3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced) (Mumford [8] ). [7] (uniruled) V,. ρ : H 0 (S, N S/V ) H 0 (C, N C S/V ) (1.1) C S. ρ, C ( ) C V, S V S, C. ρ S C 1,. 1, C S. C V I C O V (S) uler χ(v, I C (S)), C S.. 1.1. C S, χ(v, I C (S)) 1, C V ( ) S S V Kleppe V 3 P 3, S 3 S 3,. 2006 ( 18 11 3 5, )
1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1,. 1.3. V 3 del Pezzo, H. H S C V 2 : (1) χ(v, I C (S)) 1; (2) S l C good line, N l/v ( O 2 P 1 ). C stably degenerate, C ( ) C V C S H. V (:= (H 3 ) V ) n, C d g, χ(v, I C (S)) 1 g d n. g < d n C stably degenerate. S bad line l (i.e. N l/v O 2 P ) C l =, C stably 1 degenerate. ( 7 del Pezzo 3-fold V 7 P 8.) V, V Hilbert scheme Hilb sc V. 1.4. C S V 1.3 (1) (2), C g 2. 3 : (a) [C] Hilb sc V ; (b) H 1 (V, I C (S)) 0; (c) S (good) line l C. H 1 (V, I C (S)) (1.1) ρ. 1.3 (cf. 4.3), ρ C 1 ( ρ ) 2. 3 S V 1. 1 1.3. 0 k.
2 2.1 Del Pezzo 3-folds 3 V K V H K V = 2H del Pezzo 3-fold. H V, H V (H 3 ) V V. del Pezzo 3-fold Fano. Iskovskih [4] [2, 3], del Pezzo 3-fold V 1 V n (1 n 8) V 6. 1 1: Del Pezzo 3-folds del Pezzo 3-folds n ρ V 1 = (6) P(3, 2, 1, 1, 1) 1 1 6 V 2 = (4) P(2, 1, 1, 1, 1) 2 1 4 V 3 = (3) P 4 3 1 3 V 4 = (2) (2) P 5 4 1 2 2 V 5 = [Gr(2, 5) Plücker P 9 ] P 6 5 1 Grassmann V 6 = [P 2 P 2 Segre P 8 ] P 7 6 2 V 6 = [P 1 P 1 P 1 Segre P 7 ] 6 3 V 7 = Bl pt P 3 P 8 7 2 P 3 1 V 8 = P 3 Veronese P 9 8 1 P 3 2 Veronese n ρ V, Picard. H S del Pezzo. 8 P 2 P 1 P 1. good line bad line Del Pezzo 3-fold V l P 1 (l H) V = 1 (line). l, N l/v : N l/v O P 1(k) O P 1( k) (k = 0, 1, 2, 3). k = 0 l good line, k 0 bad line. S, L S. 2.1. S ι : S := S \ S. ( 2 ) S < 0 deg L 0 L L ι O S H 1 (S, L) H 1 (S, L S )
. H 1 (S, L) H 1 (S, L S ). 2.2 1 V, X V. X 1 (first order infinitesimal deformation) X V Spec k[t]/(t 2 ) Spec k[t]/(t 2 ) X. X 1 X α : I X O X. α Hom(I X, O X ) ob(α) xt 1 (I X, O X ) : δ V ob(α) = δ(α) α. 0 I X O V O X 0 (2.1) δ : Hom(I X, O X ) xt 1 (I X, I X ), xt 1 (I X, I X ) Hom(I X, O X ) xt 1 (I X, O X ). X Spec k[t]/(t 3 ) ob(α) = 0. ob(α) α (obstruction). Hom(I X, O X ) H 0 (N X/V ), α N X/V. X V, ob(α) H 1 (N X/V ) xt 1 (I X, O X ). X V ob(α). 2.2. X V. d X : H 0 (X, N X/V ) H 1 (X, O X ) (2.1) O X (V ) δ : H 0 (N X/V ) H 1 (O V ) X H 1 (O V ) H 1 (O X ). ob(α) d X (α) α.. H 1 (X, O X ) H 0 (X, N X/V ) H 1 (X, N X/V ) 3 1 V 3, S V. 2.2 N S/V S V 1.
3.1. N S/V v 1 ( v H 0 (N S/V ()) \ H 0 (N S/V )) S V 1 (infinitesimal deformation with pole). S V S := S \ V := V \. S S ι, [O S () ι O S ] N S/V H 0 (S, N S/V ()) H 0 (S, N S /V ). H0 (N S/V ()). v S V 1. 3.2. S ( 2 ) S det N /V O. 0 N /S N /V N S/V 0 (3.1), 1 v S V 1 Spec k[t]/(t 3 ). v ob(v) H 1 (S, N S /V ). 2.1 ( 2 ) S < 0 H 1 (N S /V ) H 1 (S, N S/V ) H 1 (S, N S/V ()) H 1 (S, N S/V (2)) H 1 (S, N S /V ). H 1 (O S ) H 1 (S, O S ) H 1 (S, O S ()) H 1 (S, O S (2)) H 1 (S, O S ). 2.2 d X (v) H 1 (O X ) X = S d S (v) 2, H 1 (O S (2)) H 1 (O S ). v H 0 (N S/V ()), H 1 (O S ) H 0 (N S /V ) H 1 (N S /V ) H 1 (O S (2)) H 0 (N S/V ()) H 1 (N S/V (3)) ob(v) = d S (v) v H 1 (N S /V ) H1 (N S/V (3)). ob(v) 0 ob(v) ob(v) H 1 (N S/V (3) ).. 3.3 ([7, Proposition 2.4 (2)]). v H 0 (N S/V ()) d S (v) H 1 (O (2)) d S (v) H 1 (O S (2)). H 1 (O (2)) d S (v) = (v ). : H 0 (N S/V () ) H 1 (N /S ()) H 1 (O (2)) (3.1) O S ().
v v H 0 (N S/V () ). N S/V () det N /V. (3.1) (v ) 0. 3.3 H 1 (N S/V (3) ) ob(v) = d S (v) v = (v ) v 0. 3.4. V 3 P 4 cubic 3-fold, S 3 V 3, S 3 ( P 1 ( 2 ) S = 1). V 3 good line ( N /V3 O 2 P 1 ), (3.1) 0 O P 1( 1) O P 1 2 O P 1(1) 0. v H 0 (N S/V ()) \ H 0 (N S/V ) S 3 V 3 1 Spec k[t]/(t 3 ). 1.3, 3.2. 1,..., m S,. ( ) ob : H 0 (N S/V ()) H 1 (N S/V (3)), ob ob : H 0 (N S/V ()) / H 0 (N S/V ) H 1 (N S/V (3) ), v (mod H 0 (N S/V )) ob(v).. 3.5. S H 1 (N S/V ) = 0. i (i 2 ) S < 0 det N i /V O i, 0 N i /S N i /V N i S/V 0 (3.2), ob. V del Pezzo. 3.6. V 3 del Pezzo, V H S. 1,..., m S, V good line. ob.
4 V k 3, S V, C S. 4.1. C (stably degenerate) C V ( ) C V, S S V, C S. C V, S C C. 1.1 C stably degenerate. 4.1 Hilbert-flag scheme Kleppe [6] Hilbert-flag scheme. Hilbert-flag scheme incidence scheme. [6] 2. C S V Hilbert p q. k T, V k T { CT S T V k T CT S T T Hilbert p,q }. Hilbert-flag scheme, Flag p,q V. Flag p,q V p, q ( ) Flag V. Flag V k- V C S (C, S ), C S V. Flag V V Hilbert scheme Hilb sc V ( ) pr 1 : Flag V Hilb sc V, (C, S ) C (4.1). pr 1 [C] Hilb sc V, C stably degenerate, 1.1. pr 1 (C, S) (tangential map) κ C,S : T Flag V,(C,S) T Hilb V,C = H 0 (C, N C/V ) (4.2). 4.2 (cf. [6], 2). H 1 (C, N C/S ) = H 1 (S, N S/V ) = 0. : (1) Flag V (C, S). (2) coker κ C,S coker ρ ker κ C,S ker ρ. ρ (1.1).
4.2 2.2 N C/V α C V 1 C α, ob(α) H 1 (N C/V ). N C/V π S NS/V C. H i (π S ) : H i (S, N C/V ) H i (C, N S/V C ) (i = 0, 1) 4.3. H i (π S ) (i = 0, 1) α ob(α) (exterior component), π S (α) ob S (α). 4.4. C V 1 C α π S (α) ob S (α) C S 1 C α. 4.3 1.3 V del Pezzo 3-fold, S H V del Pezzo, C S. (adjunction formula) N S/V K S N C/S K S C + K C, H 1 (N S/V ) = H 1 (N C/S ) = 0. 4.2(1) Hilbert-flag scheme Flag V (C, S). (C, S) Flag V W C,S. (4.1) pr 1 W C,S pr 1, (C, S) (4.2). κ C,S : T WC,S,(C,S) H 0 (C, N C/V ) χ(v, I C (S)) 1. α N C/V. α κ C,S W C,S α C 1 C α W C,S (C, S ). κ C,S, pr 1 : W C,S Hilb sc V [C] Hilb sc V, C stably degenerate. κ C,S. 4.5. κ C,S. C S l V good line, α H 0 (N C/V ) \ im κ C,S ob(α). α π S (α) H 0 (N S/V C ) ( 4.2 ). 4.2(2) π S (α) H 0 (N S/V ) H 0 (N C S/V ). S := l lines l s.t. l C=
. χ(v, I C (S)) 1 H 1 (N S/V ( C)). H 0 (S, N S/V ()) H 0 (C, N C S/V ). H 0 (N S/V ()) v v C = π S (α). v 3 S V 1. ob(α) ob S (α) ob S (α) = ob(v) C. good line. 3.6 ob(v) H 1 (N S/V (3)). H 1 (N S/V (3)) H 1 (N C S/V ) ( ), ob S (α). ob(α) 0. C 1, W C,S. C stably degenerate. 1.3. 4.6. 1.3 4.5. C V C l V N l/v. C C V,. C l S. S l C. 4.5 1.3. W C,S pr 1 : W C,S Hilb sc V. W C,S Hilb sc V. 4.7. C S V 1.3. W C,S (Hilb sc V ) red. Hilb sc V H 1 (V, I C (S)) = 0 W C,S, H 1 (V, I C (S)) 0 W C,S (generically non-reduced). 4.5 W C,S Hilb sc V. W C,S Hilb sc V Zariski dim W C,S h 1 (V, I C (S)). H 1 (V, I C (S)) = 0, Hilb sc V [C] W C,S. 4.7 P 3 Hilbert scheme Hilb sc P 3 (Kleppe- llia ) del Pezzo 3-fold. π S (α) N S/V 1 N S/V..
4.8 (Kleppe, llia). W [C] W 3 Hilb sc P 3,. χ(p 3, I C (3)) 1 C W Hilb sc P 3. Hilb sc P 3 H 1 (P 3, I C (3)) = 0 W, H 1 (P 3, I C (3)) 0 W. 5 1.3 4.7. 5.1. 3 del Pezzo V Hilbert scheme Hilb sc V. 8 V P 3 (cf. 1), Mumford [8] V 7. H n V. n 7. Iskovskih [5] V good line l. l del Pezzo S n H. S n Λ := 2K Sn + 2l. S n l S n+1, S n+1 n + 1 del Pezzo. Λ S n+1 2K Sn+1 S n,. Bertini Λ C, d = 2n + 2, g = n + 2. g = d n. ( 2K Sn + 2l) l = 2 2 = 0, C l. l S. 4.7 W C,Sn (Hilb sc V ). 1.4 H 1 (V, I C (S n )) 0. 4.7 Hilb sc V W C,Sn. 5.2. (1) C C S n del Pezzo S n+1 C (i.e. K C O C (1) = K Sn+1 C ) C p S n+1 \ C. (2) W C,Sn d + g + n = 4n + 4. (3) W C,Sn Hilb sc V h 0 (N C/V ) = 4n + 5., C 0 N C/Sn N C/V N Sn /V C 0, W V V Hilb sc P 3 V 3.
, N C/Sn O C (2K C ) N Sn /V C O C (K C ).. h 0 (N C/V ) = h 0 (2K C ) + h 0 (K C ) = (3n + 3) + (n + 2) = 4n + 5,.,. [1] P. llia: D autres composantes non réduites de Hilb P 3, Math. Ann. 277(1987), 433 446. [2] T. Fujita: On the structure of polarized manifolds with total deficiency one. I, J. Math. Soc. Japan 32(1980), 709 725. [3] T. Fujita: On the structure of polarized manifolds with total deficiency one. II, J. Math. Soc. Japan 33(1981), 415 434. [4] V.A. Iskovskih: Fano 3-folds. I, Math. USSR-Izvstija 11(1977), no. 3, 485 527 (nglish translation). [5] V.A. Iskovskih: Anticanonical models of three-dimensional algebraic varieties, Current problems in mathematics, J. Soviet Math. 13(1980), 745 814 (nglish translation). [6] J. O. Kleppe: Non-reduced components of the Hilbert scheme of smooth space curves in Space curves (eds. F. Ghione, C. Peskine and. Sernesi), Lecture Notes in Math. 1266, Springer-Verlag, 1987, pp.181 207. [7] S. Mukai and H. Nasu: Obstruction to deforming curves on a 3-fold, I: A generalization of Mumford s example and an application to Hom schemes, preprint math.ag/0609284 (2006). [8] D. Mumford: Further pathologies in algebraic geometry, Amer. J. Math. 84(1962), 642 648.
[9] H. Nasu: Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. Res. Inst. Math. Sci. 42(2006), 117 141 (see also math.ag/0505413). [10] H. Nasu: Obstruction to deforming curves on a 3-fold, II: Deformations of degenerate curves on a del Pezzo 3-fold, preprint math.ag/0609286 (2006). 606-8502 e-mail: nasu@kurims.kyoto-u.ac.jp