Subcanonical, Gorenstein and Complete Intersection Curves on Del Pezzo Surfaces

Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XLVII, 231-241 (2001)

Subcanonical, Gorenstein and Complete Intersection Curves on Del Pezzo Surfaces. ALBERTO DOLCETTI (*) To Mario Fiorentini

SUNTO - In questa nota si classificano le curve liscie connesse, che sono sottocanoniche, Gorenstein o intersezioni complete, tracciate sulle superfici di del Pezzo, esibendone le classi nei gruppi di Picard delle superfici stesse.

ABSTRACT - In this note we classify subcanonical, Gorenstein and complete intersection smooth connected curves lying on del Pezzo surfaces, by showing their classes in Picard groups of the surfaces.

Introduction. In the context of subvarieties of P N let us consider the following implications: complete intersection ¨ Gorenstein ¨ subcanonical Aim of this note is to study such implications in case of smooth connected curves lying on a del Pezzo surface S . Precisely we classify subcanonical curves in Theorem 10, Gorenstein and complete intersection curves in Theorem 14. As in [CDE] and in [DE] the existence of subcanonical curves is

(*) Indirizzo dell’autore: Dipartimento di Matematica «U. Dini», viale Morgagni 67/A, 50134 - Firenze (Italy). E-mail: dolcettiHmath.unifi.it

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chiefly translated in terms of the existence of fixed divisors on S , i.e. effective non-zero divisors Y such that h 0 (OS (Y) ) 4 1 (Proposition 3 and Corollary 4). k

Fixed divisors on S are of the form

! b L , where, for all 1 G j G k , b D 0 and

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j

j

j

Lj are mutually skew lines (Lemma 7). We get Gorenstein curves, by checking which subcanonical curves on S are arithmetically Cohen–Macaulay and in each case we decide which curves are complete intersections. Finally, by isomorphic projections, we get subcanonical curves in P 3 and therefore, by Hartshorne–Serre correspondence, rank 2 vector bundles on P 3 , among which we obtain some stable bundles (Proposition 16). Many results depend only on the fact that del Pezzo surfaces are Gorenstein and sometimes only on the property of being subcanonical or arithmetically Cohen–Macaulay (see for instance 3 and 12 (b)): so some generalizations to wider classes of surfaces could be possible. I like to dedicate this note to Mario Fiorentini, who, as my teacher in commutative algebra and algebraic geometry, has drawn my attention to chains of implications like the above one (see for instance [F]). He is not only able to transmit scientific knowledge, but also enthusiasm and love for doing mathematics (as well as for freedom and justice), by means of his way of working (and his life). We work over an algebraically closed field of characteristic 0, this allows a free use of Kodaira vanishing Theorem (see for instance [H1; III, 7.15]).

0. – Some definitions. Let V be a smooth connected subvariety of codimension c in P N . V is said to be l complete intersection: if its homogeneous saturated ideal 5t  Z H 0 (IV (t) ) is generated by c polynomials F1 , R , Fc . In this case we say that V is complete intersection of type (d1 , R , dc ) with di 4 deg (Fi ); l arithmetically Cohen-Macaulay (aCM for short): if h i (IV (t) ) 4 0 for any t  Z and 1 G i G N 2 c 4 dim (V); l g-subcanonical: if v V C OV (g) for some g  Z; l Gorenstein: if it is both aCM and subcanonical. If V is complete intersection of type (d1 , R , dc ), then it is Gorenstein (in

g ! d 2 N 2 1h see for instance [H1]; ex. II, 8.4), while the c

particular v V C OV

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i

converse is also true if V has codimension 2 (G. Gherardelli theorem).

SUBCANONICAL, GORENSTEIN AND COMPLETE INTERSECTION CURVES ETC.

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1. – Del Pezzo surfaces. A del Pezzo surface is a smooth connected (21 )-subcanonical surface S of degree n in P n . The list of all del Pezzo surfaces is the following: l S 4 Sn % P n , 3 G n G 9: a surface of degree n , obtained by blowing up r 4 9 2 n general points of P 2 and embedded in P n by means of the complete linear system corresponding to plane cubics through such points l S 8 4 S88 % P 8 : a surface of degree 8 obtained by the 2-tuple embedding of a smooth quadric surface in P 3 . Such surfaces are well-known (see for instance [D], [H1], [M]). The canonical divisor KS is linearly equivalent to 2H , where H 4 HS is the hyperplane section divisor; a system of generators of Pic (S) is given by the proper transform L of a line in P 2 and by the opposites of the exceptional lines 2Tj , 1 G j G r; the intersection form is determined by the relations: L 2 4 1 , L Q Tj 4 0 , Tj2 4 21 . Equivalently it is possible to choose H instead of L: we r

have H A 3 L 2

! T , H 4 n , H Q T 4 1 and H Q L 4 3 .

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i

2

j

The image of every line on a smooth quadric surface in P 3 through the 2tuple embedding is a smooth plane conic, hence Pic(S 8 ) is generated by two smooth plane conics Z1 and Z2 with Zj2 4 0 and Z1 Q Z2 4 1 . Equivalently Pic (S 8 ) is generated by H 4 HS 8 and by Z1 (or Z2). We have H A 2 Z1 1 2 Z2 , H 2 4 8 , H Q Zj 4 2 . Any del Pezzo surface X is aCM, hence Gorenstein (see [D; p. 63]). About the existence of smooth connected curves on del Pezzo surfaces (see for instance [GP], [Ha3], [R]), we only recall the following r

2. LEMMA. Let C A aL 2

! b T lying on S 4 S , n G 8 . If b F R F b F 0 and

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i

i

n

1

r

a F b1 1 R 1 bh , with h 4 min ] 3 , r(, then NCN contains a smooth connected curve, with the exception C A a(L 2 T1 ), a F 2 . PROOF. If r 4 6 , then S is a smooth cubic surface in P 3 and the result follows from [H1]; V, 4.12 and ex. 4.8 (see also [H3]; p. 303). Otherwise let W : S3 K Sn be the blowing up morphism of 6 2 r general points of Sn outside of the support of C and set C 8 4 W * (C). Note that C and C 8 are isomorphic by means of W , and it suffices to prove that NC 8 N contains a smooth connected curve, indeed if D A C 8 and E is the exceptional line over one of the new blown-up points, then D Q E 4 C 8 Q E 4 0 . Now the class of C 8 in S3 is aL 2 6

2

! b T , with b 4 0 for r 1 1 G i G 6 and we conclude with the case of smooth

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i

i

cubic surface.

i

r

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ALBERTO DOLCETTI

The following one is a key-result for the classification of subcanonical curves lying on subcanonical surfaces. Previous versions on smooth surfaces in P 3 are in [CDE; 1.2] and in [DE; 3.1], while integral surfaces in P 3 are considered in [EH; 1.8]; in any case the main point is that such surfaces are subcanonical. 3. PROPOSITION. Let C be a smooth connected curve of degree deg (C) on a smooth connected b-subcanonical surface X % P N and let a D b be an integer. Then the following statements are equivalent: i) C is a-subcanonical ii) h 0 (OX (C 2 (a 2 b) H) ) 4 1 and C 2 4 (a 2 b) deg (C). PROOF. We start with some consequences of adjunction formula of C on X . Since X is b-subcanonical, posed g(C) to be the the genus of C , we have: (1)

C 2 4 2 g(C) 2 2 2 KX Q C 4 2 g(C) 2 2 2 bH Q C 4 2 g(C) 2 2 2 b deg (C)

Again by adjunction we have: OC (C) C v C (2b), hence, by tensorizing by OX (b 2 a) the exact sequence 0 K OX K OX (C) K OC (C) K 0 becomes 0 K OX (b 2 a) K OX (C 2 (a 2 b) H) K v C (2a) K 0 Since a D b , we have h 0 (OX (b 2 a) ) 4 0 and, by Kodaira, h 1 (OX (b 2 a) ) 4 40 , hence h 0 (OX (C 2 (a 2 b) H) ) 4 h 0 (v X (2a) )

(2)

Suppose that C is a-subcanonical: v C C OC (a). Hence 2 g(C) 2 2 4 4a deg (C), so, by (1), we have: C 2 4 (a 2 b) deg (C); moreover h 0 (v C (2a) ) 4 4h 0 (OC ) 4 1 , hence h 0 (OX (C 2 (a 2 b) H) ) 4 1 by (2). Now assume conditions (ii). From (2) we have h 0 (v C (2a) ) 4 1; furthermore, remembering (1), deg (v C (2a) ) 4 2 g(C) 2 2 2 a deg (C) 4 4 C 2 1 b deg (C) 2 a deg (C) 4 (a 2 b) deg (C) 1 (b 2 a) deg (C) 4 0 . Hence v C (2a) C OC . r 4. COROLLARY. Let C be as in Proposition 3 on a del Pezzo surface S and a F 0 be an integer. Then C is a-subcanonical if and only if h 0 (OS (C 2 (a 1 1 ) H) ) 4 41 and C 2 4 (a 1 1 ) deg (C). PROOF.

S is (-1)-subcanonical.

r

SUBCANONICAL, GORENSTEIN AND COMPLETE INTERSECTION CURVES ETC.

235

5. REMARKS. i) The smooth connected a-subcanonical curves with a E 0 , on a del Pezzo surface S are not classified by the previous result but, from the relation 2 g(C) 2 2 4 a deg (C), it is easy to check that either a 4 22 , deg (C) 4 1 , g(C) 4 0 or a 4 21 , deg(C) 4 2 , g(C) 4 0 We recall that Sn contains lines (e.g. L1) or smooth plane conics (e.g. C A L 2 2T1) if and only if n G 8 and that S88 contains smooth plane conics, but not lines. Indeed S9 is the 3-tuple embedding of P 2 , hence curves on it have degrees f 0 (mod 3). Analogously curves on S88 have degrees f 0 (mod 2). ii) Look at the condition h 0 (OX (C 2 (a 2 b) H) ) 4 1 of Proposition 3 (ii): it means that either C A (a 2 b) H or there exists only one effective non-zero divisor Y on X such that Y A C 2 (a 2 b) H and h 0 (OX (Y) ) 4 1 . Such a divisor is called fixed in [CDE] and in [DE] and unique in [EH]. Here we use the first terminology: 6. – Fixed divisors. An effective non-zero divisor Y on a smooth connected projective variety X is is said to be fixed, if h 0 (OX (Y) ) 4 1 . If Y is fixed, then h 0 (OX (Y 2 H) ) 4 0 . Indeed, if not, Y 2 H A D for some (possibly zero) effective divisor D . Hence Y A D 1 H , which is not fixed. Next we classify the fixed divisors on del Pezzo surfaces. 7. LEMMA. a) Let Y c 0 be an effective divisor on S 4 Sn . Then Y is fixed if and k

only if Y 4

! b L , where, for all 1 G j G k , b D 0 and L

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j

j

j

j

are lines such that

Li Q Lj 4 0 for i c j . b) S 4 S88 does not contain any fixed divisor. k

PROOF..

a) Let Y 4

! b L be as above, we want to prove that Y is fixed.

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j

j

First of all Lj is fixed: this follows from the exact sequence: 0 K OS K KOS (Lj ) K OLj (Lj ) K 0 : indeed h 0 (OLj (Lj ) ) 4 0 , since Lj2 4 21 . h

To conclude it suffices to prove that if Y 8 4

! b 8 L , where b 8 D 0 and L

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j

j

j

j

are lines such that Li Q Lj 4 0 for i c j , is fixed and T is a line on S , such that either T Q Lj 4 0 , 1 G j G h or T 4 Li for some i , than Y 8 1 T is fixed.

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ALBERTO DOLCETTI

This follows from the sequence 0KOS (Y 8)KOS (Y 81T)KOT (Y 81T)K0: like before h 0 (OT (Y 8 1 T) ) 4 0 , since T Q (Y 8 1 T) E 0 . Vice versa let Y c 0 be a fixed effective divisor on S and let T be an integral components of Y . We prove that T is a line. From the sequence 0 K OS (2T) K OS K OT K 0 , we get h 1 (OS (2T) ) 4 0 , so, by Serre duality, h 1 (OS (T 2 H) ) 4 0 . Now T is fixed because of Y , so from 0 K OS (T 2 H) K OS (T) K OH (T) K 0 we have 1 4 h 0 (OS (T) ) 4 h 0 (OH (T) ) 4 H Q T 1 h 1 (OH (T) ) F H Q T 4 deg (T) F 1 hence deg (T) 4 1: T is a line. To conclude it suffices to prove that it T , T 8 are distinct integral components of Y , then T Q T 8 4 0 . Suppose that T Q T 8 c 0 . Since T , T 8 are lines, we have T Q T 8 4 1 , OT 8 (T 8 ) C COT 8 (21 ) and OT (T 1 T 8 ) C OT . Hence from the exact sequence 0 K OS K KOS (T 8 ) K OT 8 (21 ) K 0 , we get h 1 (OS (T 8 ) ) 4 0 , therefore from the exact sequence 0 K OS (T 8 ) K OS (T 1 T 8 ) K OT K 0 , we obtain h 0 (OS (T 1 T 8 ) ) 4 2 , but T 1 T 8 is fixed, because of Y . b) It suffices to prove that no integral divisor Y on S 8 is fixed. Since Z1 and Z2 are not fixed, we can suppose that Y is not linearly equivalent to Z1 or Z2 and write Y A aZ1 1 bZ2 . Since Y Q Zj F 0 , we get a , b F 0 , but (a , b) c ( 0 , 0 ), hence Y is not fixed. r 8. REMARKS. i) A consequence of the 7 (a) is that every Sn contains some fixed divisors with the exception of n 4 9 . Indeed S9 does not contains lines. ii) The Lemma above does not imply that, fixed a basis (H ; 2T1 , R , 2 k

2T9 2 n ) of Pic(S) all fixed divisors are of the form

! b T , but it implies that

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j

j

every fixed divisor Y has support consisting of a set of skew lines, which can be extended to such a basis of Pic (S). 9. PROPOSITION. Let L1 , R , Lk , 1 G k G r be k mutally skew lines on S 4 Sn , k

n G 8 and let a F 0 be an integer. Then N(a 1 1 ) H 1 smooth connected a-subcanonical curves. r

! (a 1 1 ) L N contains j

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r

! L we have C A 3(a 1 1 ) L 2 ! (a 1 1 ) L 1 1 ! (a 1 1 ) L 4 3(a 1 1 ) L 2 ! (a 1 1 ) L . We conclude that C is smooth PROOF. k

j41

Since H A 3 L 2 j

i41 r

i

j4k11

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j

j

connected by applying Lemma 1 (up to permutation of indeces). By Corollary 4 we conclude that C is also a-subcanonical. Indeed C 2 (a 1 1 ) H is fixed (Lemma 7), while the relation C 2 4 (a 1 1 ) deg (C) is a standard computation. r

SUBCANONICAL, GORENSTEIN AND COMPLETE INTERSECTION CURVES ETC.

237

10. THEOREM. Let C be a divisor on a del Pezzo surface S . Then C is a smooth connected a-subcanonical curve if and only if one of the following assertions holds: (a) S 4 Sn , n G 8 , a 4 22 and C is a line (b) S c S9 , a 4 21 and C is a smooth plane conic (c) a F 0 and C A (a 1 1 ) H (d) S 4 Sn , n G 8 , a F 0 and there are k F 1 mutually skew lines on S: k

L1 , R , Lk such that C A (a 1 1 ) H 1

! (a 1 1 ) L

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j

PROOF. Part «if». Of course (a) and (b) are smooth connected a-subcanonical. For (c) we remark that h 0 (OS (C 2 (a 1 1 ) H) ) 4 1 and C 2 4 (a 1 1 )2 n 4 4(a11) deg (C) and conclude with Corollary 4. Finally (d) is Proposition 9. Part «only if». If a E 0 we are in cases (a) or (b) (see 5 (i)). Suppose now a F 0 . From Corollary 4 we have C A (a 1 1 ) H 1 Y with Y 4 0 or Y fixed. If k

Y 4 0 we are in case (c). If Y c 0 , then S 4 Sn , n G 8 and Y 4

! b L , where,

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j

j

for any 1 G j G k , b j D 0 and Lj are lines such that Li Q Lj 4 0 for i c j (Lemma 7). Since C is not a line, then 0GC Q Lj4a112b j, hence b j G a 1 1 . From k

Corollary 4 we must have: C 2 4 (a 1 1 ) deg (C), which gives 2b j ) 4 0 , so b j 4 a 1 1 for any j . r

! b (a 1 1 2

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j

11. REMARK. With the same notations as in 1 we can reformulate 10 (c) and 10 (d) for S 4 Sn in the following way: r

(c8) a F 0 and C A 3(a 1 1 ) L 2 the form (L ; 2T1 , R , 2Tr )

! (a 1 1 ) T for every base of Pic (S) of j

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(d’) n G 8 , a F 0 and there is a basis Pic (S) of the form (L ; 2T1 , R , r2k

2Tr ) such that C A 3(a 1 1 ) L 2

! (a 1 1 ) T

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j

for some k F 1 .

12. – An exact sequence. a) Let D c 0 be an effective divisor on a smooth connected surface X % P N . D can be viewed as a subscheme of both X and P N , hence the ideal sheaf ID , X of D in X is OX (2D) and we have the exact sequence: (3)

0 K IX (t) K ID (t) K OX (tH 2 D) K 0

b) Let D and X be as before and suppose that X is aCM. From (3) we get h 1 (ID (t) ) 4 h 1 (OX (tH 2 D) ) for any t  Z . Moreover if D A kH 1 D 8 , where

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ALBERTO DOLCETTI

D c 0 is an effective divisor on X , from the analogous sequence for D 8 we deduce: H 1 (ID (t) ) C H 1 (ID 8 (t 2 k) ), i.e. the Hartshorne-Rao modules of D and D 8 are isomorphic (up to shift). 13. REMARK. Let C be a smooth connected curve on a del Pezzo surface S , n(n 2 3 )

then there are at least distinct integral quadric hyperfurfaces con2 taining both S and C . Remember the sequence (3) for C and S: for t 4 2 we get the injection 0 K KH 0 (IS ( 2 ) ) K H 0 (IC ( 2 ) ). Note that no hyperquadric, containing S , is the union of two hyperplanes, since S in non-degenerate. From the exact sequence: 0 K KIS ( 2 ) K OP n ( 2 ) K OS ( 2 ) K 0 , we get h 0 (IS ( 2 ) ) 4

(n11 )(n12 ) 2

92n

Now suppose S 4 S 4 Sn : we have 2 H A 6 L 2

2 h 0 (OS ( 2 ) ).

! 2 L , hence, by computing

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the dimension of the complete linear system of plane sextics with 9 2 n assigned double points in general position, we get h 0 (OS ( 2 ) ) 4 3 n 1 1 . Hence h 0 (IS ( 2 ) ) 4 0

n(n 2 3 ) 2

. We conclude with the case S 4 S 8 4 S88 . We have

0

h (OS 8 ( 2 ) ) 4 h (OQ ( 4 ) ), where Q is a quadric in P 3 , hence h 0 (OQ ( 4 ) ) 4 25 , so h 0 (IS 8 ( 2 ) ) 4 20 . 14. THEOREM. Let C be a divisor on a del Pezzo surface S , then C is a smooth connected Gorenstein curve if and only if one of the following assertions holds: i) S 4 Sn , n G 8 and C is a line: complete intersection of type (1 , R , 1)  n21

ii) S c S9 and C is a plane conic: complete intersection of type ( 2 , 1 , R , 1)  n22

iii) C A kH , k F 1 . Moreover C is complete intersection if and only if S 4 S3 : type ( 3 , k) or S 4 S4 : type ( 2 , 2 , k) iv) S 4 Sn , n G 8 and C A H 1 L , where L is a line on S . Moreover C is complete intersection if and only if n 4 3: type ( 2 , 2 ). PROOF. We look for aCM curves in the list of subcanonical curves of Theorem 10. Cases (a) and (b) are complete intersection and correspond to (i) and (ii). Case (c). The exact sequence (3) above becomes: 0 K IS (t) K IC (t) K OS (t 2 a 2 1 ) K 0 .

SUBCANONICAL, GORENSTEIN AND COMPLETE INTERSECTION CURVES ETC.

239

Hence h 1 (IC (t) ) 4 h 1 (OS (t 2 a 2 1 ) ). We conclude that C is aCM by Kodaira vanishing. The same exact sequence gives: h 0 (IC (a 1 1 ) ) 4 h 0 (IS (a 1 1 ) ) 1 1 , i.e. there is a hypersurface of degree a 1 1 , containing C , but not S . Hence C is contained in a complete intesection of type ( 3 , a 1 1 ) for S 4 S3 and of type ( 2 , 2 , a 1 1 ) for S 4 S4 . C is actually one of the complete intersections above by reasons of degree. Now suppose S 4 Sn , n F 5 or S 4 S88 . From Lemma 12, C is contained in at least n 2 1 integral quadric hyperfurfaces. Hence if it would be complete intersection, it should be of type (2 , R , 2, 1 ) for a 4 0 and  n22

of type (2 , R , 2) for a F 1 . We exclude that C A H is a complete intersection  n21

of type (2 , R , 2, 1 ) because deg (H) 4 n E 2n 2 2 as soon as n F 5 . If C A  n22

A (a 1 1 ) H , a F 1 , would be complete intersection of type (2 , R , 2), we  n21

would have (a 1 1 ) n 4 2n 2 1 . On the other hand: v C C OC (n 2 3 ), so a 4 n 2 3 , but n(n 2 2 ) E 2n 2 1 as soon as n F 5 . This completes (iii). Case (d). First of all suppose a 4 0 and k 4 1 i.e. C A H 1 T , where T is a line. In this case it is easy to chech that C is a non-degenerate elliptic curve of degree n 1 1 . From 12 (ii) we get h 1 (IC (t) ) 4 h 1 (IT (t 2 1 ) ). Hence C is aCM. If n 4 3 , C is complete intersection of type ( 2 , 2 ). If n F 4 , C is not a complete intersection, because, for instance, n 1 1 4 deg (C), 4 G n G 8 cannot be the product of n 2 1 integers greather or equal to 2 . Now suppose (a , k) c ( 0 , 1 ). To conclude it suffices to prove that C is not aCM. Since S is aCM, the exact sequence (3) for C and S gives: h 1 (IC (t) ) 4 h 1 (OS (tH 2 C) ) for any t  Z . Hence it suffices to prove: h 1 (OS ( (a 1 1 ) H 2 C) c 0 . Pose D 4 C 2 (a 1 1 ) H 4 k

4

! (a 1 1 ) L . We have h (O (2D)) 4 0 , because D c 0 is effective. Further-

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0

j

2

S

0

more h (OS (2D) ) 4 h (OS (D 1 K) ) 4 h 0 (OS (D 2 H) ) 4 0 , because D is fixed (remember 6 and 7). Now if h 1 (OS (2D) ) 4 h 1 (OS ( (a 1 1 ) H 2 C) ) 4 0 , by Riemann-Roch we would have: 0 4

D2 2

2

DQH 2

11 42

which is actually negative for (a , k) c ( 0 , 1 ).

k(a 1 1 )2 2

2

k(a 1 1 ) 2

11,

r

15. – Rank 2 vector bundle on P 3 . Let Y % P 3 be an a-subcanonical curve. By Hartshorne–Serre correspondence Y is related to a rank 2 vector bundle E on P 3 with Chern classes c1 (E) 4 a 1 4 , c2 (E) 4 deg (Y), by means of the exact sequence 0 K OP 3 K E K IY (a 1 4 ) K 0

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ALBERTO DOLCETTI

E is indecomposable if and only if Y is not complete intersection if and only if Y is not arithmetically Cohen-Macaulay. Moreover if c1 (E) D 0 , then E is stable (i.e. c1 (L) G

c1 (E) 2

, for any line subbundle L % E) if and only if s(Y) D

c1 (E) 2

,

where s(Y) is the minimum degree of a surface containing Y (for this standard material see [H2]). The a-subcanonical curves C , constructed on S88 and on Sn , n F 4 , are not in P 3 , but general projections into P 3 yeld smooth connected a-subcanonical curves C 8 on integral surfaces of degree n . The related bundles are undecomposable, because the curves in P 3 are not linearly normal. k

16. PROPOSITION. Let C A (a 1 1 ) H 1

! (a 1 1 ) L , be a smooth connected

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j

curve on Sn , n F 4 as in Theorem 10 (c) with k 4 0 or (d) and let E be the rank 2 vector bundle on P 3 related to the general projection C 8 of C into P 3 . If n(n21) n1k

21GaG2n25, then E is stable with (c1 , c2)4(a14, (n1k)(a11)).

PROOF. Since deg (C 8 ) 4 (n 1 k)(a 1 1 ) and C 8 lies on an integral surfaces of degree n , the condition deg (C 8 ) F n(n 2 1 ), assuring that s(C 8 ) 4 n , is equivalent to

n(n 2 1 ) n1k

2 1 G a , while the condition

c1 (E) 2

E s(C 8 ) 4 n is

equivalent to aG2n25. For the Chern classes we conclude with 15.

r

REFERENCES

[CDE] [D]

[DE] [EH]

[F]

G. CASNATI - A. DOLCETTI - P. ELLIA, On subcanonical curves lying on smooth surfaces in P 3, Rev. Roumaine Math. Pures Appl., 40, 3-4 (1995), pp. 289-300. M. DEMAZURE, Surfaces de del Pezzo II, III, IV, V, in M. Demazure, H. Pinkham et B. Teissier (eds), Séminaire sur les Singularités des Surfaces, LNM 777, Springer-Verlag, Berlin Heidelberg New York, 1980, pp. 23-69. A. DOLCETTI - P. ELLIA, Curves on Generic Surfaces oh High Degree Through a Complete Intersection in P 3, Geometriae Dedicatae, 65 (1997), pp. 203-213. P. ELLIA - R. HARTSHORNE, Smooth specializations of space curves: questions and examples, in Freddy Van Oystaeyen, Commutative Algebra and Algebraic Geometry, proceedings of the Ferrara meeting in honor of Mario Fiorentini, lecture notes in pure and applied mathematics Vol. 206, Marcel Dekker Ink., New York Basel, 1999, pp. 53-79. M. FIORENTINI, Esempi di curve di Buchsbaum che non sono di Macaulay, Seminario di variabili complesse, Università di Bologna, 1981, 73-84.

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[GP] [H1] [H2] [H3]

[M]

[R]

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L. GRUSON - C. PESKINE, Genre des courbes de l’espace projectif, Ann. Sci. de l’Ecole Norm. Sup. (4), 15 (1982), pp. 401-418. R. HARTSHORNE, Algebraic Geometry, GTM 52, Springer-Verlag, New York, 1977, pp. xvi-496. R. HARTSHORNE, Stable vector bundles of rank 2 on P 3, Math. Ann., 238 (1978), pp. 229-280. R. HARTSHORNE, Genre des courbes algébriques dans l’espace projectif (d’après L. Gruson et C. Peskine), Séminaire Bourbaki, 34e année, 1981/82, n. 592, 301-313. Y. I. MANIN, Cubic Forms — Algebra, Geometry, Arithmetic, North-Holland Mathematical Library volume 4, Second Edition, North-Holland, Amsterdam, 1986, pp. X-326. J. RATHMANN, The Genus of Curves in P 4 and P 5, Math. Z., 202 (1989), pp. 525-543.

Pervenuto in Redazione il 20 dicembre 1999.