[6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, B 2, B 3 A i 1 B i+1 A i+1 B i 1 P i i = 1, 2, 3 3 3 P 1, P 2, P 3 1
*1 19 3 27 B 2 P m l (*) l P P l m m 1 P l m + m *1 A N P N (N + 1) (a 0 : a 1 : : a N ) 2
m + P m m l (**) 2 1 + 2 *2 l l l l l l l l l 1 (2) *2 R 1 1 3
y l O x l 3 x y 1 ax + by + c = 0 a x + b y + c = 0 (1) a : b : c a : b : c (1) 2 a : b = a : b a : b a : b 1 x, y, z 1 ax + by + cz = 0 a x + b y + c z = 0 (2) x = y = z = 0 (p, q, r) (kp, kq, kr) p : q : r (***) a : b : c a : b : c (2) 1 x : y : z = bc cb : ca ac : ab ba (**) (***) (**) (***) (p, q) 4
(p : q : r) (p, q) (p : q : 1) (p : q : 0) (kp, kq) (p : q : 1/k) k (p : q : 0) 1 ax + by + cz = 0 1 3 A 1 B 2, A 2 B 3, A 3 B 1 x = 0, y = 0, z = 0 m x + y + z = 0 (x : y : z) l ax + by + cz = 0 3 A 2 B 1, A 3 B 2, A 1 B 3 ax + ay + cz = 0, ax + by + bz = 0, cx + by + cz = 0 3 A i B i+1 A i+1 B i 3 xyz = 0, (x + y + c ) ( a ) ( a z b x + y + z x + b ) c y + z = 0 (3) k kxyz + (x + y + c ) ( a ) ( a z b x + y + z x + b ) c y + z ( x = (x + y + z)(ax + by + cz) b + y c a) + z (4) (3) 3 9 A i, B i, P i (i = 1, 2, 3) (4) 3 2 l, m x/b + y/c + z/a = 0 P 1, P 2, P 3 1 (**) 1 (***) (4) 4 xy- F (x, y) = 0 y = f(x) F (x, y) x, y x, y 1 (x p) 2 + (y q) 2 r 2 = 0 (5) 2 ax 2 + bxy + cy 2 + dx + ey + f = 0 (6) x n + y n 1 = 0 (7) 5
3 x, y *3 3 f(x, y, z) f(x, y, z) = 0 d F (x, y) d z f(x, y, z) f(x, y, 1) = F (x, y) f(x, y, z) = 0 F (x, y) = 0 (5), (6), (7) (x pz) 2 + (y qz) 2 r 2 z 2 = 0 (5 ) ax 2 + bxy + cy 2 + dxz + eyz + fz 2 = 0 (6 ) x n + y n z n = 0 (7 ) C 1 : F 1 (x, y) = 0 C 2 : F 2 (x, y) = 0 C = C 1 C 2 F 1 (x, y)f 2 (x, y) 2 x n + a 1 x n 1 + + a n 1 + a n = 0 (8) (****) n (8) n 3 A = (a ij ) 1 i,j 3 φ A (p : q : r) = (p : q : r ) p q = a 11 a 12 a 13 a 21 a 22 a 23 p q (9) r a 31 a 32 a 33 r (genus) *3 Wiles Taylor 6
5 ( 1) 2 2 2 ( ) 2 6 3 A 2 A 1 A 3 P 3 P2 P 1 B 1 B 2 B 3 1 2 Q 6 A 1, B 2, A 3, B 1, A 2, B 3 A i B i+1 A i+1 B i P i 1 3 A 1 B 2, A 2 B 3, A 3 B 1 x = 0, y = 0, z = 0 (x : y : z) 2 C 3 (6) C : ax 2 + by 2 + cz 2 + dyz + ezx + fxy = 0 a, b, c a = b = c = 1 A 1 (0 : 1 : a 1 ), A 2 (a 2 : 0 : 1), A 3 (1 : a 3 : 0), B 2 (0 : 1 : b 2 ), B 3 (b 3 : 0 : 1), B 1 (1 : b 1 : 0) a 1 b 2 = a 2 b 3 = a 3 b 1 = 1 a 1 + b 2 + d = a 2 + b 3 + e = a 3 + b 1 + f = 0 7
3 A 2 B 1, A 3 B 2, A 1 B 3 x y b 1 = a 2 z, y z b 2 = a 3 x, z x b 3 = a 1 y k kxyz + (x a 3 y a 2 z)(y a 1 z a 3 x)(z a 2 x a 1 y) = (x 2 + y 2 + z 2 + dyz + ezx + fxy)(a 2 a 3 x + a 3 a 1 y + a 1 a 2 z) x/a 1 + y/a 2 + z/a 3 = 0 P 1, P 2, P 3 (10) 6 3 (3) x, y, z 2 l : ax + by + cz = 0, m : bx + cy + az = 0 (x : y : z) 3 ( x xyz = 0, a + y c + z ) ( x b b + y a + z ) ( x c c + y b a) + z = 0 (11) 3 3 (ax + by + cz)(bx + cy + az)(cx + ay + bz) = 0 (12) P 1, P 2, P 3 n : cx + ay + bz = 0 9 A i, B i, P i 9 A i B i±1, l, m, n, (i = 1, 2, 3) 3 x y z x a = c = 1 3 A 2, P 2, B 2 x + y + z = 0 (x + y + z){x 2 + y 2 + z 2 + (b + b 1 )(yz + xz + xy)} 2 2 6 A i, B i, P i (i = 1, 3) x + y + z = 0 6 8
A 1 A 2 A 3 P 3 P 2 P 1 B 1 B 2 B 3 b 1 ω 2 2 4 3 (x + y + z)(x + ωy + ωz)(x + ωy + ωz) = 0 (Hesse pencil) x 3 + y 3 + z 3 3λxyz = 0 (13) 1. 4 3 λ =, ω, ω, 1 3 2. ( 1 : ω i : 0), ( 1 : 0 : ω i ), (0 : 1 : ω i ) i = 0, 1, 2 (14) 3 3. 3 C : f(x, y, z) = 0 9 H(x, y, z) = det( i j f) 1 i,j 3 = 0 1, 2, 3 x, y, z f H 3 af + bh = 0, a, b k (13) 3 3 (extremal elliptic surface) 13 *4 E 8 1 1 3 2 A 2 + A 2 + A 2 + A 2 E 8 *4 E 8 T 2,3,6 9 Mordell-Weil 4 D 4 +D 4, D 6 + A 1 + A 1, A 3 + A 3 + A 1 + A 1 A 2 + A 2 + A 2 + A 2 9
7 3 (Bézout) m n mn m = n = 1 (**), (***) m = 1 (****) m, n > 1 3 y = 4x 3 3x x = 4y 3 3y (15) 9 4(4x 3 3x) 3 3(4x 3 3x) x = 0 x 9 9 P 1 (1, 1), P 2 (cos π 5, cos 3π 5 ), P 3( 1 2, 1 2 ), P 4 (cos 2π 5, cos 4π 5 ), P 5(0, 0), P 6 (cos 3π 5, cos π 5 ), P 7( 1 1, ), P 8 (cos 4π 2 2 5, cos 2π 5 ), P 9( 1, 1) mn 2 (5 ) (1 : ± 1 : 0) m, n 2 (resulatant) mn 2 C 8 m n mn + 1 4 n n 2 mn m n(n m) n m 10
n λ 1 F 1 + λ 2 F 2 = 0 n 2 mn P 1,..., P mn m G = 0 G = 0 P 1,..., P mn P λ 1 F 1 +λ 2 F 2 = 0 P λ 1, λ 2 λ 1 F 1 +λ 2 F 2 = 0 G = 0 mn + 1 λ 1 F 1 + λ 2 F 2 G λ 1 F 1 + λ 2 F 2 = GH n m H H = 0 ( 5) 2 2 Q 3 C 1 = A 1 B 2 + A 2 B 3 + A 3 B 1, C 2 = A 2 B 1 + A 3 B 2 + A 1 B 3 9 6 A i, B i, (i = 1, 2, 3) m = 3, n = 2 3 P 1, P 2, P 3 1 (10) 4 5 ( ) ABC 3 P, Q, R ARQ BP R CQP 1 7 ARQ, BP R C 1, C 2 R M 3 C 1 + BC, C 2 + AC 9 AB 3 6 C, P, Q, M 4 2 CQP A R Q M B P C 11
9 5 3 P, Q, R ABC 3 Silverman[9, Chap. IV] ADE 3 ADE 1. 2Ã2 + Ã1 2 2. 3Ã2 < xyz, (ax + by + cz)(bx + cy + az)(cx + ay + bz) > ( 3) 3. 3Ã2 + Ã1 < xyz, (x + by + z)(bx + y + z)(x + y + bz) > ( 6) 4. 4Ã2 < xyz, x 3 + y 3 + z 3 >(13) 5. 3Ã1 ( 5) 6. 4Ã1 A 1 3 ABC, A B C 3 a, b, c, a b, c 6 2 1) 3 AA, BB, CC 2) 3 a a, b b, c c (perspective) 3 27 Jordan 8 3 E 8, E 7, E 6 27 3 12
B C A A A C C B B P B ([4]) 9 9 9 3 9 3 (9 3 ) (9 3-9 3 ) *5 (9 3 ) 3 Hilbert-Cohn-Vossen[5, Chap. 3] *6 9 (14) 3 12 (12 3-9 4 ) (12 3-9 4 ) 3 F 3 = {0, 1, 2} 3 2 S 4 192 [1] 3 2 1 S 4 4 3 6 10 10 (10 3 ) 5 S 5 [3] 10 10 p ij, l ij, (1 i < j 5) i, j, k, l l kl p ij *5 *6 3 27 Reye 13
C m C 1 : F 1 (x, y) = 0 n C 2 : F 2 (x, y) = 0 N P 1,..., P N (a 1, b 1 ),..., (a N, b N ) (9) ( 7) 7 m C 1 : F 1 (x, y) = 0 n C 2 : F 2 (x, y) = 0 C[x, y] I := (F 1, F 2 ) C[x, y]/i (a 1, b 1 ),..., (a N, b N ) C R i, 1 i N, N dim C C[x, y]/i = dim C R i = mn (16) i=1 8 C 1, C 2 mn R i (a i, b i ) m i = (x a i, y b i ) (a i, b i ) C 1 C 2 dim C R i 1 C 1 C 2 (transversal) C 1, C 2 (a i, b i ) r, s dim C R i rs C[x, y]/i N i=1 R i Artin [2, Theorem 8.7] (16) dim C C[x, y]/i = mn C[x, y] (Koszul) ( ) F1 0 C[x, y] (F 2, F 1 ) C[x, y] C[x, y] F 2 I 0 (17) 1 (F 2, F 1 ) (A, B) AF 1 + BF 2 F 1 F 2 C[x, y] C[x, y]/i C[x, y] I = C[x, y]/i ( )/( ) (x : y) P 1 m C 1 m m F 1 m C 2 F 1 m f 1 F 2 n f 2 P 1 A 2 x,y (0, 0) Hilbert l *7 (f 1, f 2 )C[x, y] (x, y) (x, y) l *7 l m + n 1 14
l C C[x, y] l C (x, y) l C[x, y] l C[x, y] I C[x, y] l C[x, y] C[x, y]/i C[x, y] l /(C[x, y] l I) (17) 0 C[x, y] l m n C[x, y] l m C[x, y] l n C[x, y] l I 0 (18) dim C[x, y] l /(C[x, y] l I) = dim C[x, y] l dim(c[x, y] l I) = dim C[x, y] l dim C[x, y] l m dim C[x, y] l n + dim C[x, y] l m n = mn [1] Artebani, M. and Dolgachev, I.: The Hesse pencil of plane cubic curves, Enseign. Math., 55(2009), 235 273. [2] Atiyah, M.F. and Macdonald, I.: Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969 2006 [3] Coxeter, H.S.M.: Desargues configurations and their collineation groups, Math. Camb. Phil. Soc. 76(1975), 227 246. [4] Dolgachev, I.: Abstract configurations in algebraic geometry, Proc. Fano Conference (Torino, 2004), Torino Univ., 2005. [5] Hilbert, D. and Cohn-Vossen, S.: Anschaulich Geometrie, Verlag von Julius Springer, Berlin, 1932 [6] Kempf, G.: Algebraic Varieties, Cambridge University Press, 1993. [7] Mumford, D.: Algebraic geometry I, Complex projective varieties, Springer-Verlag, 1976. [8] Reid, R.: Undergraduate Algebraic Geometry, Cambridge Univ. Press, 1988 1991 [9] Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves, Springer Verlag, 1994 2003 15