平成 22 年度 ( 第 32 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 22 月年 58 日開催月 2 日 ) V := {(x,y) x n + y n 1 = 0}, W := {(x,y,z) x 3 yz = x 2 y z 2

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1 V := {(x,y) x n + y n 1 = 0}, W := {(x,y,z) x 3 yz = x 2 y z 2 = xz y 2 = 0} V (x,y) n = 1 n = 2 (x,y) V n = 1 n = 2 (3/5,4/5),(5/13,12/13)... n 3 V (0,±1),(±1,0) ( ) n 3 x n + y n = z n, xyz 0 (x,y,z)

2 y = sinx V 1 p Q,R,C C := {(x,y) xy + x 3 + y 3 = 0}, D := {(x,y) xy = 0} C,D V xy W xyz k n A n := {(x 1,...,x n ) x 1,...,x n k} n f 1 (x 1,...,x n ),..., f r (x 1,...,x n ) V := {(x 1,...,x n ) A n f 1 (x 1,...,x n ) = = f r (x 1,...,x n ) = 0} n P n P n := {[x 0 :... : x n ] x 0,...,x n k 0} k n + 1 [x 0 :... : x n ] x i 0 k 0 t [x 0 :... : x n ] = [tx 0 :... : tx n ] P n x 0 0 U 0 U 0 [x 0 :... : x n ] 1/x 0 [1 : x 1 /x 0 :... : x n /x 0 ] U 0 [1 : x 1 /x 0 :... : x n /x 0 ] (x 1 /x 0,...,x n /x 0 ) A n

3 U 0 A n U 0 A n U 0 A n x i 0 P n U i U i A n [x 0 :... : x n ] x i 0 P n U 0,...,U n P n = U i 0 i n P n n + 1 n = 2 V 2 := {(x,y) A 2 x 2 + y 2 1 = 0} C u = x + y 1, v = x y 1 V 2 = {(u,v) A 2 uv 1 = 0} V 2 (u,v) u A 1 V 2 A 1 o A 1 \{o} V 2 V 2 P 2 V 2 := {[U : V : W] P 2 UV W 2 = 0} V 2 (u,v) [u : v : 1] V 2 P 1 V 2 [U : V : W] [U : W] [W : V ] P 1 C V 2 C u = x + y 1, v = x y 1 R C C W := {(x,y,z) x 3 yz = x 2 y z 2 = xz y 2 = 0}

4 (t 3,t 4,t 5 ) 1 Y := {(x,y,z,w) xy zw = 0} 4 3 W A 1 t (t 3,t 4,t 5 ) W C := {(x,y) xy + x 3 + y 3 = 0} B B := { (x,y),[x : Y ] A 2 P 1 xy yx = 0} A 2 P 1 (x,y),[x : Y ] [X : Y ] (x,y) B A 2 π B (x,y),[x : Y ] (x,y) A 2 A 2 o x,y π B\π 1 (o) A 2 \{o} x = y = 0 [X : Y ] xy yx = 0 o π 1 (o) P 1 B A 2 P 1 A 2 C B C C C π C C π : B A 2 A 1 W B A 2 3. X

5 X X X 2 P 1 P 1 X P 1 X X X P 1 X ( ) C C P 1 (C 2 ) = 1 X C,D C D C,D (C D) X Z 1 (X) := { r i C i r i R} (C 2 ) C C C C C C D C C D (C C +C ) = (C D) = 0 (C 2 ) = (C C ) < 0 n X C (C D) D n 1 X n 1 Z 1 (X) := { d i D i d i R} Z 1 (X) Z 1 (X) R N 1 (X) = Z 1 (X)/ N 1 (X) = Z 1 (X)/ N 1 (X) N 1 (X) R C X X K X X X n x 1,...,x n n dx 1 dx n

6 f (x 1,...,x n ) ω = f dx 1 dx n dx dx = dx dt dt ω ω X P 1 = {[X : Y ]} P = [0 : 1] Y 0 x := X/Y ω = dx Q = [1 : 0] ω Q y := Y /X x = y 1 ω = dx dy dy = y 2 dy Q 2 ( 2 ) y 2 ω P 1 K P 1 = 2Q X ω D i n i divω := n i D i X K X ω P 1 dy K P 1 = 2P X f f D i f i div f := f i D i C (K X C) ω K X C (K X C) X X X (i) X K X K X X X (ii) X S f : X S f K X X P 1 f 4. N 1 (X) NE(X) NE(X) K X P 1

7 ( ) NE(X) = NE KX 0(X) + R 0 [C i ] C i (K X C i ) < 0 R 0 [C i ] π i : X Y i R 0 [C i ] X K X K X X Y Y 3 Y 3 x x 3 X π : X Y π X E Y K Y π X Y K X = π K Y + ae E π E a K Y E π K X E a π : X Y Y K Y π : X Y π + : X + Y π K X + X + X,X + Q- 2 3 Y := {(x,y,z,w) xy zw = 0} Q := {[X : Y : Z : W] XY ZW = 0} B Q {l t : [X : Z] = [W : Y ] = t} t P 1 {l t + : [X : W] = [Z : Y ] = t} t P 1 B X B X + X Y, X + Y

8 K X,K X + 0 (x,y,z,w) ( x,y,z, w) X Y X + S Q- X S (i) K X X (ii) K X K X π : X Y (ii-1) Y X π X (ii-2) π Y S X Y (ii-3) π Y / S π + : X + Y X + S X X + ( ) N 1 (X) ( ) X (X, ) X Q- R- R- 0 (X, ) K X K X 1

9 D H 0 (X,D) := { f div f + D 0} π : X Y Y P r π : X P r X f 0,..., f r [ f 0 :... : f r ] P r H = P r 1 π H π H 0 (X,π H) f 0,..., f r V K X H i (X,K X + ) ( ) X A H i (X,K X + A) = 0 (i 1) X S (X,S + B) S (S,B S ) K X + S + B S = K S + B S B 0 H 0 (X,K X + B) H 0 (X,K X + S + B) H 0 (S,K S + B S ) H 1 (X,K X + B) (X,S +B) (S,B S ) pl (X,S +B) pl 2005 pl ( ) n 1 n pl (X,S + B) (S,B S ) H 0 (X,m(K X + S + B)) H 0 (S,m(K S + B S ))

10 m(k X + S + B) m pl B Q- (X,S + B) R X := m 0H 0 (X,m(K X + S + B)) R X H 0 (X,m(K X + S + B)) m H 0 (X,m(K X + S + B)) H 0 (X,m (K X + S + B)) H 0 (X,(m + m )(K X + S + B)) R S := m 0H 0 (X,m(K S + B S )) ρ : R X R S pl R X ρ(r X ) R S ρ R X m X,S X m,s m S m + B m H 0 (X,ml(K X + S + B)) = H 0 (X m,ml(k Xm + S m + B m )), H 0 (X m,ml(k Xm + S m + B m )) H 0 (S m,ml(k Sm + B m,sm )) l l(k X + S + B) m X m m S m m S m T R l X := H 0 (X,ml(K X + S + B)) R l T := m 0 m 0H 0 (T,ml(K T + B m,t )) R l T ρ(rl X ) ρ(r l X ) ρ(r X) 7. R- R- R- R-

11 ( ) (X, ) (i) ( ) K X + R- R- (X, ) (ii) ( ) (X, ) Y Y (iii) ( ) K X + R- R- R- R- (i) (iii) K X + ( ) (ii) (X, = A + B ) A R- B pl X X S X Y X Y (i) ( ) X S ( ) (X, ) Q- m 0 H 0 (X,m(K X + )) ( ) (pl n ) n pl (i n ), (ii n ), (iii n ) (i), (ii), (iii) n (ii) (ii ) (i) (i n 1 ), (ii n 1 ), (iii n 1 ) (pl n )

12 (i n 1 ), (ii n 1 ), (iii n 1 ) (ii) (ii n 1 ) (ii n) (ii n) (X,S + B ) (S,B S ) (iii) (pl n ), (ii n) (i n ) pl (ii n) (iv) (iii n 1 ), (i n ), (ii n) (iii n ) R- Q- (v) (i n ), (iii n ) (ii n ) [0,1] [0,1] R- X Q- (X, ) C t (X, tc) K X + tc R- R- (X, ) C (i) K X + tc t (ii) t = 0 K X + X (iii) t > 0 NE(X) K X + K X + tc 0 π (iii-1) π (iii-2) π (X, ) C C ( ) C (X, ) C (X, ) K X + ( ) K X + C (X, ) C

13 8. R- A R- B A ( ) K X + (X, ) ( ) (X, ) 5 H 0 (X,m(K X + )) Φ m : X P r m m(k X + ) m Φ m ( ) K X + R- R- 3 0 ( ) 3 9. J. Kollár

14 61 (2009), C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc. 23 (2009), C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type II, J. Am. Math. Soc. 23 (2009),

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +