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1 Contents References , 1

2 2,., ,. 8.,,,..,.,, ,....,..,... 4.,..

3 3.,,,,,. 4.,, 3.., , Z. Z = {, 2, 1, 0, 1, 2, 3, }. Q, R. R >0, R 0., ,.

4 4 1 ( ) ( ). n R n. R n A, A = (a 1,, a n ) a i. 3 ( ). n R n... 4 ( ). n + 1 n., n , 3. 1 Yes , ( ). R n A, A 2 A. 6 ( ). R n A, A R n, A..

5 5 7 ( ). R n H n H = {(x 1,, x n ) R n a i x i = b} R n.,. i=1 H + = {(x 1,, x n ) R n H = {(x 1,, x n ) R n n a i x i b}; i=1 n a i x i b}. 8 ( ). H P R n. (i) P H + P H ; (ii) = P H P.. 9 ( ). P R n, P H P., H..., P F. 11. P F F, F F. F F i= [3] (13.11).. 1. P, α 1, α 2,, α v. F, F α i i ω(f). ω(p) = α 1.

6 6 2. P Ψ : F 0 F 1 F d 1 P, F i P i, Ψ P., d P., Ψ, 1 i d, ω(f i ) F i Ψ, ω(f 0 ), ω(f 1 ),, ω(f d 1 ), ω(p) d (Ψ)., 0 = (Ψ)., Ψ. 0 P..., ,.. r Z- N Z r.. N R R- NR := N Z R. 14 ( ). NR σ, N e 1, e 2,, e s σ = R 0 e R 0 e s = {a 1 e a s e s, a i 0 for every i}. σ ( σ) = {0}.. {e i } σ. dim σ σ R- σ + ( σ) = Rσ. σ = e 1,, e s. e i. e i N Re i Z-.., 1 e i e i...

7 7 15 ( ). σ. f : NR R. f σ. f(σ) R 0. τ := σ {f} = {x σ; f(x) = 0} σ. 8.,.. 16 ( ). N NR. (i) σ σ. (ii) σ, τ σ τ σ τ. (iii). := σ σ. = NR,. 17 ( ).. = NR ,..., ,,. 19 ( ).. h : NR R, N, r σ., r σ l σ : NR R, σ h = l σ., r σ, l σ (n) h(n), n σ, h..

8 8 20 ( ) P R n R n+1 P {1} {0}.,.. 23 ( ). σ, σ σ σ. = , Q-,., 21, Q-. 1 Q-., Q-. Q ( ). N. M := HomZ(N, Z). σ σ MR. X(σ) := Spec C[σ M] σ. σ X(σ), X( ).

9 9 26 ( ). σ = {0}. 0. X(σ) = Spec C[X 1, X1 1,, X r, Xr 1 ] (C ) r. (C ) r r toric ( )., σ. N Z r Z- {e 1,, e r } s r σ = R 0 e R 0 e s ( ) ,,

10 , -. -.,.. n., = NR R n. (n 1) σ = e 1,, e n 1. σ n τ n = e 1,, e n τ n+1 = e 1,, e n 1, e n+1. σ = τ n τ n+1. n n+1 a i e i = 0 i=1., a n+1 = 1. a n > 0.. a i < 0, 1 i α a i = 0, α + 1 i β a i > 0, β + 1 i n α β n ( ). e. σ e. σ e = 0 e {e i } n+1 i=1 σ e = mult(σ) mult(τ n+1 ) > 0 e = e n+1 σ e = a i σ e n+1 e = e i., mult. mult(σ) := [N σ : Ze Ze n 1 ]. N σ σ Ze Ze n 1 {e i } n 1 i=1. mult. mult = 1.,.... Z 1 ( ) = Rσ, Z 1 ( ) = Re., σ (n 1), e 1..,

11 11 : Z 1 ( ) Z 1 ( ) R.. 32 ( ). σ 1, σ 2 Z 1 ( ) e Z 1 ( ) σ 1 e = σ 2 e. σ 1 σ 2. Z 1 ( ). N 1 ( ) := Z 1 ( )/, N 1 ( ) := Z 1 ( )/. σ Z 1 ( ) N 1 ( ) [σ]. Z 1 ( ).. N 1 ( ) N 1 ( ) R 33 ( - ). n. NE( ) := R 0 [σ] -. σ (n 1). NE( ) N 1 ( ) ( - ). S n. S. S. NE( / S ) := R 0 [σ] -., σ (n 1) S ,.. 36 ( ). E := e i. e i \ S ( ). E. a NE( / S ) E a 0...

12 12 38 ( ).., NE( / S ) = R 0 [σ] ( ). R NE( / S ). R. a, b NE( / S ) a + b R a R b R ( ). R. R = R 0 [σ]. σ 1 σ 2 R 0 [σ 1 ] = R 0 [σ 2 ]. R (n 1) S..., (n 1). 41.., S R = R 0 [σ] 3. α 13. (1) α = 0,.... (2) α = 1,...

13 13 (3) α 2,.... -,. S , -., -., ( ) , ,,. 45 ( )

14 [5] ,. E = e i. E. E R < 0 R. -. E., ( )..., ,...

15 15. E E E = 0. M., 48. S. S. E,. S M. M. (i) M S. (ii) M S 1. (iii) M.. M 49. M. 2, ,.. 18.,.,..! [1]. 2.5.,. [2]..... [4]

16 16,,,.,. [3]. 20..,...,, [3]...,. References [1],. [2],. [3],. [4], Introduction to the Mori Program. Springer [5] M. Reid, Decomposition of toric morphisms