, 2, K 2 X = 4 MWG(f), X P 2 (cf [8]) 6 (cf [8], [9]) X, f : X P 1 2, K 2 X = 4 MWG(f) υ 0 : X P 2, f F 1 : (1) 4l 2e 1 (2) 6l 2 (3) 7l 3e 1 2 (4) 9l 3 13 e i i=2 12 e i 8 e i i=1 i=9 11 e i i=2 8 e i 2e 9 2e 10 e 11 i=1 10 (5) 13l 5e 1 4 i=2 e i (K 2 X = 4 ), (K 2 X = 3 ), (K 2 X = 2 ), (K 2 X = 2 ), (K 2 X = 1 ) l = υ0o P 2(1) e i υ 0 ( 1) X f ( 1) (1), (2) (4),, υ : X (P 2 ) deg υ 0 (F ) deg υ (F ) (X, F ) #- (cf [2], [4]), f (cf [7]) K 2 X = 1, (5), 8 [3],, 6 υ 0 (2) e 9, e 10, e 11, e 12 (4) e 11, υ 0, (cf [11, 1]) 6
7 3 31 K 2 X = 2 f ( 1) X K 2 X = 2, f : X P 1 ( 1) 2, υ 0 : X P 2 6, e i υ 0 ( 1) X (X, F ), e 11 f ( 1) (cf [11, 1]),, υ 0 (e j ) υ 0 (e i ), i < j 7 υ 10 : X X 10 e 11, υ 9 : X X 9 e 11 e 10 F υ i F i X 9 υ 10 (e 9 ) ( 1) C 9, C 9 F 9 3, υ 0 (e 10 ) υ 0 (e 9 ) υ 11 (e 10 ), υ 11 (e 9 ) X 10 ( 1) C 10 C 10 F 10 3 υ 0 (e 10 ) υ 0 (e 9 ) X 10 υ 11 (e 10 ) ( 1) C 10, υ 11 (e 9 e 10 )C 10 0 υ 11 (e 10 )C 10 0 C 10 F 10 = C 10 ( 3K X10 + υ 11 (e 9 e 10 ) + 2υ 11 (e 10 )) 3, 6 υ 0, e 11 ( 1), e 10 e 9 e 10 e 9, e 11, 5 8 f Φ KX +F Z : X W := P 1 P 1 B B [0], B (III k ), (IV k ) (V) (I) B (I 1 ), W Γ [1] Γ [2], [0] P i [i], i = 1, 2 [1] [2] σ π, 6 e 9 e 10 ϕ (II) B W, (II 1 ), [0] P 3 [ ] P 3, W (E 3 E 4 ) 7
8 σ π, 6 e 10 ϕ, (E 3 E 4 ) σ π, 6 (e 9 e 10 ) ϕ f ( 1) e 11 X ê 11 (σ π) ê 11 Γ = e 11 F = 1 σ π ê 11, (σ π) ê 11, e 11 f, (σ π) ê 11 0 = e 11 (K X + F ) e 11 Z ê 11 E e 11 (2K X + 2F ) = m i (σ π) ê 11 (σ ) 0 4 i=1 m ie i, (σ π) ê 11 ( K W ) = e 11 (2K X + 2F ) = 0, #{m i m i = 1} = 2 (σ π) ê 11 B (σ π) ê 11 (B l (σ π) ẽ l ) = 2, [3] 9 X K 2 X = 2, f : X P 1 ( 1) e 11 2 f (II 1 ), F ( 1) e 10, e 10 (II 1 ) (e 9 e 10 ) f (II 1 ), F ( 1) e 9, e 10 K X + e 9 + e 10, Φ KX +e 9 +e 10 ɛ : S P 1, MWG(f) MWG(ɛ) MWL(f) MWL(ɛ) (32), ɛ : S P 1, 6 (4) (32), (I 1 ) f : X P 1 f (II 1 ), (II 1 ) 8, f (I 1 ) 8 [0] Γ [1] Γ [2] P 01 P 02 σ : W W P 01, P 02, P 1, P 2, 8
9 W ( 1) E 01, E 02, E 1, E 2 Γ [i], i = 1, 2 σ Γ [i] [i], i = 1, 2 σ [i] W ( 1) ( 1), Γ [i], ( 1), ( 1), ς : W W Σ 2 := P(O P 1 O P 1(2)) B ς B B B W B ϖ : S W S, X e 9 e 10 ν 2 : X S W Γ ν 2 ϖ K X + e 9 + e 10 K X + e 9 + e 10, S ɛ, [0] σ [0] ς W, E i, i = 1, 2 ς W Γ i Γ[i] ς Q i, i = 1, 2 ς E 3 Q 1 B, Q 2 B, 2 W, ς E 4 Q 2 B, Q 1 B, 2 W (Γ 1 + ς E 3 ) (Γ 2 + ς E 4 ) ς (σ ) Γ Q 1 Q 2 B, 4 W 0, ϖ ς E i, i = 3, 4 ν 2, f e 11 ν 2, B S ɛ T ɛ T/(Ze 9 Ze 10 ) = T ɛ ϖ ς E 1, ϖ ς E 3, ϖ ς E 2 ϖ ς E 4 ν 2 Θ 1,0, Θ 1,1, Θ 2,0 Θ 2,1 0 e 9 = 2Θ 1,0 Θ 2,1 + 2e 11, e 10 = 2Θ 2,0 Θ 1,1 + 2e 11 T = T ɛ Ze 9 Ze 10, NS(X) = NS(S) Ze 9 Ze 10 [15, 1 9] (32), (O) ɛ ɛ : S P 1 2K S + 2(O) ɛ,, (O) ɛ P 3 v 0, S v 0 v 0 W Σ 2 Σ 2 0, Γ S Σ 2, Γ B, 9
10 No T ɛ MWL(ɛ) MWG(ɛ) tor 42 A 6 1 A 2 1 (Z/2Z) 2 57 D 4 A 3 1 A 1 (Z/2Z) 2 60 A 3 2 A 1 1/6 Z/3Z 71 D 6 A (Z/2Z) 2 73 D (Z/2Z) 2 74 (A 3 A 1 ) 2 0 Z/4Z Z/2Z 1 ϖ, ɛ : S P 1 Γ Σ 2 P 1 ϖ B Q 1, Q 2 Q i, i = 1, 2 Σ 2 Γ i Q 1 B, Q 2 B, 0 + 2Γ Σ 2 C 3, Q 2 B, Q 1 B, 0 + 2Γ Σ 2 C 4 (Γ 1 + C 3 ) (Γ 2 + C 4 ) Σ 2 Λ Λ Q 1 B Q i, i = 1, 2 ϖ R i ϖ Λ R 1 R 2, 4 ν 2 : X S R 1 R 2 K 2 X = 2, ϖ Λ ν 2, 2 f, 0 X f ( 1), (Γ 1 + C 3 ) (Γ 2 + C 4 ) f (I 1 ) (32) 10 f : X P 1 ɛ : S P 1 9 f (II 1 ), MWG(ɛ) MWG(ɛ) tor, T ɛ MWL(ɛ) MWG(ɛ) tor, [6, ], 1 ɛ : S P 1 (O) ɛ 2K X + (O) ɛ S Σ 2 B T ɛ MWL(ɛ) MWG(ɛ) tor, 1,, B Σ 2 0, 0 + 2Γ Σ 2 Q 2 10
11 , f (II 1 ), (I 1 ), f : X P 1 B Σ 2, 32, ς (σ ) Γ 0 + 3Γ, B ( 0 + 3Γ )( 0 + 2Γ ) = 2, B 0, (II 1 ) f 9 MWG(f), 11 f : X P 1 1 (4) MWG(f) f 2 7, 1, f (O) K X 2 = 3, X K X 2 = 3, f : X P1 2 (21), f (0) 11
12 , (I 1 ) (III 1 ), (V) 6 υ 0 : X P 2 e i υ 0 ( 1) X 8 9, K X + e 12 + e 11 + e 10 + e 9, ɛ : S P 1 ɛ (O) ɛ 2K S + 2(O) ɛ ϖ : S Σ 2, B F Σ Γ Λ : (32A) (32B) (32C) f (I 1 ) (I 1 ), Σ 2 Γ 0 Γ 1 0 Γ 0 Γ 1 Λ,, B f (III 1 ) (III 1 ), Σ 2 2Γ 0 0 Γ 0 B Λ f (V) (V), Σ 2 2Γ 0 0 Γ 0 B, B Λ (V)
13 , Γ 0 B 1, (O) = e 12, ϖ Γ 0 X K X + e 9 + e 10, (I 1 ) f : X P 1 T = ZF Z(O) Z( K X + e 9 + e 10 ) [15, 9] 1 (2) 9, (32C) : 12 X K 2 X = 3, f : X P 1 (V) 2 K X + F ɛ : S P 1 MWG(f) MWG(ɛ) MWL(f) MWL(ɛ) (33), ɛ : S P 1, 6 (2) (33), (V) f : X P 1, (32A) (32B), (I 1 ) (III 1 ) f, 12, MWG(f), 13 P 1 P 1 t, x (α, β, γ, δ) (C 4 \ {γ = 0}) \ {α 2 4β = βδ + γ = 0} tx(x 5 + γt 3 + βt 2 x + αtx(x δt) 2 δtx 2 (3x 2 3δtx + δ 2 t 2 )) = 0 (34), P 1 P 1, X P 1 P 1 X 2 f : X P 1 X K X 2 f t = 0 = 3, MWG(f) (V), t = (0) 13
14 δ 0 δ = 0 β β =
15 f, 2 8, 9, 10, 11 β 0,, α 2 4β 0,, α 2 4β = 0, β = 0, ( 3), ( 2), α 0, α = 0, 2 f : X P 1, MWG(f) KX 2 = 3 X, f Φ KX +F : X P 1 P 1,, (34) X K X 2 = 3, f : X P1 2 MWG(f) f Φ KX +F : X W := P 1 P 1 B P 1 P 1 t, x, P 1 P 1 Γ t, x 13, f t = 0 (V) B (0, 0) B Γ 0 (0, 0), (B Γ 0 ) 0 = 3 B 0, 9, B 0 Γ 0,, (, ) B 0 Γ 0 x 5 + t 3 = 0 (34) 33 f ( 1) f : X P (1), (2) (4) f ( 1), 6 X P 2 υ 0 ( 1), NS(X) f : X P 1 (3) (5), 3, f ( 2) f 6 υ 0,, NS(X) = Zl i Ze i MWG(f) MWL(f), [15, 1 9], 14 D 1 D 2 F 1 X NS(X) 15
16 , D i T i NS(X)/T 1 NS(X)/T 2,, (T1 ) NS(X) (T 2 ) NS(X) L i = (T i ) NS(X), i = 1, 2 Z α : L 1 L 2, H 1 L 1 α(h 1 ) = H 1 (D 1 H 1 )F β : L 2 L 1 β(h 2 ) = H 2 (D 2 H 2 )F, β α α β, L 1 L 2, α β, L 1 L 2 NS(X), NS(X) V T i = ZD i V T i = (T i Q) NS(X) V = (V Q) NS(X) NS(X)/T i V T i /T i V /V, NS(X), ( ) NS(X)/T 1 T1 /T 1 L 1 ( ) T2 /T 2 L 2 NS(X)/T 2, f 1 (3), 6 e 11 e 10, Φ KX +e 10 +e 11 Φ F +2KX : X P 1 P 1, e 10 e 11, K X + e 11, T, f 1 (3), ( 2) (O), T = ZF Z(O) Z(e 10 e 11 ) Z( K X + e 11 ) [15, 9] 1 (3) 15 P 1 P 1 t, x, pr 1 : P 1 P 1 P 1 x 5 + tx 4 + tx 3 + t 2 x + t 3 = 0 A x + t = 0 pr 1 C pr 1 t = 0, Γ [0], Γ [ ], B = A + C + Γ [0] + Γ [ ], B Γ [0] Γ [ ] (V) B P 1 P 1, X pr 1 16
17 X 2 f : X P 1 X K X 2 = 2, MWG(f) f (O) (O) 2 = 2, C X 2(O) f t = 0, (V), 12, f f 1 (5) Φ F +4KX : X P 1, 6 e 10 e 9 e 8, K X + e 10 + e 9 + e 8, P 2 F P 2,,, f, f, e 8 e 9, e 9 e 10, 2K X + e 10 T, f 1 (5), ( 2) (O), T = ZF Z(O) Z(e 8 e 9 ) Z(e 9 e 10 ) Z( 2K X + e 10 ) [15, 9] 1 (5) 16 X K 2 X = 3, f : X P 1 (V) 2 f, MWG(f), f 2 f, (V) , (V) 2, f : X P 1 1 (5), MWG(f) 17
18 [1] A Beauville: Complex algebraic surfaces London Mathematical Society Student Texts, 34, Cambridge University Press, Cambridge, pp x+132 (1996) [2] R Hartshorne: Curves with high self-intersection on algebraic surfaces Publ Math IHES 36, (1969) [3] E Horikawa: On algebraic surfaces with pencils of curves of genus 2, in Complex Analysis and Algebraic Geometry, a Volume Dedicated to Kodaira K, pp (Cambridge, 1977) [4] S Iitaka: On irreducible plane curves Saitama Math J 1, (1983) [5] K V Nguyen: On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces J Math Sci (New York), 102, (2000) [6] K Oguiso, T Shioda: The Mordell-Weil lattice of a rational elliptic surface Comment Math Univ St Paul 40, (1991) [7] S Kitagawa: Maximal Mordell-Weil lattices of fibred surfaces with p g = q = 0 Rend Sem Mat Univ Padova, 117, (2007) [8] S Kitagawa: Exceptional hyperelliptic fibrations on rational surfaces in preparation [9] S Kitagawa: Pencils of genus two curves on rational surfaces preprint (2008) [10] S Kitagawa: Genus two fibrations on rational surfaces and frame lattices in preparation [11] S Kitagawa, K Konno: Fibred rational surfaces with extremal Mordell-Weil lattices Math Z 251, (2005) [12] M-H Saito: On upper bounds of Mordell-Weil ranks of higher genus fibrations Proc meeting and workshop on Algebraic Geometry and Hodge Theory at Hokkaido Univ, June 1994, pp [13] M-H Saito, K Sakakibara: On Mordell-Weil lattices of higher genus fibrations on rational surfaces J Math Kyoto Univ 34, (1994) [14] Shioda, T: Mordell-Weil lattices of type E 8 and deformation of singularities Lecture Notes in Math, 1468, Springer, Berlin, (1991) [15] T Shioda: Mordell-Weil lattices for higher genus fibration over a curve New trends in algebraic geometry (Warwick, 1996), , London Math Soc Lecture Note Ser, 264, Cambridge Univ Press, Cambridge,
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