2-Spanned surfaces of sectional genus six

Annali di Matematica pura ed applicata (IV), Vol. CLXV (1993), pp. 197-216

2-Spanned Surfaces of Sectional Genus Six(*). ANTONIO LANTERI

- Un fibrato lineare olomorfo su una superficie algebrica proiettiva complessa non singolare S ~ detto k-spanned se le sue sezioni separano opportuni O-cicli di S di lunghezza k + 1 - - da (k + 1)-uple di punti distinti sino a getti di ordine k - - e forniscono pertanto una immersione proiettiva di S di ordine superiore. Si fornisce un contributo alla classificazione proiettiva delle superfici determinando quelle di genere sezionale 6 immerse da un fibrato 2spanned e quelle di genere sezionale 7 immerse da un fibrato 3-spanned.

Sunto.

Introduction.

The notion of k-spanned line bundle L on a complex projective smooth surface S has been introduced in [BFS] in connection with the concept of higher order embedding. Essentially L is k-spanned if F(L) separates 0-cicles of length k § 1 on S, going from (k + 1)-tuples of distinct points up to jets of order k (for the precise definition see Sect. 0). We say that S is k-spanned to mean that it is embedded via a k-spanned line bundle. Afterwards this notion has been extensively studied in [BS]. In particular, in [BS] k-spanned surfaces (k I> 2) of sectional genus g ~<5 are classified. The aim of this paper is to do the same in case g = 6. The problem looks more complicated than in the previous cases since in some instances to decide whether some admissible surfaces are in fact 2-spanned is not easy and requires non'standard arguments. Here we state the main result of this paper. For the notation we refer to Sect. 0. THEOREM 1. - Let S c P ~ be a complex smooth surface, let L = O~(1) s and assume that g(L) -- 6. The pairs (S, L) such that L is k-spanned, k I> 2, are listed in the Table I except possibly case 8, i n which however L is generically 2-spanned (see (0.6)). Points to blow-up are supposed in general position.

(*) Entrata in Redazione il 14 febbraio 1991, ricevuta versione finale il 20 febbraio 1993. Indirizzo dell'A.: Dipartimento di Matematica ~,F. Enriques~ delrUniversit~, Via C. Saldini 50, 1-20133 Milano, Italy.

198

ANTONIOLANTERI: 2-spanned surfaces of sectional genus s i x TABLE I.

Case

k

S

L

1

5

p2

~ (5)

2

2

elliptic PLbundle with e = - 1

[3~ + f ]

3

3

F0

[3z + 4f]

4

2

elliptic pl-bundle with e = - 1, or 1

[2z + (e + 5) f ]

5

2

Fe, e~<5

[2~+(e+7) f ]

6

2

Del Pezzo surface of degree 5

Ks-2

7

2

Bpl..... p6(Fe), e ~<1

r:*[4a+(2e+5)f]|

+E6] -2

8

2

Bpl .....

pg(Fe), e <~2

zz*[4~r+(2e+6)f]|

... +Eg] -2

9

2

Bp(S'), S' Del Pezzo surface of degree 2

Ks-2 @ 7~rKS~ 1

In particular it turns out there are no 2-spanned surfaces with sectional genus 6 of nonnegative Kodaira dimension. I recall that 2-spanned surfaces are exactly those embedded by a 2-very ample line bundle [BS1]. So Theorem 1 also provides the list of 2-very ample surfaces of sectional genus 6. Here is a sketch of the proof. Since 2-spannedness implies that h~ is large enough, from the Castelnuovo bound for the genus of a space curve we get the inequality d 1> 2g - 2. So, apart from a special case we deal with separately, we have that X is ruled and then adjunction provides a useful tool. Actually, let H = Ks | L; after checking some well known exceptions, we can assume that H is a very ample line bundle. Then we look at the surface S embedded by [HI. Since h ~ = g - q, there are some possibilities according to the values of q, the irregularity of S. The most intricated case is that of rational surfaces in the projective 5-space. However in this case it turns out that the sectional genus g(H) is very low, due to the 2-spannedness Of L. This allows us to keep to a minimum the use of the classification results of surfaces with small sectional genus in finding pairs (X, H), which are candidates for L = H | Ks-1 being 2-spanned, Then the result follows by combining some classification results of LIVORNI[Lil], [Li2], with basic properties of k-spannedness [BS] and some (,ad hoc, 2-spannedness results proven for some special surfaces. I would like to contrast this procedure with the hard job of checking the very long list of surfaces of sectional genus 6 (e.g. see [Li2]). As a consequence of Theorem 1, we get the following

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

199

COROLLARY. - Let S r be a complex smooth surface of degree d ~< 10, and assume that L = Op~ (1)s is 2-spanned. Then (S, L) is one of the pairs listed in the table II.

TABLE

d

g(L)

4

0

8

9

h o(L)

II.

S

L

6

p2

O~ (2)

1

9

F0

[2~ + 2f]

3

7

Del Pezzo surface of degree 2

Ks-2

5

6

general complete intersection of type (2, 2, 2)

1

10

p2

~2 (3)

The proof is very simple. Actually, apart from the first and the fourth cases in the table above, we can assume that n >I 6 (see (0.4)). So, by the Castelnuovo inequality [GH, p. 252]

g(L) <~[(d - 2)/(n - 2)](d - n + 1 - [(d - n)/(n - 2)]. (n - 2)/2) where [ ] is the greatest integer function, we get the following bound for the sectional genus: g(L) < 6. Since all surfaces in Theorem 1 have degree d/> 12, the assertion follows by checking the list of 2-spanned surfaces of sectional genus ~< 5 in [BS]. The same procedure proving Theorem 1 works also for classifying 3-spanned surfaces of sectional genus 7 and gives THEOREM 2. - There are only two pairs (S, L) where L is a 3-spanned line bundle with g(L) = 7 on a complex projective smooth surface S; they are 1) (F1, [3~ + 6f]) and 2) S = Del Pezzo surface of degree 2, L = Ks-3. In the same way as the above Corollary, it thus follows that (p2, ~p~ (3)) is the unique 3-spanned surface of degree ~< 12. The paper is organized as follows. In Sect. 0 some background material is collected and the k-spannedness of some surfaces occurring in Theorem 1 is proven. Sections 1 and 2 are devoted to the proof of Theorem 1, while Section 3 contains the proof of Theorem 2. I would like to thank M. BELTRAMETTI for many helpful discussions about k-spannedness.

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

200

0. - B a c k g r o u n d material and 2-spannedness results.

The word surface will always mean complex projective smooth surface; line bundles are holomorphic line bundles. For a line bundle L on a surface we write L ~ instead of L | and L . L to indicate the self-intersection index. All notation and definitions not specified are the standard ones in algebraic geometry. Let X be a surface: ruled means birationaily ruled. If X is a PLbundle over a smooth curve, then ~ and f will denote a fundamental section (i.e. a section attaining the minimal self-intersection) and a fibre of X, and e = - o. z will stand for the invariant [Ha, p. 372]. In particular F~ = (t~l @ Gp1( - e)). Let L be an ample line bundle on X. I recall the names of some classes of pairs (X, L) frequently occurring in the paper; (X, L) is a scroll if X is a P~-bundle over a smooth curve and L .f= 1 for every fibre f of X; assume that (X, L) is not a scroll: then (X, L) is a conic bundle if X is ruled and L . F = 2 for every fibre F of the ruling. (X, L) is a Del Pezzo pair if X is a Del Pezzo surface (i.e. Kx 1 is ample) and L = K:~ 1. By the way recall that Kx'Kx is said the degree of the Del Pezzo surface X. The symbol Bpl ..... p, (X) will denote the surface obtained by blowing-up X at the points Pl, ..., P~. When not specified, = will stand for the blowing-up and E~ for the exceptional curve =-1 (Pi), i = 1, ..., s. In order to insure the ampleness or the very ampleness of some line bundles we consider on the blown-up surface, the general position assumption on the points p~,...p~ is always implicit. A pair (X, L), where X = Bp~..... p~(p2), L = ='Or2 (4) | [El + ... + E~]- 1 and Pl, ..., P~ (s ~< 10) are in general position is said a Bordiga surface. I briefly recall the notion of k-spannedness [BFS], [BS]. Let X be a surface. A 0-cycle on X is called admissible if the Corresponding 0-dimensional subscheme is locally supported on smooth curves of X. A line bundle L on X is said to be k-spanned (k I> 0) if for any admissible 0-cycle ~ of length k + 1 on X, the restriction homomorphism (0.0.1)

F(L) --, F(L | G~)

is surjective. Recall that 0-spanned is equivalent to L being spanned by global sections and 1-spanned is equivalent to being very ample. I also recall that for 2-spanned line bundles (0.0.1) is a surjection for any 0-dimensional subscheme of length 3 regardless any assumption on the locally supporting curves [BFS, (3.1)]. This can be rephrased by saying that 2-spanned line bundles are the same as 2-very ample line bundles (see [BS1]). For the basic properties of k-spannedness I refer to [BS, Sects. 0, 1]. Here I simply recall some facts including a slight improvement with respect to [BS]. Let X be a surface. (0.1) Let L~ be a ki-spanned line bundle on X, i = 1, .... n. Then L1 | ... | L~ is (kl + ... + k.)-spanned. Let L be a line bundle ~on X.

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

201

(0.2) If L is k-spanned (k I> 2), then L . C 1> k for every smooth rational curve CoX. (0.3) LEMMA. - If L is 2-spanned then L . C ~>4 for every effective divisor C of X

with p~(C) = 1. PROOF. - If C is reducible the assertion follows from (0.2) for k = 2. So we can assume that C is an irreducible plane cubic of (X, L). But then taking a 0-cycle Z cut out on C by a line we see that (0.0.1) is not a surjection, contradicting the 2-spannedness of L. 9 Note that for C a smooth curve, (0.3) is a consequence of[BS, (0.5.2) and (1.4.1)]. The following result proven by BALLmO[Bal], [Ba2] is very important in the sequel. (0.4) If L is k-spanned (k I> 2), then h ~(L) I> k + 5 unless k = 2 and X embedded by ILl is either the Veronese surface or a general complete intersection of three quadrics of p s . k-spannedness results for some special Pezzo surfaces can be found in [BS, Sect. will refer to them throughout the paper, which we need for cases 7, 9 and 8 in the (0.7)].

classes of surfaces like pl-bundles and Del 3 and (0.8)] (see also [BFS, (2.6)]) and we Here we prove two 2-spannedness results Theorem. The first one is inspired by [BS,

(0.5) PROPOSITION. - Let X be one of the following surfaces polarized by the corresponding very ample line bundle H:

i) Bpl ..... p~(Fe), e ~< 1, H = r~*[2z + (e + 3 ) f ] | [ E 1 + ... + E6] - 1 , ii) Bp (Y), where Y is the Del Pezzo surface of degree Ky. Ky = 2 (p a general point), H = rc*Ky 2 | [E] -1 , where, in both cases 7: denotes the blowing up and E = r~-1 (p), Ei = =-1 (Pi). In both cases the line bundle L = H | K~ 1 is 2-spanned. PROOF. - To prove the 2-spannedness of L we use the main result of[BFS]. Assume that L can be written as (0.5.1)

L = Kx | M ,

with M numerically effective and such that M. M/> 13. Then, if L is not 2-spanned X must contain an effective divisor D satisfying the following numerical inequalities: (0.5.2)

M . D - 3 <~D. D < (M.D)/2 < 3.

202

ANTONIO LANTERI: 2 - s p a n n e d s u r f a c e s o f s e c t i o n a l g e n u s s i x

CASE i). Note that X is a Del Pezzo surface. Actually, recalling that F1 = Bq (p2), while Bpl (F 0) = Bql ' q2(p2), we see that X is gotten by blowing up p2 at seven points in general position. Hence K x 1 is ample. Recalling that K x = r : * [ - 2 o - (e + 2 ) f ] | | + ... + E6] we have L = K x e | r:*f. Then (0.5.1) holds .with M = K x s | r:*f. Moreover M is ample, since K x 1 is so and r:*f is spanned; in addition M . M = 9Kx" K x - 6Kx" r:*f = 30.

Assume that L is not 2-spanned; then we get from the last inequality in (0.5.2) 6 > M.D

= -3Kx'D

+ r : * f . D >I - 3 K x ' D

in view of the spannedness of r:*f. Hence - K x ' D < 2 and so, the ampleness of K~/1 implies - K x ' D = 1. F r o m the first equality in (0.5.2) we have (0.5.3)

D . D >I M . D - 3 = - 3Kx" D + r~*f. D - 3 = rc*f" D >t 0 ;

furthermore, by the genus formula we see that (0.5.4)

D - D = 2pa (D) - 2 - Kx" D = 2pa (D) - 1

is odd. Putting together (0.5.3) and (0.5.4) we thus get D - D I> 1. On the other hand, since M . D ~< 5, by (0.5.2), the Hodge index theorem implies that 25 i> (M. D) ~ I> (M. M)(D-D) = 30D-D, giving D .D < 1, a contradiction. CASE ii). As K x = ~ v * K y | with

we have L = K x 2 |

1. Thus (0.5.1) holds

M = K x 8 | r:* K y 1 . Note that r:*Ky 1 is numerically effective and that K x 1 is ample, since p is general; so that M is ample; in addition M . M = 9Kx" K x + 6Kx" r:* K y + K y . K y = 23.

Assume that L is not 2-spanned; then the last inequality in (0.5.2) gives 6 > M.D

= - 3 K x ' D - r : * K y . D >I - 3 K x ' D ,

in view of the numerical effectivity of r:*K:71 . Therefore - K x ' D < 2 and then -Kx'D = 1, as before, since K x 1 is ample. The same argument as in case i) thus shows that D . D I> 1. Then, by the Hodge index theorem, 1 = (Kx" D) 2 >I (Kx" K x ) ( D " D) >I Kx" K x = 1,

so we get [D] - K x 1 (in fact [D] = K~/1, since Pic (X) is torsion free). Then, recalling

ANTONIO LANTERI: 2 - s p a n n e d s u r f a c e s o f s e c t i o n a l g e n u s s i x

203

the first inequality in (0,5.2) we have 1 = K x ' K x = D . D >t M . D

- 3 = 3Kx'D - 3 + r~*K~'.D = =*Ky1.D.

On the other hand r:* K ~ I ' D

a contradiction.

= r~* K y ' K x =

r~* Ky.7~* K y = 2,

[]

Now consider the conic bundle (X, H), where (0.6.0) X = Bpl ..... pg(F~),

e~<2 and H = r~*[2~ + (e + 4) f ] Q [EI + ... + Eg] 1.

The line bundle L =H|

I = =*[4~ + (2e + 6 ) f ] |

[El + ... + E g ] -2

is very ample [Lil, p. 169]. Unfortunately the same argument used in the proof of the above lemma does not work in this case. In fact I have not been able to prove that L is 2-spanned, but only that it is generically 2-spanned, in the sense explained below. (0.6) DEFINITION. Let 2 be a very ample line bundle on a surface S and let p e S. Consider the surface S~ = Bp (S) and let 2~ = o* 2 Q [E]- 1, where ~: S' ~ S denotes the blowing-up and E is the corresponding exceptional curve. 2 is said generically 2-spanned if for every p e S, s is ample and spanned and the morphism associated to F ( ~ ) is birational. -

(0.7) LEMMA. Let things be as in (0.6). If (S, 2) does not contain lines through p, then 2~ is ample and spanned. -

PROOF. First of all 2~ is spanned; this follows from the very ampleness of 2, in view of the bijection between 12~ I and 12 - P l. Actually, were q' e S~ a base point of 12~ I, then letting q = a(q') we would have 12 - P l = 12 - P - q l (q infinitely near to p, ff q' ~ E), contradicting the very ampleness of 2. To prove the ampleness we use the Nakai-Moishezon criterion. First of all note that 2~.2~ = 2" 2 - 1 > 0; otherwise it would be (S, 2) = (p2, Op~ (1)), but this possibility is ruled out by our assumption on (S, 2). Now let C be any irreducible curve in S~ ; then either C = E or C = ~* F - m E , F being an irreducible curve on S, with multiplicity m = F. E / > 0 at p. In the former case 2~. C = 2p" E = 1, while in the latter one we have 2~-C = 2. F - m I> 0, 2~ being numerically effective. Assume that 2~. C-- 0; then F would be an irreducible curve of degree m in (S, 2) having a point of multiplicity m. Then 0 ~
ANTONIO LANTE~I: 2-spanned surfaces of sectional genus six

204

Let things be again as in (0.6) and assume that 2 is 2-spanned. Then (S, 2) cannot contain lines, by (0.2), and so 2~ is ample and spanned for every p e S, in view of Lemma (0.7). Assume that there exists a 0-cycle Z' of length 2 on S~ such that

is not surjective and let Z denote the 0-cycle of length 3 on S induced b y Z' and p. Then, in view of the bijection between 12 - E I and 12~ - Z'I we have that for Z the surjectivity of the homomorphism (0.0.1) fails. This proves that (0.7.1)

any 2-spanned line bundle is generically 2-spanned.

(0.8) PROPOSITION. - Let (X, L) be as in (0.6.0). The line bundle L is generically

2-spanned. PROOF. Assume that L is not 2 spanned and let p be a pont in the support of a 0-cycle of length 3 on X for which the surjectivity of the homomorphism (0.0.1) fails, Consider X~ = Bp (X) and L~, as in (0.6) and for simplicity let (X', L ' ) = (X~, L~). Note that (X,L) does not contain lines, since L E 2 Pic(X); hence L' is ample and spanned in view of Lemma (0.7). Let r be the finite morphism associated to F(L') and set Z = r Since L " L ' = L .L - 1 = 11 and h ~ ') = h~ - 1 = 7 (e.g. see ILl2, table in Sect. 2]), from -

l l = degr degZ >I degr176 we conclude that degr = 1.

') - 2)

9

I conclude this section with the following Lemma, which will play a relevant role in Sects. 2, 3. (0.9) LEMMA.- Let L be a 2-spanned line bundle of genus g on a rational surface X with K x ' K x >I O. Let H = Kx | L and set d' = H . H . Then d ' ~< 2g - 6 + 16/(2g + 2), equality implying both that H and L are linearly dependent in Num (X) and L . Kx = = - 4 . In addition Kx. Kx >~ d' + 6 - 2g.

PROOF.- Since X is rational we have h 2 ( K ~ 1) = h~ theorem gives h~

= 0. So the Riemann-Roch

~) >1 z(g:~ t) = t + K x ' K x .

Therefore there exists an effective divisor D e IKxI]. Note that the arithmetic genus

ANTONIO LANTERI: 2 - s p a n n e d surfaces o f sectional g e n u s six

205

of D is 1. So by L e m m a (0.3), we have that (0.9.1)

4 <<.D . L = - K x ' L = d - (2g - 2),

where d = L . L . gives

Thus d i> 2g + 2. L e t d ' = H . H ;

then the Hodge index theorem

(2g - 2) 2 = ( H . L ) 2 >I d d ' >1 d ' ( 2 g + 2) and this proves the former inequality. Moreover, i f equality holds, then both b, for some a , b ~ Z and equality holds in (0.9.1), i . e . L . K x = - 4 . Now,

L~-H as

d ' = Kx" K x + 2(2g - 2) - d,

we get from (0.9.1) 4 <<.D . L = d - (2g - 2) = K x . K x - d ' + 2g - 2,

which gives the latter inequality.

9

1. - P r o o f o f T h e o r e m 1: first part.

We are assuming that L is a k-spanned (k t> 2) line bundle on a surface X and that g = g ( L ) = 6. By (0.4), since g = 6, we have

(1.0.1)

h ~ (L) I> 7.

L e t N = h e (L) - 1; by the Castelnuovo inequality [GH, p. 252], we have I N ~ 2 + (2(N - 2)g + 1/4) 1/2 , d >t I N ~ 2 + (2(N - 2)g)1/2,

if N - 4 is odd, if N - 4 is even,

(e.g. see [BS, (0.4)]). Hence, by (1.0.1) we get (1.0.2)

d ~> 10.

(1.1) LEMMA. - E i t h e r X is ruled, or d = 10 and X is a K3 surface embedded by ILl in P~. PROOF. F r o m (1.0.2) and the genus formula we have L . K x = 10 - d ~< 0. So, if d I> 11, no positive multiple of K x can be effective and then X is ruled in view of the Enriques criterion. On the other hand, if d = 10 and X is not ruled, then K x is numerically trivial, hence X is a minimal surface of Kodaira dimension zero. In this case L| 1 is ample and then, h i ( L ) = h i ( K x | 1 7 4 1) = 0 for i = 1,2, by the Kodaira vanishing theorem. Thus the Riemann-Roch theorem gives -

h ~ (L) = z ( L ) = X(Ox) + d / 2 = Z(Ox) + 5,

206

ANTONIO LANTERI: 2-spanned surfaces of sectional genus si~

hence, by (1.0.1) and recalling the Enriques-Kodaira classification we conclude that Z(t~x) = 2, i.e. X is a K3 surface and h~ = 7. 9 Now let H = K x | have

and put d' = H . H .

F r o m the Hodge index theorem we

(1.2.1)

100 = (2g - 2) 2 = ( L ' H ) 2 >i d d ' .

(1.2) LEMMA. - We have d' ~< 10, and if equality holds, then X is a K3 surface embedded by ILl in p6. PROOF. - The inequality d' ~< 10 comes from (1.2.1) and (1.0.2). Now assume that d' = 10; then d ~< 10, which combined with (1.02) shows that d = 10. So (1.2.1) is an equality and then by the Hodge index theorem there exist integers a, b such that L a= ( K x | b. So, in N u m ( X ) | we have (1.2.2)

Kx = L (~ - b)/b .

Note also that from 0 = d ' - d = (Kx | L). (Kx | L) - L . L we get Kx" K x = - 2Kx" L. Recalling (1.2.2), this immediately gives a = -+ b. If a = b, we have that K x is numerically trivial and then the same argument proving (1.1) shows that X is a K3 surface and h~ = 7. Now assume that a = - b; in this case K x | 2 is numerically trivial. This would mean that X is a Del Pezzo surface of index two; in other words it would be X = P ~ • and d = K x ' K x / 4 - - 2 , a contradiction. This proves the Lemma. 9 The proof will proceed with a case by case analysis. CASE A: pg > 0. - In this case it must be L . Kx >I O. By the genus formula, recalling (1.0.2) this gives d = 10. Since X cannot be ruled, L e m m a (1.I) says that X embedded by ILl is a K3 surface of degree 10 in p6. We have the following (1.3) PROPOSITION. not 2-spanned.

-

Let S r p6 be a K3 surface of degree d = 10. Then Op (1)s is

PROOF. We have pg (S) = 1 and one immediately checks that this is the maximum geometric genus for a surface of degree 10 in p6. So S is a Castelnuovo variety in the sense of Harris and it follows from [H, p. 65] that it is either -

i) a divisor inside a rational normal scroll T of P~, or ii) the complete intersection of the cone over the Veronese surface of p5 with a hypersurface not containing the vertex of the cone.

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

207

Case ii) cannot occur, since the degree 10 is not divisible by 4. In case i) we have that S - 32c - 2~, where ~ and # denote a hyperplane section and a fibre of T respectively [H, p. 56]. Therefore S is residual to two planes in the complete intersection of T with a cubic hy: persurface of p6 (case (i) in [H, p. 65]). In particular

13 c 1 = Iop2(3)1 and this shows that the fibres of T cut out on S a pencil of plane cubics, whose general element C is a smooth curve of genus 1, by Bertini's theorem. Then the assertion follows from L e m m a (0.3). " CASE B: pg -- 0, H not v e r y ample. - The exceptions to the v e r y ampleness of H for a v e r y ample line bundle L are known after [SV]. Recalling that g = 6 and that (X, L) cannot contain lines due to (0.2), such exceptions reduce to the following one (see also [BS, (4.1)]): (X, L) is a conic bundle and X is a pl-bundle. Recall that conic bundles are characterized by the condition d' = 0. Then, by the genus formula, recalling also that Kx" Kx = 8(1 - q), we immediately get (1.4.1)

d = 28 - 8q,

and taking into accouiit (1.0.2), this implies that q ~< 2. Now let a and f be a minimal section and a fibre of X and let e denote the invariant, i.e. e = - a. ~; since (X, L) is a conic bundle we have L - [2~ + bf], for some integer b and then (1.4.1) gives b = e + + 7 - 2q. As a first thing assume that q = 2. In this case z is a smooth curve of genus two and so, since L is 2-spanned it must be 5 < d e g L : = 3 - e; this, combined with Nagata's inequality e I > - q, gives e = - 2, and L N [2~ + f ] . In fact this case cannot occur. To prove it, note that L can be written as L = Kx | M, where M - [4a - 3f] is ample, by [Ha, p. 382]. Thus the Kodaira vanishing and the Riemann-Roch theorems

give h~

= h~174

M) = z(L) = 6,

contradicting (1.0.1). When q = 0 or 1, we have numerical conditions for the kspannedness of a line bundle [BS, Sect. 3]. It turns out that the line bundle [2~ + (e + + 7 - 2q) f ] is 2-spanned exactly in the following cases: q=0,

with

0~<e~<5

q=l,

with

-1~<e~<1

[BS, (3.1)]

and [BS, (3.3), (3.4)].

Moreover L cannot be 3-spanned, by [BS, (1.1.1)]. So we have

208

ANTONIO LANTERI: 2-spanned surfaces o f sectional genus six (1.4) pg -- 0 and H not v e r y ample give rise to cases 4, 5 in Theorem 1.

In view of the above we can assume f r o m now on that pg = 0 and H is v e r y ample. Note that from the Kodaira vanishing and the Riemann-Roch theorems we get (1.5.1)

h ~ (H) = 6 - q.

Moreover, by L e m m a (1.1) we know that X is a ruled surface. Since surfaces in p3 are regular, in view of (1.5.1) we have to consider only the following possibilities: i) h ~

= 5; X embedded by IHI is an elliptic ruled surface of p4,

ii) h ~

= 6; X embedded by IHI is a rational surface of p s .

We will consider the f o r m e r possibility in this section, while the latter one will be dealt with in the next section. CASE C. - H is v e r y ample and X embedded by IHI is an elliptic ruled surface of p4. As a first thing, in view of L e m m a (1.2) we can assume that d' ~< 9. Moreover we can also assume that d' i> 5, since there are no irregular surfaces of degree <~ 4. Note that H ' K x = H ' ( H | L -1) = d ' - H . L = d ' - (2g - 2) = d ' -

10.

In addition we have X(Ox)= 0; so the double point formula for surfaces in p4 [Ha, p. 434] gives (1.6.1)

d'2 _ 1 5 d ' + 50 = 2Kx" K x .

Note also that d'-

K x ' K x = (H | K x 1 ) . ( H | K x ) = L . ( K ~ |

L) = 2 L ' K x + d = 20 - d;

hence, recalling (1.0.2), we get (1.6.2)

K x . K x ~ d ' - 10.

By combining (1.6.1) with (1.6.2) we thus see that d' ~< 7. So (1.6.3)

4 ~< d ' ~< 7.

Now assume that (X,H) is not a scroll. Then, from the inequality ( K x | 9(Kx | H ) >I 0 [So, (2.1)], we get (1.6.4)

Kx" K x >! 20 - 3 d ' .

This, combined with (1.6.1), provides the inequality d' I> 9, which contradicts (1.6.3). It thus follows that (X, H) is a scroll. On the other hand there is a unique elliptic scroll in p4 (e.g. see [La]): X is the pl-bundle of invariant e = - 1 over an elliptic curve and H N [~ + 2f]. It turns out that L N [3~ + f ] , which is in fact 2-spanned, as one can

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

209

prove by using the same argument of[BS, (3.4)]. However, since d e g L : = 4, L cannot be 3-spanned, according to [BS, (1.4.1)]. So we have proved (1.6) For pg = 0, q = 1 and H very ample we get case 2 in Theorem 1.

2. - P r o o f of Theorem 1: second part. In this section we deal with the last and most intricated CASE D. - H is very ample and X embedded by IHI is a rational surface of pS. In view of (1.2) we know that d' ~< 9. Now let g' = g(H). F r o m (2.0.0)

g' = g + Kx" (Kx | L) = Kx" K x + 3g - 2 - d,

(2.0.1)

d'= (Kx| L).(Kx|

L) = K x ' K x + 4g - 4 - d,

we immediately get (2.0.2)

d' -- g' + 4.

Hence (X,/4) can only have the following characters: (d', g ' ) = (4, 0), (5, 1), (6. 2), (7, 3), (8, 4), (9, 5). Now we look at the various possibilities by using the classification of rational surfaces with small sectional genus [Lil], [Li2]. As a first thing, let (d',g') = (4, 0). In this case (X,H) is either (p2, Op2(2)) or a scroll of degree 4 in P~, i.e. (X, H ) = (F0, [z + 2f]) or (F2, [~ + 3f]). Since L = H | K / 1 , this gives

(p2, Op2 (5)), !

(Z, L) = i(F0, [3z + 4f]), [(F2, [3~ + 7f]). In the first case L is 5 spanned by (0.1). In the second case L = [a + f ] 2 | [~ + 2f], hence it is 3 spanned in view of (0.1). This gives cases 1 and 3 of Theorem 1. In the third case, note that d e g L : = 1, so L cannot be 2-spanned. Let (d', g') = (5, 1). In this case X is the Del Pezzo surface of degree 5, i.e. X = = Bpl ..... p4(p2) is the blow-up of p2 at 4 points in general position and H = K;/1 . Then L = K x e is 2-spanned by [BS, (0.8)]: this gives cases 6 of Theorem 1. Let (d',g') = (6,2). In this case X = B p l ..... p6(Fe), is the blow-up of Fe (e ~<2) at 6 points on distinct fibres and H = r:*[2a+ (e + 3 ) f ] Q [ E 1 +... +E6] -1, where 7: stands for the blowing-up and Ei = =-l(p~) [Lil]. Thus L = 7:*[4~ + (2e + 5) f ] | [El + . . . + E6] -2. L e t z' = r: -1 (~) and note that degL~, ~< 5 - 2e, hence L cannot be 2-spanned for e = 2. On

210

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

the other hand, for e = 0, i we have L = K~/2 @ =*[f] and then the 2-spannedness of L follows from Proposition (0.5), i). This gives case 7 of Theorem 1. From now on we can assume that g' i> 3. From the Hodge index theorem, recalling (2.0.2), we have (2.0.3)

(g' - 6) 2 = (2g' - 2 - d ,)2 = (H. Kx) 2 >t d' gz" Kx = (g' + 4) gx" g x .

On the other hand, by (2.0.1) we have s d ' = K x ' K x + 2 0 - d; recalling (1.0.2) and (2.0.2) this gives (2.0.4)

Kx" Kx >1g' - 6.

By combining (2.0.3), (2.0.4) we get (2.1.1)

-3<.Kx.Kx<.I

for g ' = 3 ,

(2.1.2)

-2
for g ' = 4 ,

(2.1.3)

-I<<.Kx.Kx<.O

for g ' = 5.

Now note that since g' ~> 3, (X, H) can be neither a scroll, nor a Del Pezzo pair; this implies that (Kx | H ) . (Kx @ H ) >i 0, equality holding iff (X, H) is a conic bundle [So, (2.1)]. Recalling (2.0.2) this yields (2.1.4)

K x ' K x >I 8 - 3g'

with = iff (X, H ) is a conic bundle.

So (2.1.1) can be replaced with the sharper inequality (2.1.1')

-I<<.Kx.Kx~
for g ' = 3 , w i t h = o n

the left iff (X,H) is a conic

First of all we deal with the following case: (X,H) is a conic bundle with ( d ' , g ' ) = (7,3). In this case it follows from[Lil] that X = Bpl ..... p9(Fe) is the blow-up of Fe (e ~< 3) at 9 points lying on distinct fibres and H = ~*[2o + (e + 4 ) f ] | + ... + E g ] -1, where r: stands for the blowing-up and E~ = =-1 (Pi). Moreover the 9 points must satisfy some more restrictions on their position in order H to be very ample[Li2, p. 169]. Thus L = r:*[4o+ (2e + + 6) f ] | [El + ... + Eg] -2 9Let o' = =-1 (0) ad note that degL:, ~< 6 - 2e, hence L is not even a m p l e for e = 3. For e ~< 2, the generic 2-spannedness of L follows from Proposition (0.8) and this gives case 8 of Theorem 1. Now note that due to (2.1.2), (2.1.3), (2.1.4), (X, H) cannot be a conic bundle for g ' = 4, 5. So, from now on, we can assume that (X, H) is not a conic bundle. Then from [LP, (5.1)] (see also [So, (2.6.1)]), (2.1.4)can be sharpened giving (2.2.1)

Kx" Kx I> 6 - 2g' cubics.

with = fff (X, H) is either a Bordiga surface or ruled by

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

211

This implies the following facts: as a first thing (2.1.1') can be further sharpened, giving (2.2.2)

0 <~Kx' Kx ~< 1

with = on the left iff (X, H) is a Bordiga surface or ruled

by cubics. As a second thing, recalling also that Bordiga surfaces have sectional genus three, (2.1.2) can be rewritten as (2.2.3)

- 2 ~< Kx" Kx <<-0

with = on the left iff (X, H) is ruled by cubics.

Finally note that when Kx'Kx >>-0 it can only be d' = 7 and Kx'Kx = 1, due to L e m m a (0.9) and (2.2.2). This fact, recalling (2.2.2), (2.2.3), (2.1.3) shows that the only possible cases to consider are those listed in the table III. TABLE III.

Subcase

g'

Kx" Kx

a

3

1

b

4

-2

c

4

-1

5

-1

d

:

Comments

(X, H) ruled by cubics

Now we look at Livorni's classification [Lil] for a detailed study of the above subcases. In particular look in table at p. 96 at surfaces with h~ = 6 and the corresponding value of Kx" Kx. X is always described as Bpl,. p~(X A), the blow-up of another surface X A at some points Pl, ..., P~ in general position; = will denote the blowing up and Ei = 7~-1 (Pi). For the proof of the very ampleness of H and of the corresponding L, see [Li2]. - L e t X A be the Del Pozzo surface with KxA'KxA = 2. Then and H = r ~ * K ~ | -1. It follows that L = 7 ~ * K ~ | 1 7 4 | I = K:~ 2 | =* Kj/2 and then L is in fact 2-spanned in view of Proposition (0.5),ii). This gives case 9 of Theorem 1. SUBCASE

a).

X = B p ( X A)

SUBCASE b). - We have (X,H) = (Bpl, ..pl0(XA), rv*HA~)[E1 + ... + E l o ] - l ) , where ( X A , H A ) = ( Q , Op~(3)| Q being a smooth quadric of pS. L e t h e IOw(1)| Then we have L = [ r ~ * 5 h - 2 E 1 - . . . - 2 E l o ] and so h~ = = h ~(5h) - 30 = 6, contradicting the 2-spannedness of L, in view of (0.4). So this case does not occur.

212

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

SUBCASE C). - We have (X, H ) = (Bpl ' . p4(XA), r~*H A | [El + ... + E4]-I), where (X A , H A) = (S, Op~ (2) | Os), S being a smooth cubic o f P ~ . Let h e 10~(1) | ~sl and note that L = [=* 3h - 2E1 - ... - 2E4]. L e t F be the proper transform via = of the hyperplane section of S with the plane spanned by the three points Pl, P2, P~. Then Pa([')=l

and

rel~*h-E1-...-E~l,

so t h a t / ' . L = 3h. h - 6 = 3, contradicting the 2-spannedness of L in view of Lemma (0.3). So Subcase c) does not occur as well. SUBCASE d ) . - (Unfortunately the table in [Lil] omits this case which we need to consider; see[Lie, case 2) in the table at p. 161]). We have ( X , H ) = = (Bpl, ..pl0(p2), r~* Op~ (7) | [El + ... + Ei0]-2). Let h 9 I Op2 (1) 1; then L = [7:* 10h - 3 E 1 - . . . - 3E10] and so we have dim ILl = dim]10h I - 6 0 = 5, contradicting the 2-spannedness of L, in view of (0.4). So Subcase d) does not occur.

This concludes the proof of Theorem 1.

3. - 3 - s p a n n e d s u r f a c e s o f s e c t i o n a l g e n u s s e v e n .

The same procedure as in Sections 1, 2, works also for proving Theorem 2. Let L be a 3-spanned line bundle on a surface X and assume that g = g(L) = 7. In this case we have h~ >I 8, by (0.4), and then the Castelnuovo inequality gives (3.0.1)

d >I 12,

equality implying h ~( L ) = 8. The same argument proving Lemma (1.1), now gives (3.1) Either X is ruled or d = 12 and X is a K3 surface embedded by ILl in pT. Letting a g a i n H = K x | and d ' = H . H , the Hodge index theorem gives 144 ~> dd', and in the same way as in Sect. 1 we get (3.2) d' <~ 12, equality implying that X is a K3 surface embedded by ILl in pT. Now, following the same procedure as in Sect. 1, we deal as a first thing with CASE A': p~ > 0. - By (3.1) we thus see that X is a K3 surface embedded by ILl in p7 and the same argument as in Proposition (1.3) rules out this case. CASE B': p~ = 0 and H not very ample. - Since L is 3-spanned (X, L) can contain neither lines nor conics, by (0.2). Hence, as g = 7, it follows from [SV] that H is very ample. So this case does not occur.

ANTONIO LANTERI: 2 - s p a n n e d surfaces o f sectional g e n u s s i x

213

In view of the above we can assume that pg = 0 and H is very ample. Then by reasoning as in Sect. 1 we get the following possibilities: C') X is a ruled surface with q = 2 embedded by IHI in p4; C") X is a ruled surface with q = 1 embedded by IHI in p s ; D') X is a rational surface embedded by IHI in p6. We analyze the above possibilities. CASE C'. - We have H ' K x = d ' - 12, Z(~gx) = - 1, K x ' K x <- 8(1 - q) = - 8. So the double point formula for surfaces in p4 gives the inequality d '~ - 15d' + 48 = 2 K x ' K x <<- - 16, which can never be satisfied. So this case does not happen. CASE C". - Let g' = g(H); then from (2.0.0), (2.0.1) we get (3.3.1)

d' = g ' + 5.

Moreover, in view of (3.2) we have d' ~< 11, so that (3.3.2)

g' ~< 6.

(3.4) REMARK. If L is 3-spanned and (X, H) is a conic bundle, then X is a pl-bundle. Actually all the fibres of the ruling of X must be irreducible, in view of (O.2). -

Now, looking at the classification of irregular ruled surfaces of sectional genus ~< 6 [Li2, table in Sect. 3] and taking into account (3.3.1), (3.3.2), (3.4) and the equality h ~ --- 6, we see that X can only be the elliptic PLbundle of invariant e = - 1 and H is numerically equivalent to either [~ + 2f] or [3z]. However, in both cases we have degL~ = 4, contradicting the 3-spannedness of L in view of[BS, (1.4.1)]. It remains to consider CASE D'. characters:

By (3.2) and (3.3.1), (X,H)

can only we have the following

(d', g ' ) - (5, 0), (6, 1), (7, 2), (8, 3), (9, 4), (10, 5), (11, 6). Since g' ~< 6, we can look at the various possibilities by using the classification of rational surfaces with small sectional genus [LU, [Lie]. As a first thing, let (d', g') = (5, 0). In this case (X, H) is a scroll of degree 4 in p s , i.e. (X, H) = = (F~, [~ + 3f]) or (Fs, [~ + 4f]). Note that in the latter case L = [3~ + 9f] is not even ample since degL~ = 0. The former case leads to the pair (X, L) = (F1, [3~ + 6f])

214

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

and L is in fact 3-spanned in view of (0.1), since L = [~ + 2f] 3 . This gives case 1 in Theorem 2. (d',g') = (6, 1). In this case X is the Del Pezzo surface of degree 6, i.e. X = = Bpl ..... p3(p2) is the blow-up of p2 at 3 points in general position and H = K ~ 1 . Then L = K~ 2 is 2-spanned by (0.1), but not 3-spanned, since L ' E i = 2, for i = 1,2,3. (d',g') = (7,2). In this case (X,H) is a conic bundle, with X = B p I ..... ps(Fe), e ~< 2. So this case cannot occur due to Remark (3.4). More generally, since X is never an Fe, Remark (3.4) allows us to exclude conic bundles from our consideration. So, from now on we can assume that (X, H) is not a conic bundle. Then, recalling (3.3.1), from the inequality ( K x | 1 7 4 >t g ' - 2 and the characterization of the corresponding equality[LP, Th. 5.1], we get (3.5.1)

Kx" K x >I 7 - 2 g ' , with = iff (X, H) is either ruled by cubics, a Bordiga surface (g' = 3) or some special surface with g ' = 6.

On the other hand, as in Sect. 2, by the Hodge index theorem, we get the inequality (3.5.2)

(g' - 7)2 = (2g' - 2 - d ' ) 2 = ( H . K x ) 2 >I d ' K x ' K x = (g' + 5 ) K x ' K x .

Now we can deal with case (d',g') = (8,3). In view of (3.5.1), (3.5.2), when g' = 3 we have 1 ~< Kx.Kx<~ 2, with = on the left iff (X, H) is either a Bordiga surface or ruled by cubics. Now look at [Li2, tables at pages 161, 169]. Due to (3.5.1), an inspection of the invariant ]'0 = = d' - K x " K x shows t h a t (X, H) cannot be ruled by cubics. Moreover ~(Bpl ..... ps (P2), 7c* Op2 (4) ~ [E 1 + ... + Es] -1)

if K x ' K x = 1,

(X, H ) = [(Bp~ ..... pT(P2), 7:* Op2(6) | [El + ... + ET] -2)

if K x ' K x = 2.

In the former case we have L "Ei = 2 for i = 1, ..., 8, hence L cannot be 3-spanned in view of (0.2). In the latter case X is a Del Pezzo surface of degree 2 and L-= Kx~; hence L is 3-spanned by [BS, (0.8.3)]. This gives case 2 in Theorem 2. To deal w i t h the remaining cases, we first prove the following (3.6) LEMMA.

-

If d' >/9, then Kx" K x < O.

PROOF. Assume, by contradiction, that Kx.Kx>~ O. Then' d ' < 9, by L e m m a (0.9), equality implying that L a - ( K x | b for some a,b e Z and L ' K x = - 4 . L e t t = (a - b)/b; then -

(3.6.1)

td = - 4.

ANTONIO LANTERI: 2-spanned surfaces o f sectional genus six

215

On the other hand, from (3.6.2)

9 = d ' = H . H = K x ' K x + 2(2g - 2) - d = K x ' K x + 24 - g,

we have (t 2 - 1)d = - 15.

(3.6.3)

Now (3.6.1), (3.6.3) give t = - 1/4 or 4. Of course, it cannot be t = 4, otherwise K x would be ample, contradicting the fact that X is a rational surface. Hence L - K~ 4 . In particular X is a Del Pezzo surface, L = Ks/4, since Pic (X) is torsion free, and (3.6.2) gives Kx" K x = 1. Recalling the general properties of Del Pezzo surfaces we thus conclude that IK~/~l contains a smooth curve F of genus 1. But F. L = 4Kx" K x = 4,

which contradicts the 3-spannedness of L in view of[BS, (1.4.1)]. So case d' = 9 cannot occur. 9 Now let (d',g') = (9,4). In this case Lemma (3.6) and (3.5.1) imply that K x . K x = - 1 and (X, H) is ruled by cubic. F r o m [Lie, p. 169] we see that X is not a Pl-bundle and so this case cannot occur in view of the following (3.7) REMARK. If L is 3-spanned and (X, H) is ruled by cubics, then X must be a PLbundle. Actually, were F = F1 + F2 a reducible fibre of the ruling of X, then for the irreducible component F~ satisfying H-F~ = 1 we would get L . F i = H ' F i - K x " 9Fi ~< 1 + 1 = 2, since Fi or another irreducible component of F is an exceptional curve. But this contradicts the 3-spannedness of L, by (0.2). -

Now, taking into account (3.0.1) and (3.3.1), we get from (2.0.1) Kx. K x = d ' + d - 24 ~ d' - 1 2 = g ' - 7.

L e m m a (3.6) and the above inequality show that (3.8)

If ( d ' , g ' ) = (10, 5) then K x ' K x = - 1

or - 2 , while if ( d ' , g ' ) = (11,6) then

gx.gx = -1.

Now let (d', g') = (10, 5). By (3.8) we have to look in [Li2, pp. 161, 169] only at surfaces with •o = 11 and 12 and we thus see that (X,H) alwasy contains a ( - 1)-line E. Then L . E = H . E - K x ' E = 2, which contradicts the 3-spannedness of L, by (0.2). Finally let (d', g') = (11, 6). By (3.8) we have to look in [Li2, pp. 161, 169] only at surfaces with ~'0 = 12. Then X = Bp, ..... pl0 (p2), with H = 7~* Op2 (9) | [3E1 + ... + 3E6 + 2Es +

216

ANTONIO LANTERI: 2-spanned surfaces of sectional genus six

+ ... + 2E10] -1 . However, for the corresponding line bundle L we have h~ = 7 [Li2, Case 19 in the table at p. 161], and this contradicts the 3-spannedness of L, by (O.4). This concludes the discussion of case D' and the proof of Theorem 2.

A d d e n d u m ( F e b r u a r y 1993). After the submission of this paper and its circulation as a preprint further progress on the subject was made. I r e f e r to the paper by M. ANDREATTA, Surface8 of sectional genus <~8 with no trisecant lines, Arch. Math., 60 (1993), pp. 85-95.

REFERENCES BALLICO E., On k-spanned projective surfaces, Algebraic Geometry, Proc. L'Aquila, 1988, Lect. Notes Math., 1417, Springer (1989), p. 23. [Bae] BALLICO E., A characterization of the Veronese surface, Proc. Amer. Math. Soc., 105 (1989), pp. 531-534. [BS] BELTRAMETTIM. - SOMMESEn. g., On k-spannedness for projective surfaces, Algebraic Geometry, Proc. L'Aquila, 1988, Lect. Notes Math., 1417, Springer (1989), pp. 24-51. Essll BELTRAMETTIM.- SOMMESEA. J., O-cycles and k-th order embeddings of smooth projective surfaces, Problems in the Theory of Surfaces, Proc. Cortona, 1988, to appear. [BFS] BELTRi~METTIM. - FRANCIA P. - SOMMESEt . J., On Reider's method and higher order embeddings, Duke Math. J., 58 (1989), pp. 425-439. [GH] GRIFFITHS PH. - HARRIS J., Principles of Algebraic Geometry, J. Wiley & Sons, New York (1978). HARRIS J., A bound on the geometric genus of projective varieties, Ann. Sc. Norm. Sup. [H] Pisa,~(4), 8 (1981), pp. 35-68. [Ha] HARTSHORNE R., Algebraic Geometry, Springer-Verlag, Berlin, New York (1977). LANTERI t., On the existence of scrolls in p4, Acc. Naz. Lincei, Rend. C1. Sc. Fis. Mat. [La] Nat. (8), 69 (1980), pp. 223-227. [LP] LANTERIA.- PALLESCHIM., On the adjoint system to a very ample divisor on a surface and connected inequalities, II, Acc. Naz. Lincei, Rend. C1. Sc. Fis. Mat. Nat. (8), 71 (~981), pp. 166-175. [Lil] LIVORNI L. E., Classification of algebraic surfaces with sectional genus less than or equal to six. - I: Rational surfaces, Pac. J. Math., 113 (1984), pp. 93-114. [Li2] LIVORNI L. E., On the existencv of some surfaces, Algebraic Geometry, Proc. L'Aquila, 1988, Lect. Notes Math., 1417, Springer (1989), pp. 155-179. SOMMESEA. J., Hyperplane sections of projective surfaces. - I: The adjunction mapping, [S] Duke Math. J., 46 (1979), pp. 377-401. [SV] SOMMESE Ao J. - VAN DE VEN A., On the adjunction mapping, Math. Ann., 278 (1987), pp. ~593-603. [Bal]