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
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1 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 for Scientic Research (KAKENHI No ) and the Grantin-Aid for JSPS fellows.
2 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 cone (=movable ) Movable 2 ( ) n 3 X P n+1 Lefschetz Pic (X) = ZO X (1) X
3 2.3. *1 X P 1 P 3 (2, 4) Lefschetz Pic (X) = ZO X (1, 0) ZO X (0, 1) X 3 Calabi- Yau f : X P 3 2 generically finite X X0 2 f 0 (Y 0,..., Y 3 ) + X 0 X 1 f 1 (Y 0,..., Y 3 ) + X1 2 f 2 (Y 0,..., Y 3 ) f 0, f 1, f 2 4 f fiber (Y 0 : : Y 3 ) P 3 f 0, f 1, f 2 fiber P 1 ( 4 3 = 64 ) f Stein fiber g : X Y K X g 64 P 1 flopping contraction g flop g : X Y f : X P 3 ι X 2 [Og, Theorem 3.3] ι O X (0, 1) = O X (0, 1), O X (1, 0) + ι O X (1, 0) = O X (0, 4) X = X g = g ι Mov (X) O X (1, 0) ι O X (1, 0) = O X ( 1, 4) Mov (X) Nef (X) = R 0 O X (1, 0) + R 0 O X (0, 1) Nef (X ) = ι Nef (X) = R 0 O X (0, 1) + R 0 O X ( 1, 4) Mov (X) ι O X (1, 0) = O X ( 1, 4) Nef(X ) Nef(X) O X (0, 1) = ι O X (0, 1) O X (1, 0) X 2.4. Q toric *1
4 Cox ( 2.13) = D D- D- D-nef D- [HK, Proposition 1.11] D ( [GOST] ( K X )- ) K X K3 Enriques 1. X 2. X nef cone 3. X Proof. 1 2 [AHL] X nef cone nef semi-ample Riemann-Roch log mmp log abundance K3 Enriques nef semi-ample (R ) ( [Og, Theorem 3.1 (3)] ) 3 2 Morrison ([Kaw, Theorem 2.1]) Calabi-Yau ( )
5 2.2 Cox VGIT 2.6. X X Weil Γ Wdiv (X) Γ R X (Γ) Γ k- R X (Γ) = H 0 (X, O X (D)). D Γ Weil D R(X, D) = m 0 H 0 (X, O X (md)) 2.7. X Q Cl (X) X Weil Γ Γ Q Cl (X) Q ( Q = Z Q) Γ R X (Γ) X Cox 2.8. Cl (X) torsion free Γ Cl (X) Cox Cl (X) Cox (log terminal,log canonical ) [GOST, Remark 2.17] 2.9. Example X Picard ZO X (1) O X (1) R(X, O X (1)) = k[x 0,..., X n+1 ]/(F (X 0,..., X n+1 )) X Cox (F X ) X toric [Co] X Cox X torus 1 1 Cox Γ Γ torus T = Hom gp (Γ, k ) s H 0 (X, O X (D)) R X (Γ) g T g s = g(d)s
6 T V = Spec R X (Γ) GIT( ) D Γ T ev D ev D (g) = g(d) k V ss (ev D ) V V ss (ev D ) T GIT V ss (ev D ) T (categorical quotient)v ss (ev D ) V ss (ev D )//T Proj (R(ev D )) R(ev D ) ev D semi-invariant R X (Γ) f ev D semi-invariant m g f = ev D (g) m f ( g T ) m R(ev D ) X 2.2 V = Spec k[x 0,..., X n+1 ]/(F (X 0,..., X n+1 )) = V (F ) A n+2 V \ V ss (ev A ) = {(0,..., 0)} V (A = O X (1)) V ss (ev A )//k = Proj (k[x 0,..., X n+1 ]/(F (X 0,..., X n+1 ))) = X D R(ev D ) GIT VGIT(variation of GIT quotients) R(ev D ) = R(X, D) Cox VGIT A 1 D V ss (ev A ) V ss (ev A ) V ss (ev D ) V ss (ev D ) /T /T //T V ss (ev A )/T V ss (ev A ) V ss (ev D )/T V ss (ev D )//T = ( ) X φ D Proj R(X, D) =
7 /T (geometric) 1 1 ( ) φ D Proj R(X, D) 2 D, E φ D φ E ([Ok, Proposition 6.8]) X D E X 1. V ss (ev D ) = V ss (ev E ) 2. φ D φ E B(D) = B(E) B(D) = m Bs md D stable base locus Pic (X) R 2 (=[HK, Proposition 2.9]) X Q Cl (X) X X Cox toric Cox Fano X Fano effective Q (X, ) klt (K X + ) 0 Fano nef cone nef semi-ample [BCHM] ([BCHM, Corollary 1.3.2]) Fano Cox Fano Cox R X (Γ) Γ [HK] (
8 ) Γ ([CL]) GKZ X Cox VGIT Γ Q Cl (X) Q Γ Q χ(t ) Q (χ(t ) T ) D ev D Cl (X) Q χ(t ) Q effective cone ( semi-invaraint ) cone effective cone 3.1. X effective cone 2.12 ( 2 ) [Ok, Proposition 6.8] toric Gelfand-Kapranov-Zelevinsky(GKZ) [OP] [Ok] Zariski Cox VGIT GIT [Hau] [Ok, Example 9.1] X Fan (X) 3.2 ([Ok, Theorem 1.1]) 3.2. X f : X Y (Y Q ) Y
9 Y Cox X Cox f f : Pic (Y ) R Pic (X) R X Y Y X ([Ok, Theorem 1.2]) Fan (Y ) = Fan (X) Pic (Y )R Fan (X) Pic (Y )R Fan (X) Pic (Y ) R ([Ok] ) 3.3 global Okounkov body [LM] (global) Okounkov body [LM] X n L big X Y = (Y 0 = X Y 1 Y n = {pt}) ( ) R n Y (X, L) L (Y )Okounkov body L Y (X, L) Euclid ( n! ) L Okoukov body L ( ) Seshadri [I] toric Okounkov body moment polytope Okounkov body global Okounkov body R n N 1 (X) R big L N 1 (X) fiber L Okounkov body ( Y ) (global) Okounkov body toric toric strata global Okounkov body [LM, Proposition 6.1
10 (ii)] toric 3.4. X global Okounkov body [LM, Problem 7.1] [Ok3] 3.5. X N 1 (X) R Zariski ( Fan (X) ) ([Ok3, Lemma 1.2]) 3.6. global Okounkov body Okounkov body Y (X, L) Y 1 Okounkov body ( ) ([Ok3, Lemma 4.1]) X 3.7 X (2 ) global Okounkov body ( [Ok3]) [S] X Cox ( affine V ) GIT V us (ev A ) = V \ V ss (ev A )(A ample ) V 3 X Picard 1 X toric 2 1 ([S]) toric
11 3 toric ample [Kaw][Og] Fano Cox toric Cox ([HK, Corollary 2.10]) toric Fano Fano - - ( ) [GOST] 3.8. X Fano X Cox log terminal [GOST] log terminal F ([HW, Theorem 3.9]) 3.8 ([GOST, Theorem 1.2]) 3.9. X Fano X F F F F ([GOST, Proposition 2.10]) F F (of globally F -regular type) [GOST, Definition 2.13] [SS] Fano F [SS, Theorem 1.2] 3.9 [Ok2] X F ( ) X F X [Ok2]
12 [ADHL] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, arxiv: [AHL] M. Artebani, J. Hausen, and A. Laface, On Cox rings of K3 surfaces, Compos. Math. 146 (2010), no. 4. [BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2. [CL] A. Corti, V. Lazic, New outlook on Mori theory, II, arxiv: v2. [Co] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, [GIT] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer- Verlag, Berlin, [GOST] Y. Gongyo, S. Okawa, A. Sannai, and S. Takagi, Characterization of varieties of Fano type via singularities of Cox rings,. [HW] N. Hara and K.-i. Watanabe, F -regular and F -pure rings vs. log terminal and log canonical singularities, J. Algebraic. Geom. 11 (2002), no. 2, [Hau] J. Hausen, Cox rings and combinatorics II. Mosc. Math. J. 8 (2008), no. 4. [HK] Y. Hu and S. Keel, Mori Dream Spaces and GIT, Michigan Math. J. 48 (2000). [I] A. Ito, Okounkov bodies and Seshadri constants, preprint. [Kaw] Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997). [LM] R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5. [M] J. Mckernan, Mori dream spaces, Jpn. J. Math. 5 (2010), no. 1. [OP] T. Oda and H. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. (2) 43 (1991), no. 3. [Og] K. Oguiso, Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups, arxiv: [Ok] S. Okawa, On Images of Mori Dream Spaces, arxiv: [Ok2] S. Okawa, Surfaces of globally F -regular type are of Fano type,.
13 [Ok3] S. Okawa, On global Okounkov bodies of Mori dream spaces, in the proceedings of the Miyako-no-Seihoku Algebraic Geometry Symposium (2010). Also available on [S] J. Shin-Yao, A Lefschetz hyperplane theorem for Mori dream spaces, Math. Z. 268 (2011), no [SS] K. Schwede and K. Smith, Globally F -regular and log Fano varieties, Adv. Math. 224 (2010), no. 3.
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References tll A. Hurwitz, IJber algebraischen Gebilde mit eindeutige Transformationen Ann. in sich, Math. L27 A. Kuribayashi-K. Komiya, On Weierstrass points of non-hyperelliptic compact Riemann surfaces
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3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]).
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