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3 Introduction. [6], I. Simon (), J.-E. Pin. min-plus ().,,,. min-plus. (min-plus ). a, b R,, { a b := min(a, b), a b := a + b.. (R,, ) (, ). ( min-plus min, max ).,., NR := R r, MR := HomR(NR, R)..2. NR, ϕ: NR R ϕ = min(m + c,..., m s + c s ) m,..., m s MR c,..., c s R..3. V ( NR), ϕ,..., ϕ s : NR R V = {p NR : ϕ i p affine }.

4 ϕ dq dq ξ ϕ.: ϕ (.4),..4. r = NR = R. ξ : NR = R R, p R ξ(p) = p (). ϕ := min( ξ,, ξ). min,.., ϕ (), {, } R. ϕ := ξ. ϕ..5. r = 2. ξ, η : NR = R 2 R ξ : R 2 (a, b) a R η : R 2 (a, b) b R, ϕ := min(ξ, η, 0).,.4., ξ, η, ξ 0, η η 0 η, ξ ξ η 0, ξ ϕ.2: (.5 ), ϕ,.2. 2

5 . ξ, η, ϕ = min(2ξ +, ξ + η, 2η +, ξ, η, ).,.3., ϕ. η ϕ ξ 2ξ + η ξ ξ + η 2η +.3: (.5 ).6., ξ, η.5, ϕ := min(aξ, bη) (a, b 0)., ϕ,.4. η aξ ( b ) a bη ξ.4: ϕ (.6) 2. (() [], [3] ). () () (fan) (2) 3

6 (3) (4), 2. (). (C ) 2 (x + y + = 0) R 2 (x, y) ( log x, log y ), 2. (). C \ {0, } P C \ {0,, } ppppp pppppppppppppppppppppppppppppppppppp ppppppppppp p ppppppppp ppp 0 2.:,.2. 3 () K C((t)) C((t)) = n C((t/n )). (C )., Puiseux. v K : K R v K : K i Q a it i min{i : a i 0} R. v K (0) :=, v K K R := R { }. R 0 R x log x,, R. T := (K ) r (r ), v T : T R r. v T : T = ( K ) r v r K R r (x,..., x r ) (v K (x ),..., v K (x r )) 4

7 3.. V T., f,..., f s K[X ±,..., X ± r ] V = {(x,..., x r ) : f i (x,..., x r ) = 0, i}., V V trop. V trop := v T (V ) R r V trop,, V := {(x, y) (K ) 2 : x + y + = 0} (K ) 2, V trop? x + y + = 0, 0, (x, y) V, v K (x)(=: ξ), v K (y)(=: η), v K () = 0. V trop min(ξ, η, 0). V trop, = ().,.2, 3.. v K (y) v K (x) V trop 3.: I K[X ±,..., X r ± ]. V(I), : a m x m I \ {0} m M V(I) trop = (ξ,..., ξ r ) R r : m (), m (2) M (m () m (2) ) s.t. v K (a m (i)) + (ξ, m (i) ) v K (a m ) + (ξ, m) ( m M, i =, 2) ( =: V trop (I) ). M = Z r, ξ = (ξ,..., ξ r ) R r, m = (m,..., m r ) M (ξ, m) := i ξ im i. 5

8 3.2, monomial, 0 monomial,,., monomial a m x m v K (a m ) + (ξ, m). 3.2, V, V trop min( )., V (I),, V := V(tX 2 + XY + ty 2 + X + Y + t) (K ) r V trop =. (, monomial )., V = V(X 2 + Y 2 + ) V trop = X 2 Y 2., 3.2, V trop [5] R r affine. (R, R r affine ). R r+ /R(,..., ) r., R r (T v K R r ) R r affine. ()., P 2,trop.,, P 2,trop. T trop. M = Z r. T = ( K ) r = HomK alg (K[M], K) NR. = HomZ(M, K ) v T HomZ(M, R) = NR. 6

9 , A r K := K r = Hom (N r, K) v K Hom(N r, R)., N = {0,,...}, K., A r,trop := Hom (N r, R) = R r. ([4, Proposition.8]), A,trop, A 2,trop. A,trop = R ( = R 0 ) r A 2,trop = R 2, P,trop = A,trop A,trop r r r r P,trop = =,. P 2,trop = A 2,trop P 2,trop = A = B C A B C. P 2,trop. ()., P 3,trop. 3.7.,, N r. P r,trop, P r P v P r r,trop V V trop, V trop. 7

10 NR 3.2: x + y + = 0 P 2,trop 3.3: X a + Y b 3.8. P 2,trop NR, x + y + = 0, 3.2. X a + Y b, 3.3. NR 3.2 P 2,trop, boundary, 3.3, P 2,trop boundary. 4 Tropical Bezout Theorem on P 2,trop (d ).(e ) = = de (Bezout Theorem). tropical. 4. NR, node. 4. (Tropical Bezout Theorem [2, Corollary 4.4]). f d, g e. V trop (f) V trop (g) node,. V trop (f).v trop (g) = de 8

11 q node 4.:, node 4.2. [5], generic tropical curves. generic, f d, boundary. [5] tropical P 2 boundary, boundary., binomial curve X a + X b, X c + X d, 4.2, boundary. boundary c c 4.2: boundary., (), d e de., f g ( x, y, z ), x, y, z f, g. boundary.,. 4..,, ([2, Section 5]).,.,., 3.2 V = V(X + Y + ) 3.4 V = V(X 2 + Y 2 + ), Bezout. 9

12 4.3. x m, x m, m m = w (primitive vector) w X 2 + Y 2 +, X + Y +, X Y 2 X Y X 2 + Y 2 + X + Y + 4.3:, X 2 + Y 2 + X 2 ( ) ( ) ( ) ( ) = = , 2.,., 4.4. C e prim(e) D w t e w e e prim(e P ) 4.4: 4.4, C D P i(c.d; P ) i(c.d; P ) := w e w e M/(Z prim(e) + Z prim(e )) = w e w e det(prim(e), prim(e ))., M monomial. prim(e) = ( ) a b, prim(e ) = ( ) c d, i(c.d; P ) = w e w e ad bc. 0

13 C e tp e D 4.5:, boundary. boundary. P 2,trop (), 4.5. i(c.d; P ) = w e w e. P 2,trop ( 4.6),, 4.6 base (, ) ( ) ( ) a prim(e) =, prim(e c ) = b d, i(c.d; P ) = w e w e min(ad, bc). prim(e) = ( ) a b C tp prim(e ) = ( ) c d D 4.6: P 2,trop 4.5. C D C.D. C.D = P : i(c.d; P ), , P, (( ) ( )) det 0, 0 =.

14 , =. 4.7, boundary Q. primitive vector ( ), Q.,. t ( 0 ) ( ) P 0 4.7: tx + Y X + Y tq 4.7., 2 ( 4.8). 2 = 2. d d d 4.8: X 8 +X 4 +X 2 Y +XY 2 +Y 5 +Y 8 tx 2 +Y, 4.9, d. 4d ( ) ( 2 ) ( d 3 ) tx 2 + Y 3 X 8 + X 4 + X 2 Y + XY 2 + Y 5 + Y :. 2

15 4.9., 4.0 X a + Y b X c + Y d (a, b, c, d 0, a b, c b). 4.0 P, Q, R, P ( b a) Q ( d ( ( ) ) c) n P 0 R 4.0: X a + Y b X c + Y d ( det d c ) b a, Q min(bc, ad). R,, base ( ) ( ), 0. base, ( ) ( ) ( ) b = (a b) + a a 0, R min((a b)c, a(c d))., ( ) det d b + min(bc, ad) + min((a b)c, a(c d)) = ac c a, Tropical Bezout Theorem. [] T. Kajiwara, Tropical hypersurfaces and degeneration of projective toric varieties, 2005 [2] T. Kajiwara, Tropical toric varieties, preprint [3] G. Mikhalkin, Enumerative tropical algebraic geometry in R 2, math.ag/

16 [4] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 5, Springer-Verlag, Berlin, 988 [5] J. Richter-Gebert, B. Sturmfels, and T. Theobald, First steps in tropical geometry, Idempotent mathematics and mathematical physics, , Contemp. Math., 377, Amer. Math. Soc., Providencs, RI, 2005 [6] D. Speyer and B. Sturmfels, Tropical mathematics, math.co/

k k Q (R )Z k X 1. X Q Cl (X) 2. nef cone Nef (X) nef semi-ample ( ) 3. 2 f i : X X i X i 1 2 movable cone Mov (X) fi (Nef (X i)) 3 movable

k k Q (R )Z k X 1. X Q Cl (X) 2. nef cone Nef (X) nef semi-ample ( ) 3. 2 f i : X X i X i 1 2 movable cone Mov (X) fi (Nef (X i)) 3 movable 1 (Mori dream space) 2000 [HK] toric toric ( 2.1) toric n n + 1 affine 1 torus ( GIT ) [Co] toric affine ( )torus GIT affine torus ( 2.2) affine Cox ( 3) Cox 2 okawa@ms.u-tokyo.ac.jp Supported by the Grant-in-Aid