Some Very Ample and Base Point Free Linear Systems on Generic Rational Surfaces

Math. Nachr. 245 (2002), 45 – 66

Some Very Ample and Base Point Free Linear Systems on

Generic Rational Surfaces

By Stephane Chauvin of Angers and Cindy De Volder∗) of Ghent

(Received June 12, 2001; revised version March 13, 2002; accepted May 16, 2002)

Abstract. We prove a conjecture of J. Alexander concerning the base point freeness and veryampleness of linear systems on general blowings–up of the projective plane, whose standard form hasat most 9 multiple points.

1. Introduction

The study of linear systems on general blowings–up of the projective plane hasreceived considerable attention. When the number of points blown up is at most nine,the situation is well understood (see [12] and Corollary 2.6), but, for ten or morepoints, the basic questions of the dimension, base point freeness and very amplenessremain unillucidated. Harbourne and Hirschowitz have given a conjecture for thedimension and Alexander has given a general conjecture for separation properties.Here we are concerned with base point freeness and very ampleness and we provethat Alexander’s conjecture holds in a special, but non–trivial, case. This caseencompasses two known cases, the first, which we just mentioned, is for at most ninepoints and the second is a result of J. d’Almeida and A. Hirschowitz [7] whichconsiders linear systems coming from a smooth base locus.

Let k be an infinite field and let P2k be the projective plane over k.

To better deal with the notion general it is useful to consider the so–called genericblowing–up of P

2k which is defined as follows. For an integer r ≥ 0, let Λr = Λ be the

field of functions of the variety(P

2k

)r . Then there is a tautological set of r points inP

2Λ, called r generic points of P

2k. Throughout we let Xr be the blowing–up of P

2 inr generic points P1, . . . , Pr and we let Ei denote the exceptional fiber over Pi. Recallthat an exceptional curve on Xr is a divisor E P

1 on Xr with E2 = −1. We let E0

2000 Mathematics Subject Classification. Primary: 14C20, 14E25, 14J26.Keywords and phrases.∗) Corresponding author/[email protected]

c© 2002WILEY-VCHVerlagGmbH &Co. KGaA,Weinheim, 0025-584X/02/24511-0045 $ 17.50+.50/0

46 Math. Nachr. 245 (2002)

denote the pull back of the class of lines in the plane. Note that the same symbols areused regardless of the value of r.

By abuse of notation we will let a divisor D also denote its class in the divisor classgroup. It should always be clear from the context how the symbol is being viewed.Any property attributable to an effective divisor is said to hold for a divisor class D ifthis property holds geometrically for the generic member of the linear system |D|. Thearithmetic genus p(D) of a divisor class D satisfies by definition D2+D . K = 2p(D)−2,where K = Kr is the canonical class on Xr . An effective class E satisfying E2 = 0,E . K = −2 (resp. E2 = 1, E . K = −3) is called a pencil class

(resp. a line class

). An

effective class D is called non–special if Hi(O(D)) = 0 for i > 0.

Definition 1.1. An effective class E on Xr satisfying E2 = a− 1 = E . Kr is calledan isolated class of genus a ≥ 0.

An exceptional configuration on Xr is a sequence E = (E0, E1, . . . , Er), where theEi are effective classes on Xr , Ei is exceptional for i = 1, . . . , r, E0 is a line classand Ei . Ej = −δij . This is a natural basis for the divisor class group of Xr that isorthogonal for the intersection form. Given such an exceptional configuration, |E0|induces a birational morphism of Xr to P

2 with exceptional fibers Ei.The study of a divisor class D on Xr is greatly facilitated by making an adapted

choice of an exceptional configuration. A divisor class H is said to be standard ifH . E ≥ 0 for all exceptional classes, pencil classes and line classes E. When r ≥ 3this is equivalent to H . E ≥ 0 for all exceptional classes and, in all cases standardimplies that H . E ≥ 0 for any effective rational class (see [2]). It can be shown(loc. cit.) that H is standard if and only if there exists an exceptional configurationE = (E0, E1, . . . , Er) such that

H ≡ mE0 − l1E1 − . . .− lrEr

where m ≥ l1 ≥ . . . ≥ lr ≥ 0 and m ≥ l1 + l2 + l3. When H ≡ mE0 − l1E1− . . .− lrEr

satisfies these latter conditions it is called E–standard.

Definition 1.2. On Xr , r ≥ i, we let Ci ≡ 3E0 − E1 − . . . − Ei and we supposethat Ci is a smooth member of the linear system |3E0 − E1 − . . .− Ei| for 0 ≤ i ≤ 9.

It is known that for a > 1 there are only finitely many isolated E–standard classesof genus a, whereas when a = 1 there are all the multiples of C9 of which only C9

is reduced. When a = 2 there are three such classes (see 1.1). These classes play animportant role in Alexander’s conjecture as we now explain.

Clearly, for a divisor class H , being standard is a necessary condition for |H | tobe base point free. In fact H.E < 0, for an exceptional curve E, makes E a fixedcomponent of |H |. It’s natural then, in the present context, to work entirely withE–standard classes for some fixed exceptional configuration E.

When r ≥ 3, Alexander’s conjecture is that an E–standard class H ≡ mE0 −l1E1 − . . .− lrEr

1. with χ(O(H)) ≥ 3 is base point free if H . C9 ≥ 2 and,

Chauvin and De Volder, Very Ample and Base Point Free Linear Systems 47

2. with χ(O(H)) ≥ 6 is very ample if lr ≥ 1, H . C9 ≥ 3, and H . E ≥ 5 for the(following) three isolated E–standard classes of genus two

(1.1) C12 + E0 − E1 , C11 + C8 , −2C10 + C8 .

After making a slight modification to deal with r = 0, 1 or 2, the conjecture then takesthe following form.

Conjecture 1.3. ([2]) On Xr, for an E–standard class H ≡ mE0−l1E1−. . .−lrEr

1. satisfying χ(O(H)) ≥ 3, |H | is base point free if and only if H . C9 ≥ 2,2. satisfying lr > 0, m > l1 + l2 if r = 2, m > l1 if r = 1, and χ(O(H)) ≥ 6, |H | is

very ample if and only if H . C9 ≥ 3 and H . E ≥ 5 for the isolated E–standard classesE of genus 2.

We prove the following,

Theorem 1.4. If H ≡ mE0 − l1E1 − . . .− lsEs is an E–standard class on Xs withli = 1 for i ≥ 10, then

1. for χ(O(H)) ≥ 3, |H | is base point free if and only if H . C9 ≥ 2;2. for ls > 0, m > l1 + l2 if s = 2, m > l1 if s = 1, and χ(O(H)) ≥ 6, |H | is very

ample if and only if H . C9 ≥ 3.That is Conjecture 1.3 holds for |H |.

Remarks 1.5. 1. It is possible that the theorem can be refined to include allDel Pezzo surfaces in the following sense: if X is Del Pezzo and H0 is very ample ofprojective dimension m ≥ 5, then for generic points p1, . . . , ps on X, H0−F1− . . .−Fs

is very ample on the blowing–up of X in the pi provided m−s ≥ 5. The authors havenot investigated whether the proofs can be adapted to this more general case.

2. The test on genus two classes in the conjecture, which is a necessary conditionfor H to be very ample, is automatically satisfied in the context of our theorem.

3. Using the definition, it is easy to see that a class H ≡ mE0 − l1E1 − . . .− lsEs isE–standard if and only if

H ≡ aE0 + b(E0 − E1) + c(2E0 − E1 − E2) +r∑

i=3

αiCi

where Ci ≡ 3E0 − E1 − . . . − Ei and a, b, c, αi ≥ 0 (see e. g. [11]). We say that E0,E0 −E1, 2E0 −E1 −E2 and the Ci (i ≥ 3) are the generating classes. This expressionis frequently used throughout this paper.

4. If H is an E–standard class with r ≥ 3 then lr is the minimum value of H . E forall exceptional classes E and even for all effective rational classes on Xr (see [2]).

5. Any H satisfying dim |H | < 2 (resp. < 5) will only be base point free if it satisfiessome supplementary, Porteous type, numerical condition (e. g. H2 = 0 for base pointfree). As such, for non–special H , the conditions χ(O(H)) ≥ 3 and ≥ 6 are the naturalconditions for |H | to be base point free and very ample respectively. All standard Hin this paper are non–special and it is even conjectured by Hirschowitz that allstandard H are.

48 Math. Nachr. 245 (2002)

6. Though C9 is not defined on Xr for r < 9, we define H . C9 in this case by pullingH back to X9. This abuse, which will be frequently used throughout the paper,is essentially a convenient, geometrical way of writing down numerical inequalitiesinvolving m and the li. The same idea is implicit in the conditions involving isolatedE–standard classes E of genus two.

The base point free case is relatively simple compared to the very ample part. Inthe following section we prove a number of elementary results and give our own proofof the case where s ≤ 9 (see also [12]). In Section 4 we divide the very ample case intotwo parts and deduce them from two other results, Theorem 5.1 and Proposition 6.1.The first is a more general criterion for very ampleness than is required in this paper,while the second proves exactly one of the cases. One should note that in both cases,the same elementary Lemma 2.2 is applied and that the difficulties are in verifyingthat the hypotheses of this lemma are indeed satisfied.

This work was initially motivated by a result of J. d’Almeida and A. Hirschowitz[7] which proves the theorem for divisor classes of the form H ≡ mE0 −E1 − . . .−Es

(i. e. all li = 1). Coppens [5] gave another proof which inspires our main criterion5.1. One other case is well–known. This is when H ≡ mE0 − l1E1 − . . .− lsEs withs ≤ 9 (see for example [12]), for which we give a direct proof in Corollary 2.6. For thesake of completeness, we have endeavourd to make our proof self contained.

Both authors proved the results of this paper, independently and simultaneously,using slightly different techniques (see [6] and [8]); with exception of the results of § 6,which are due to S. Chauvin.

2. Some elementary results

This section contains a few rather elementary results. Though Proposition 2.5 andits Corollary 2.6 are essentially well–known, we include our own proof.

Definition 2.1. A 2–cluster on a scheme Y of finite type over an algebraicallyclosed field k, will be any zero dimensional closed subscheme of Y of degree two.For V ⊂ H0(Y, L) a finite dimensional linear subspace of the global sections of theinvertible sheaf L, we say that V separates a 2–cluster Z ⊂ Y if V → H0(Z, OZ ⊗L)is surjective.

Recall that in the notation of the definition, V is very ample on Y if V separates all2–clusters on Y . The same is true over any field k provided V separates all 2–clusterson Y × k where k is the algebraic closure of k. If V is base point free then V is veryample if and only if for each 2–cluster Z on Y , there exists a section f ∈ V vanishingon some point of Z but not on Z.

Lemma 2.2. (Alexander) Let L be an invertible sheaf on a smooth surface X andlet V ⊂ H0(X, L) be a linear system. Let C be an effective divisor on X (including C ≡0) and let V (−C) be the canonical pullback of V to H0(X, L(−C)). Let P ⊂ V (−C)be a linear pencil. Then, if the linear system |V | separates all 2–clusters contained in


C and all 2–clusters entirely contained in any effective divisor D of the linear systemP, then |V | is very ample on X.

Proof . Since |V | has no base points in any curve in the pencil P, it is base pointfree on X. We must show that if Z is a 2–cluster on X not lying in C, or in any curveof the pencil P, then Z is separated by |V |. Clearly any such Z does not meet thebase of the pencil P. Consequently, if Z meets C in a point, the general curve D inP does not meet Z and Z is separated by C + D ∈ |V |. If Z does not meet C then aunique curve D in the pencil P meets a given point p in Z and C + D ∈ |V | separatesZ.

In the following lemma, we recall a well–known result.

Lemma 2.3. Let C be a smooth and irreducible projective curve of genus g and letL be an invertible sheaf on C of degree d. Then for n ≥ g, the canonical map

Cn −→ Picd−n(C) ; (x1, . . . , xn) −→ L(−x1 − . . .− xn)

is surjective. In particular, if x1, . . . , xn is a generic sequence, then L(−x1 − . . .−xn)is a generic invertible sheaf of degree d − n.

Lemma 2.4. Let π : X → S be a smooth family of projective surfaces over the regu-lar irreducible Noetherian scheme S and let C ⊂ X be a relative Cartier divisor. Sup-pose that over some point (not necessarily closed ) s ∈ S, the fiber Cs is geometricallya nodal curve, that the singular locus of Cs is irreducible and that the normalization ofCs is disconnected. Then if the generic fiber Cξ is geometrically irreducible it is alsosmooth.

Proof . We can clearly suppose that S = Spec A is the spectrum of a regular localring with closed point s corresponding to the maximal ideal m. One has the relativeJacobian B ⊂ C whose ideal in OC is defined to be the image of the canonical mapΩˇ

X/S(−C)|C → OC. This commutes with base change. We will suppose that thegeneric fiber Cξ is singular. Since Cs is nodal, Bs is zero dimensional and smooth. Bythe semi–continuity theorem of Chevalley, all the fibers have dimension zero and Bis finite over S, hence affine, and B = Spec B. Since Bs is irreducible, B/mB is a fieldand B is a local ring with maximal ideal mB.

We thus have A and B/mB regular and

dimA = dimB + dim(B/mB)

hence B is a regular local ring also, flat over A ( [10, IV.1, Chap. 0, 17.3.3]) of constantdegree dimA/m B/mB. In particular, all the fibers of C over S are nodal curves and thenumber of nodes is constant. Since the formation of higher infinitesimal neighborhoodscommutes with base change, if B(n) is the n + 1–th infinitesimal neighborhood of B

in C then B(n) is finite of constant degree over S depending only on the number ofnodes in a fiber, hence is flat over S. It follows that the blowing–up µ : C → C ofC in B commutes with base change and is flat over S. This gives, fiber by fiber overS, the normalization of C, so that the generic fiber is geometrically connected while

50 Math. Nachr. 245 (2002)

the special fiber is disconnected (both by hypothesis), but this contradicts ([10, IV.3,15.5.7]).

Proposition 2.5. On Xr (r ≤ 9), a standard divisor class D is non–special, andif D ≡ 0 then |D| contains a smooth and irreducible curve unless, in an exceptionalconfiguration E = (E0, E1, . . . , Er) for which D is E–standard, either,

(i) D ≡ λ(E0 − E1) (λ > 1) or,(ii) D ≡ λC9 (λ > 1).

In the case (i) the general member of |D| is smooth but reducible and in case (ii) it isirreducible but not reduced.

If ∆, D ≡ 0 are standard divisor classes on Xr then D . ∆ ≥ 0, with equality if andonly if ∆ and D are both of type (i) or both of type (ii). In particular, if D ≡ 0 is astandard divisor with D2 > 0, then a general member of |D| is smooth and irreducible.

Proof . If D is standard, choose an exceptional configuration E = (E0, E1, . . . , Er)for which D is E–standard. Then D can be written (see Remark 1.5.3) as

(2.1) D ≡ aE0 + b(E0 − E1) + c(2E0 − E1 − E2) +r∑

i=3

αiCi

with a, b, c, αi ≥ 0. In particular D is effective. Elementary considerations show thatif ∆, D ≡ 0 are standard divisor classes on Xr , then ∆ . D > 0 unless ∆ and D areboth of type (i) or both of type (ii). Also, if D is not as in case (ii), D . Kr < 0, sothat D2 ≥ 2p(D)−1, showing that D is non–special if D is an irreducible and reducedclass.

Claim 1. A standard divisor class D ≡ 0 is non–special, and its general member isreduced and irreducible unless either, (i) D ≡ λ(E0 − E1) (λ > 1) or, (ii) D ≡ λC9

(λ > 1).

Let q(D) = a+b+c+∑r

i=3 αi. Clearly when q(D) = 1, the general member of |D| isnon–special, reduced and irreducible. Let q > 1 and suppose that all standard divisorclasses D′ with q(D′) < q, satisfy Claim 1. If the generic curve of |D| with q(D) = qcan geometrically be decomposed as a sum A + B of effective divisors A and B, thenthese are E–standard classes because they can be specialized to a sum of effectivedivisors associated to the E–standard generating classes in (2.1). Also q(A) < q andq(B) < q, so that the induction hypothesis and Riemann–Roch for surfaces gives

dim (|A|) + dim (|B|) ≥ dim (|D|)≥ χ(O(D)) − 1

= χ(O(A)) − 1 + χ(O(B)) − 1 + A.B

= dim(|A|) + dim (|B|) + A . B .

There is therefore equality everywhere, D is non special and A . B = 0. This showsthat all standard divisor classes D are non–special because the general member of |D|is reduced and irreducible unless D is of type (i) or (ii).

We will now prove


Claim 2. For D ≡ 0 a standard class not of type (ii), the generic member of |D| issmooth.

Firstly we make the following observations which will allow us to apply Lemma 2.4.If D is a standard divisor class of the form D ≡ aE0 + b(E0 −E1)+ c(2E0 −E1 −E2)with a = 0 or c = 0, then the morphism induced by |D| at worst contracts a finiteset of orthogonal exceptional curves, hence is the pull back of a very ample linearsystem by a birational map. Its generic member is thus smooth and meets any givenreduced and irreducible standard divisor in a smooth and irreducible locus. On theother hand, if b > 0, the generic member of the class |b(E0 − E1)| is smooth and|b(E0 −E1)| induces a tamely ramified map on Cr, so that the generic member of theclass |b(E0 − E1)| meets Cr (hence the Ci for i < r) in a smooth, irreducible locus.

If D ≡ aE0+b(E0−E1)+c(2E0−E1−E2)+∑r

i=3 αiCi with αr = 0 and D is not oftype (ii) in Claim 1, then D −Cr is standard, hence non–special, and |D| induces thecomplete linear system on Cr of degree D . Cr > 0. As such, the generic member of|D| meets Cr in a smooth irreducible locus; the generic divisor of

∣∣H0(Cr, OCr (D))∣∣.

Now let α(D) = α3 + . . . + αr. We argue by induction on α. When α = 0 we arein the context treated in the first paragraph of the proof of this claim. Suppose thatα > 0 (i. e. r ≥ 3 and αr > 0) then either D = Cr (D − Cr ≡ 0), hence D is smoothand irreducible, or, by the induction hypothesis, the generic member G of |D− Cr| issmooth. In the latter case, we have just shown that G meets Cr in a smooth irreduciblelocus so that G +Cr is a nodal curve with irreducible singular locus and disconnectednormalization, while the generic member of |D| is geometrically irreducible. HenceLemma 2.4 applies and we conclude that the generic member of |D| is smooth.

Corollary 2.6. An E–standard divisor class D on Xr (r ≤ 9)1. satisfying D2 > 0 and D . Cr ≥ 2 is base point free,2. satisfying D2 > 0, D . Er > 0, D . (E0 − E1 − E2) > 0 and D . Cr ≥ 3 is very

ample.

Proof . For (1) we have D2 = 2p(D) − 2 + D.Cr ≥ 2p(D) so this follows from thefact that |D| contains a smooth and irreducible curve.

For (2) we have D2 ≥ 2p(D) + 1, so it would suffice (Lemma 2.2) to show that thegeneric pencil in |D| contains only integral curves or that the locus of non–integralcurves in |D| is in codimension ≥ 2. This locus is the finite union of the images of themaps ϕA : |A| × |B| → |D| as A varies over the effective A ≡ 0 such that B = D − Ais effective. If A (or B) is not standard then |A| has an exceptional curve E as fixedcomponent and the image of ϕA is contained in the image of ϕE . The hypothesesimply that H . E > 0 for all exceptional curves on Xr so that |D| induces a base pointfree, degree > 0 linear system on E, showing that h0(O(D − E)) ≤ h0(O(D)) − 2.Consequently, the image of ϕE is in codimension ≥ 2.

When A and B are both standard, we have as in the proof of the proposition

dim |D| − A . B = dim |A| + dim |B| .Since A . B ≥ 0 and A . B = 0 is excluded by the hypothesis D2 > 0 we need only dealwith the cases where A . B = 1.

52 Math. Nachr. 245 (2002)

If B2 = 0 then B is a pencil class or C9, but in the latter case we would haveD . C9 = A . C9 = 1 < 3. So B is a pencil class and if r > 1, there will be anexceptional curve E = B − F , F exceptional, such that D . E = 0. As such r = 1,B = E0 − E1 and A = aE0 − (a − 1)E1 with a > 0. As such D = (a + 1)E0 − aE1,a > 0, and since E0 separates all 2–clusters not entirely in E1, while E0 − E1 inducesa very ample system on E1, |D| is very ample.

When both A2 > 0 and B2 > 0, the inequality 1 = (A . B)2 > A2B2 is impossible,so, by the algebraic index theorem, A is a rational multiple of B. In the present contextthis implies A = B and A2 = 1. Then A is E–standard and A = E0 or A = C8, butin the latter case 2C8 . C9 = 2 < 3.

Proposition 2.7. On Xr (r ≤ 8), let

H0 = mE0 − l1E1 − . . .− lrEr

= aE0 + b(E0 − E1) + c(2E0 − E1 − E2) + α3C3 + . . . + αrCr

be an E–standard class with lr ≥ 1, m > l1 +1, m > l1 + l2 and H0 . Cr ≥ 6, where thenotation is that of (2.1). Then if H0 ≡ E0, H0 − E0 is standard of self–intersection> 0. In particular |H0 − E0| contains a smooth and irreducible curve.

Proof . Clearly H0 − E0 is E–standard if a > 0 and, by Proposition 2.5, H0 − E0

has self–intersection > 0 unless H0 ≡ E0 + b(E0 − E1) which is excluded by thehypotheses. Note that when r ≤ 2, the hypothesis m > l1 + l2 implies a > 0 sohenceforth we suppose r ≥ 3. This means that lr = αr is the minimum value of H0 . Eon all exceptional classes E (see Remark 1.5.4).

Suppose now that a = 0 and r ≥ 3 so that m = l1+l2+l3. To show that H ′0 = H0−E0

is standard (but not necessarily E–standard), it suffices to show that H ′0 . E ≥ 0 for

all exceptional classes E on Xr. This however is clear for the classes Ei, i = 1, . . . , r.When r ≤ 8 there are only finitely many exceptional classes on Xr . These are the

Ei and those in the following list up to a permutation of the indices

(2.2)

E0 − E1 − E2 ,

2E0 − E1 − . . .− E5 ,

3E0 − 2E1 − E2 − . . .− E7 ,

4E0 − 2E1 − 2E2 − 2E3 − E4 − . . .− E8 ,

5E0 − 2E1 − . . .− 2E6 − E7 − E8 ,

6E0 − 3E1 − 2E2 − . . .− 2E8 .

Since the sequence l1, . . . , lr is non–increasing, it suffices that H ′0 . E ≥ 0 or equivalently

H0 . E ≥ E0 . E for all exceptional classes E in this list. Clearly 1 ≤ E0 . E ≤ 6 for allE in (2.2). If E0 . E ≥ 4 then r = 8 and, by a direct verification G . E ≥ G . C8 for Ga generating class of (2.1). Hence H0 . E ≥ H0 . C8 ≥ 6 ≥ E0 . E.

As H0 . E ≥ lr ≥ 1, we can assume 2 ≤ E0 . E ≤ 3. Moreover, for any exceptionaldivisor E from the list (2.2) we have H0 . E ≥ b + c + α3 + . . . + αr. So, if H0 . E = 1then H0 = Cr and H0 . Cr = C2

r ≥ 6, showing that H0 = C3, but then H0 . E ≥ 3. If


H0 . E = 2 and E . E0 = 3, then r = 7 or 8 and H0 = (E0−E1)+Cr , or H0 = Ci +Cr ,7 ≤ i ≤ r ≤ 8, but then H0 . Cr < 6.

To see that H0−E0 has self–intersection > 0 when a = 0, suppose that the contraryholds. Then H0 − E0 ≡ λP where λ > 0 and P is a pencil class; i. e. P is standard,P 2 = 0 and P . Kr = −2. By hypothesis P ≡ E0 − E1 so that P = n0E0 − n1E1 −. . . − nrEr where n3 ≥ 0. By Noether’s inequality, n0 < n1 + n2 + n3, but m =λn0 + 1 ≥ λ(n1 + n2 + n3) since H0 is E–standard. This implies λ = 1 and H0 . Cr =P . Cr + E0 . Cr = 5 < 6!

Corollary 2.8. Let H = mE0 − l1E1 − . . .− l8E8 be an E–standard class on X8

with m > l1 + 1, m > l1 + l2 and H . C8 ≥ 5. Then for α ≥ 2, H − E0 + αC9 isstandard of self–intersection > 0 on X9 and as such the linear system |H −E0 +αC9|on X9 contains a smooth and irreducible curve. In particular, on X9, if H0 = mE0 −l1E1 − . . .− l9E9 is an E–standard class with l9 ≥ 2 and H0 . C9 ≥ 5, then |H0 − E0|contains a smooth and irreducible curve.

Proof . Firstly, let H1 = H + C8, then H1 ≡ E0 is standard and satisfies theconditions of the proposition, so that H1−E0 is standard of self–intersection > 0. LetF = (F0, F1, . . . , F8) be an exceptional configuration for which H1−E0 is F –standard.Then

F ′ = (F0, F1, . . . , Fs, E9, Fs+1, . . . , F8)

with s := maxi : (H1−E0) . Fi > 0, is an exceptional configuration and H1−E0−E9

is F ′–standard unless H1−E0 ≡ (a+ b)F0 −aF1 − bF2 or a(F0 −F1) with a, b > 0. Inthe first case H1−E0−E9 is nonetheless standard as one sees by making the quadratictransformation centered on F1, F2, E9. In the second case H1−E0−E9+C9 is standardby the same argument (note that the form of the canonical class C9 is independentof the chosen exceptional configuration). We have thus shown that H − E0 + αC9 =H1 − E0 − E9 + (α − 1)C9 is standard.

Finally, since (H1 − E0)2 > 0 and we have

(H1 − E0) . C9 = (H + C8 − E0) . C9 = H . C9 − 2 ≥ 3

we conclude that

(H − E0 + αC9)2 = (H1 − E0 − E9 + (α − 1)C9)2

≥ (H1 − E0)2 − 1 + 2(α− 1)(H1 − E0) . C9 − 2(α− 1)E9 . C9

≥ 4(α − 1)

> 0

so that Proposition 2.5 applies.Finally, if H0 = mE0 − l1E1 − . . . − l9E9 is an E–standard class with l9 ≥ 2 and

H0 . C9 ≥ 5, we can write H0 = H + αC9 with α ≥ 2 and H satisfies the hypothesesof the corollary.

54 Math. Nachr. 245 (2002)

3. Proof of Theorem 1.4.1

This is the base point free part.We consider H as being constructed from a class H0 = mE0 − l1E1 − . . . − lrEr;

r ≤ 9 and lr ≥ 2 if r > 0; so that H = H0 − Er+1 − . . .− Es. By Corollary 2.6, wecan suppose s ≥ 10 and consequently m ≥ l1 + 2, because H is E–standard. Alsoχ(O(H0)) = χ(O(H)) − (r − s) ≥ 10 − r + 3. Clearly H0 satisfies the hypothesesof Theorem 1.4, H0 ≡ λ(E0 − E1) and H0 ≡ λC9 so that by Proposition 2.5, |H0|is non–special and contains a smooth and irreducible curve of genus g let’s say. Letn = χ(O(H0))−3 and let C be the generic curve in |Hext| = |H0−Er+1 − . . .−Er+n|,we have r + n ≥ 10. By hypothesis, Hext . C9 = H0 . C9 ≥ 2, so it clearly suffices toshow that |Hext| is base point free. Since n = g − 3 +H0 . Cr (Riemann–Roch and theadjunction formula), if H0 . Cr ≥ 3 then, by Lemma 2.3, OC(H) is a generic memberof Pic g+1(C), hence is base point free. Since Xr,n is rational, |H | induces the completelinear system on C and it follows that for H0 . Cr ≥ 3, |H | is base point free.

If H0 . Cr = 2, then, using the decomposition (2.1) and lr ≥ 2, H0 would be 2C8

treated by Corollary 2.6 or one the following classes E0 − E1 + λC9, C7 + λC9 or2C8 + λC9 where, l9 = λ ≥ 2. In the latter cases H0 − C9 is standard and has selfintersection > 0. By Proposition 2.5, the generic curve D in |H0 − C9| is smooth andgeometrically irreducible of genus g(D) = g − 2.

Let P1, . . . , Pn be generic points of D, let Yr,n be the blowing–up in the Pi withexceptional fiber Fi over Pi, and let D also denote the strict transform of D on Yr,n.Let H ′ ≡ H0−F1− . . .−Fn. Since n = g(D)+1, OD(H ′) is generic of degree g(D)+1,hence base point free. Since D and C9 are non–special, |H ′| induces a complete, basepoint free, linear system on D and C9, with D +C9 ∈ |H ′|. Consequently |H ′| is itselfbase point free.

4. Proposition 6.1 and Theorem 5.1 imply Theorem 1.4.2

The following two results will be proven in the following sections. In this section wewill show that they imply Theorem 1.4.2.

Proposition 6.1. Let H0 = mE0− l1E1− . . .− l9E9 be an E–standard divisor classon X = X9 with l9 ≥ 1 and l8 ≥ 2. We suppose n = χ(O(H0))−6 ≥ 0 and H0 . Cr = 3or 4. Then |H | = |H0 − Er − . . .− Er+n| is very ample on Xr+n.

Theorem 5.1. Consider the linear system |H0| = |mE0 − l1E1 − . . .− lrEr| on Xr,r ≥ 0.

Suppose that the linear system |H0−E0| = |(m−1)E0− l1E1− . . .− lrEr| contains asmooth and irreducible curve C. Suppose that either l1 ≤ 3 and 4m ≥ l1 + . . .+ lr + 9or l1 ≥ 4 and 4m ≥ 2l1 + l2 + . . .+ lr +10. Suppose further that n = χ(O(H0))−6 ≥ 0.Let R1, . . . , Rn−1 be generic points on C and let Rn be a generic point of Xr. LetXr,n be the blowing–up of Xr in the Ri with exceptional fibers Fi, then |H | = |H0 −F1 − . . . − Fn| is a non–special, very ample, linear system on Xr,n. In particularmE0 − l1E1 − . . .− lrEr − Er+1 − . . .− Er+n is non–special and very ample.


To prove that these imply the theorem, we consider H as being constructed froma class H0 = mE0 − l1E1 − . . . − lrEr; r ≤ 9 and lr ≥ 2 if r > 0; so that H =H0 − Er+1 − . . .− Es. By Corollary 2.6 we can suppose that s ≥ 10, so that

(4.1) H0 . Cr ≥ 3 + (9 − r) and χ(O(H0)) ≥ 6 + (10 − r) = 16 − r ≥ 7 .

Let n = χ(O(H0)) − 6 and define Hext = H0 − Er+1 − . . . − Er+n. By hypothesisHext . C9 = H0 . C9 ≥ 3 and it clearly suffices to show that Hext is very ample.

Claim. If r ≤ 8, then H0 . Cr ≥ 4 and if H0 . Cr = 4 or 5 then r = 8.The first part follows from (4.1). If r ≤ 6, then H0 . Cr ≥ 6 by (4.1). When r ≥ 3,

we can write H0 = A + lrCr with A E–standard. If r = 7, then H0 . C7 ≥ 5 by (4.1)and H0 . C7 = 5 implies A . Cr = 1, but this is only possible for A = C8 contradictingr = 7.

We thus have two cases where we must show that Hext is very ample.1. r = 9 and H0 . Cr ≥ 5, or r ≤ 8, H0 . Cr ≥ 6,2. r = 9 and H0 . Cr = 3, 4, or r = 8, H0 . Cr = 4, 5 and χ(O(H0)) ≥ 7.In the second case, we can conclude by Proposition 6.1. Note that in the case of

r = 8, it is H0 − E9 that plays the role of H0 in the statement of that proposition.In the first case |H0−E0| contains a smooth and irreducible curve, by Proposition 2.7

and Corollary 2.8. As such, by Theorem 5.1 it suffices to show that if H0 ≡ 2E0, then4m ≥ 2l1 + l2 + . . .+ lr +10 if l1 ≥ 4 and 4m ≥ l1 + l2 + . . .+ lr +9 if l1 ≤ 3. In fact ifr = 0, 1 or 2, this follows respectively from m ≥ 3, m ≥ l1+2 and m ≥ l1+l2+1. When8 ≥ r ≥ 3 we have m ≥ l1 + l2 + l3, 3m ≥ l1 + . . . + lr + 6 (because H0 . Cr ≥ 6) andl2 ≥ l3 ≥ 2, giving the result. When r = 9, we have m ≥ l1+l2+l3, 3m ≥ l1+. . .+l9+5(because H0 . C9 ≥ 5) and l2 ≥ l3 ≥ 2, so 4m ≥ 2l1 + l2 + . . . + lr + 10, unlessl2 = 2 (= l3 = . . . = l9), m = l1 + 4 and 3l1 + 12 = l1 + 16 + 5, but then 2l1 is odd.

5. Principal criterion

In this section we prove what we call the principal criterion for very ampleness. Thiscriterion has a wider application than simply proving the results of this paper and canbe applied when the number of points of multiplicity ≥ 2 is greater than nine, butits application requires knowledge about the existence of smooth curves in a certainlinear system. For example, Theorem 5.1 can be applied directly to obtain resultsabout systems of nodal curves with not too many nodes.

While the required criterion is Theorem 5.1, we will deduce it from the very similarProposition 5.3 by a specialisation arguement. Before discussing the proof in moredetail we give the

Theorem 5.1. Consider the linear system |H0| = |mE0 − l1E1 − . . .− lrEr| on Xr,r ≥ 0.

Suppose that the linear system |H0−E0| = |(m−1)E0− l1E1− . . .− lrEr| contains asmooth and irreducible curve C. Suppose that either l1 ≤ 3 and 4m ≥ l1 + . . .+ lr + 9or l1 ≥ 4 and 4m ≥ 2l1 + l2 + . . .+ lr +10. Suppose further that n = χ(O(H0))−6 ≥ 0.Let R1, . . . , Rn−1 be generic points on C and let Rn be a generic point of Xr. Let

56 Math. Nachr. 245 (2002)

Xr,n be the blowing–up of Xr in the Ri with exceptional fibers Fi, then |H | = |H0 −F1 − . . . − Fn| is a non–special, very ample, linear system on Xr,n. In particularmE0 − l1E1 − . . .− lrEr − Er+1 − . . .− Er+n is non–special and very ample.

N.B. Note that the strict transform of C on Xr,n is isomorphic to C by the projec-tion to Xr. By abuse of notation we will use the same symbol C to denote its stricttransform on Xr,n. We let g be the genus of C.

We want to prove Theorem 5.1 using Lemma 2.2 applied to the pencil |E0 − Fn|.Everything comes down to showing that |H | separates all 2–clusters in a curve of thispencil. This is done by specializing the point Rn to C. In the latter situation |H |is no longer very ample, but it is possible to locate the unseparated 2–clusters in Cand show that none are contained in a curve of the pencil. Since the latter is anopen property when dim |H | and dim |E0 − Fn| remain constant, we get the requiredinformation about separation in curves of the pencil |E0−Fn| in the original situationof Theorem 5.1.

The principal difficulty will be separating 2–clusters in the exceptional fibers Ei

when li ≥ 4. In this case, |H | does not induce the complete linear system on Ei sothat simple degree arguments are not enough. In [6], the first author got around thisproblem by specializing some of the Ri to C ∩Ei and making a close study of the thenreducible curve Ei. What happens in this case is that |H | does induce the completelinear system on all the irreducible components and this is enough. The strategy usedhere is that of the second author and was inspired by the work of Coppens [5].

Remark 5.2. In Theorem 5.1 a simple calculation gives H . C = g+3 and n ≥ g+1.Also, H0 . C ≥ 2g + 4 + l1 if l1 ≥ 4.

We will firstly deduce Theorem 5.1 from the following proposition.

Proposition 5.3. Let n, Xr, C and H0 be as in the Theorem 5.1. Suppose that|H0−E0| contains a smooth and irreducible curve. Let x1, . . . , xn be generic points onC and let X′

r,n be the blowing–up of Xr in the xi with exceptional fibers F ′i . As usual

we denote the strict transform of C on X′r,n by C also. Let |H ′| = |H0−F ′

1− . . .−F ′n|.

Suppose that either l1 ≤ 3 and 4m ≥ l1 + . . . + lr + 9 or l1 ≥ 4 and 4m ≥ 2l1 + l2 +. . . + lr + 10, then |H ′| is a non–special, linear system that separates all two clusterson X′

r,n that are contained entirely in a curve of the linear system |E0 − F ′n|.

Remark 5.4. In Proposition 5.3. When li ≤ 3 one has the exact sequence

0 −→ H0(E0 − Ei) −→ O(H ′ − Ei) −→ H0(OC(H ′ − Ei)) −→ 0

with (H ′ − Ei) . C ≥ g (see Remark 5.2). By Lemma 2.3, OC(H ′ − Ei) is a genericmember of its component of the Picard scheme, hence is effective and non–special.Then O(H ′ − Ei) is non–special and one easily sees that |H ′| is non–special andinduces the complete linear system on Ei. The same argument can also be applied tothe Fi.

Proof of Theorem 5.1. Clearly in the notation of Proposition 5.3,(X′

r,n, H ′) isa specialization of

(Xr,n , H

). By semi–continuity, since |H ′| is non–special so is |H |.


Similarly, since |H ′| separates all 2–clusters contained in a curve of the pencil |E0−F ′n|,

|H | separates all 2–clusters contained in a curve of the pencil |E0−Fn|. By Lemma 2.2it suffices to show that |H | separates all 2–clusters in C.

On Xr,n, we have the exact sequence on global sections

0 −→ H0(O(E0 − Fn)) −→ H0(O(H)) −→ H0(OC(H)) −→ 0 .

Since H . C = g + 3 with OC(H) a general element of Pic g+3(C) (see Lemma 2.3),OC(H) is very ample on C and |H | as well.

5.1. The technical ground work for the proof of Proposition 5.3

The proof of Proposition 5.3 which will be presented below, actually reduces thestatement to the following lemma using Lemma 5.7.

Lemma 5.5. In the notation of Proposition 5.3, let Ei be an exceptional fiber onXr with li ≥ 4. Let X′′

r,3 be the blowing–up of Xr in three generic points x1, x2, x3 ofC with corresponding exceptional fibers F ′′

i . On X′′r,3, let |H ′′| = |H0−F ′′

1 −F ′′2 −F ′′

3 |,let Q ⊂ Ei \ C be a 2–cluster, and let IQ be the ideal of Q as a subscheme of X′′

r,3.Then the linear system H0(IQ(H ′′)) separates all two clusters in C.

Proof . We have (H0−Ei) . C ≥ 2g+4, so that on X′′r,3 we have (H ′′−Ei) . C ≥ 2g+1

and OC(H ′′ − Ei) is non–special and very ample on C. Then, since |E0 − Ei| is non–special, so is |H ′′−Ei| and it induces the complete linear system on C. If Z ⊂ C ∩Ei

is a 2–cluster, we have the exact sequence

(5.1) 0 −→ H0(O(H ′′ − Ei)) −→ H0(IQ(H ′′)) −→ H0(OEi (H′′)(−Q)) −→ 0 .

So H0(IQ(H ′′)) induces a complete linear system of degree l1 − 2 on Ei, i. e. if Z ⊂C ∩ Ei, then H0(IQ(H ′′)) separates Z.

If Z ⊂ C \ Ei is a 2–cluster, then |H ′′ − Ei| + Ei ⊂ H0(IQ(H ′′)) and |H ′′ − Ei| isvery ample on C, so that H0(IQ(H ′′)) separates Z.

Any other 2–cluster Z in C has a point p as trace on Ei and a point q ∈ C as residualwith respect to Ei. We see that Z is separated by

∣∣H0(IQ(H ′′))∣∣ using the diagram

(5.2)

0 H0(O(H ′′ − Ei)) H0(IQ(H ′′)) H0(OEi(H′′)(−Q)) 0

0 H0(Oq(H ′′ − Ei))

H0(OZ(H ′′)(−Q))

H0(Op(H ′′)(−Q))

0

where the surjectivity of the left–hand vertical arrow follows from the fact that|H ′′−Ei| induces, as we saw above, a complete, very ample system on C and q ∈ C.

Definition 5.6. Let C be a smooth, irreducible, projective curve and let L be aninvertible sheaf on C. Let V ⊂ H0(C, L) be a linear subspace. We define V e−f

e ⊂ Ce

to be the locus of points (x1, . . . , xe) where, letting W be the divisor x1 + . . . + xe asa subscheme of C, the rank of the map V −→ H0(L ⊗OW ) is ≤ e − f .

Given S = (x1, . . . , xe) ∈ Ce we also let S denote the divisor x1 + . . . + xe on C.Given a divisor S on C we let |V |−S denote the linear system of divisors correspondingto the pullback of V to H0(C, L(−S)).

58 Math. Nachr. 245 (2002)

Lemma 5.7. Let V ⊂ H0(C, L) be a linear subspace of the space of sections ofthe invertible sheaf L on the smooth, irreducible, projective curve C. Suppose thateither V = H0(C, L) or V is a hyperplane in H0(C, L). Suppose also that V is veryample and that dimV e−1

e ≥ e − 2, where 5 ≤ e ≤ d − 1 and d = dim(V ) − 1, thendim

(V 4

5

) ≥ 3.

Proof . If the characteristic is zero, or V is complete, this is exactly [5, Corollary1.4, or Remark 1.7]. So we assume that V is a hyperplane in H0(C, L). The proofof [5, Corollary 1.4] relies on [5, Proposition 1.2] in which it is supposed that thecharacteristic is zero in order to prove the statement

(5.3) dim (|V | − 2S) ≤ dim |V | − 2(e − 2) , if S is a general element of V e−1e .

Recall that dim V e−1e ≥ e − 2 implies the existence of a component of V e−1

e dominat-ing Ce−2 by projection to the first e − 2 factors and that general S means generalin this component. To see that (5.3) holds in the case where V is a hyperplane inH0(C, L), suppose the contrary. Then for general S = (x1, . . . , xe) ∈ V e−1

e the se-quence (x1, . . . , xe−2) is general on C and we must have dim (|V |−2Si) = dim |V |−2iand

dim (|V | − 2Si+1) ≥ dim |V | − 2(i + 1) + 1 = dim∣∣H0(C, L(−2Si+1))

∣∣for some i; 0 ≤ i ≤ e−3; where Si = (x1, . . . , xi). This means that |V |−2Si, and hence|V | − 2Si+1, is composite with the Frobenius map, but it also shows that |V | − 2Si+1

is the complete linear system∣∣H0(C, L(−2Si+1))

∣∣ on C, which, as explained in [3],gives a contradiction.

5.2. The main body of the proof

Proof of the Proposition 5.3. The idea is to show that a 2–cluster on X′r,n that is

not separated by H ′ is one of the finitely many 2–clusters on C that are not separatedby

∣∣H0(OC(H ′))∣∣ and finally, that these do not lie in a curve of |E0 − F ′

n|. As a firststep we show that |H ′| is base point free and any unseparated 2–cluster of |H ′| liesentirely in C or one of the exceptional fibers.

Clearly |E0| is base point free and separates all 2–clusters on X′r,n that are not

contained entirely in one of the exceptional fibers Ei or F ′i . In particular |H ′| = |C+E0|

has no base points outside C. Since |H ′ − C| = |E0| is non–special, |H ′| induces thecomplete linear system

∣∣H0(OC(H ′))∣∣ on C . Noting that n ≥ g (see Remark 5.2), the

sheaf OC(H ′) is generic of degree H ′ . C = g +2 on C (see Lemma 2.3), one concludesthat |H ′| is non–special, has no base points in C either

(hence is base point free on

X′r,n

)and separates all but finitely many 2–clusters in C. It also follows immediately

that the subsystem C + |E0| = C + |H ′ − C| of |H ′| separates all 2–clusters that areneither contained in C, nor entirely contained in one of the exceptional curves Ei orF ′

i . The same is thus true for |H ′|.We will now show that any 2–cluster in C that is not separated by |H ′|, does not lie

in a curve of the linear system |E0 − F ′n| and does not meet the exceptional fibers Ei,

F ′i . Afterwards we will deal with the problem of 2–clusters in the exceptional fibers.


Firstly, for a fixed point p of C on Xr (i. e. before choosing the xi) and for x1, . . . , xn−1

general points of C, the linear system∣∣H0(OC(H0 −x1 − . . .−xn−1 − p))

∣∣ is a genericinvertible sheaf on C of degree g +2 so that there are only finitely many points q on Csuch that

∣∣H0(OC(H0 −x1 − . . .−xn−1 − q− p))∣∣ has base points. Applying this with

p ∈ C∩Ei, we see that∣∣H0(OC(H0−x1−. . .−xn−p))

∣∣ has no base points. IdentifyingC on Xr with its strict transform on X′

r,n, we have OC(H0 −x1 − . . .−xn) = OC(H ′)and, since |H ′| induces the complete linear system on C, |H ′| separates all 2–clustersin C meeting an exceptional fiber Ei.

Secondly, n ≥ g + 1 so that dim |x1 + . . . + xn| ≥ 1 as a linear system on C. Asx′

1 + . . .+x′n varies in the linear system |x1+ . . .+xn| on C, the unseparated 2–clusters

Z ⊂ C of∣∣H0(OC(H0 − x′

1 − . . .− x′n))

∣∣ =∣∣H0(OC(H0 − x1 − . . .− xn))

∣∣remain fixed and, consequently, for generic xi, none of the finitely many 2–clustersin C that is not separated by |H ′|, meet F ′

i and none are contained in a curve ofthe linear system |E0 − F ′

n|. The latter because any 2–cluster in C not meeting theexceptional fibers determines a unique curve in the linear system |E0| and, using thesame variation argument, x′

n varies out of any finite set of these curves.

Separating in the exceptional fibers. We are left with showing that any 2–clustercontained in an exceptional fiber, but not contained in C is separated by |H ′|. For thefibers F ′

i , or the fibers Ei with li ≤ 3, |H −F ′i | and |H −Ei| are effective, non–special

linear systems (see Remark 5.4), so that |H ′| induces a complete very ample linearsystem on the exceptional fiber.

In the remaining case of those Ei with li ≥ 4, the induced linear system is notcomplete. We will now show that |H ′| separates in these fibers as well.

We begin by noting that there are at most finitely many unseparated 2–clusters inany such Ei. For this it suffices to show that for some (general) point p of Ei, p + q isseparated for all q ∈ Ei. Choose some p ∈ Ei ∩ C. Then we have already seen abovethat if q ∈ Ei ∩C then p + q is separated. If q ∈ Ei \C then |H ′ −C|+ C = |E0|+ Cseparates p + q.

To finish the proof we will suppose that |H ′| does not separate all 2–clusters in Ei

and get a contradiction with Lemma 5.5 using Lemma 5.7.Considering C on Xr, let Λ ⊂ E2

i × Cn be the closure of the locus of points

(q1, q2, x′1, . . . , x

′n) ∈ E2

i × Cn

such that OC(H0−x′1−. . .−x′

n) is non–special and has only finitely many unseparated2–clusters in Ei, and q1 + q2 is such an unseparated 2–cluster. The projection to Cn

is then generically finite as we just showed. Let Λ be an irreducible component of Λdominating Cn and let p1, p2 be the projections of Λ to E2

i and Cn respectively. Forthe generic point ξ ∈ Λ, the scheme Λ0 = p−1

1 (p1(ξ)) has dimension ≥ n − 2 as doesits image p2

(Λ0

)in Cn. Let Q be the subscheme q1 + q2 of Xr and let V be the linear

system on C coming from the image of the map H0(IQ(H0)) −→ H0(OC(H0)), sothat V has codimension two in

∣∣H0(OC(H0))∣∣. Then, in the notation of Lemma 5.7,

p2

(Λ0

)is contained in V n−1

n and dimV n−1n ≥ n−2. Moreover, because of Lemma 5.5,

we know that V is very ample; and also∣∣H0(OC(H0))

∣∣ is very ample, as it is the

60 Math. Nachr. 245 (2002)

complete linear system induced by |H0|. Noting that 5 ≤ n ≤ dim (V ) − 1 , it followsfrom Lemma 5.7 that dim V 4

5 ≥ 3, but this contradicts Lemma 5.5.

6. Special criterion

In this section we prove the

Proposition 6.1. Let H0 = mE0 − l1E1 − . . . − l9E9 be an E–standard divisorclass on X = X9 with l9 ≥ 1 and l8 ≥ 2. We suppose n = χ(O(H0)) − 6 ≥ 0 andH0 . Cr = 3 or 4. Then |H | = |H0 − Er − . . .− Er+n| is very ample on Xr+n.

Firstly, we will prove that Proposition 6.1 results from the following proposition,then in the next section we will decompose Proposition 6.2 into it’s three constituentparts and prove each of them in the subsequent sections.

Proposition 6.2. With the notation of Proposition 6.1. Let ∆ be the union of thepoints xi = pr+i for i = 1, . . . , n. Let P be the generic pencil of curves in |H0 − C9|on X8 if l9 = 1 and on X9 otherwise. Then P is a Lefschetz pencil and for each curveD in P , OD(H0 − ∆) is very ample.

Proof of Proposition 6.1. Let H ≡ H0 −E9+1 − . . .−E9+n and let P be the genericpencil of curves in |H − C9|. Since P is Lefschetz, if l9 ≥ 2, all curves C in P areisomorphic to their image in P and if l9 = 1, the same is true for all curves in P exceptfor the unique curve that is a sum of effective divisors D + E9. By Proposition 6.2it follows that OD(H) = OD(H0 − ∆) is very ample for all curves D in P except forD +E9 (l9 = 1). Suppose that we have also proven very ampleness in this latter case,then Cr being non–special, |H | is very ample on all curves in P. As well, H − Cr isa standard, hence non–special, divisor class and H . Cr ≥ 3 so that |H | is very ampleon Cr. We then conclude with Lemma 2.2.

We must now show that for D = D + E9 (l9 = 1), OD(H) is very ample on D. Wehave the exact sequence

0 −→ H0(X, O(C9 + E9)) −→ H0(X, O(H)) −→ H0(D, OD(H)) −→ 0

where X = X9+n. Also H. D = p(D) + 3 and n ≥ p(D), so that Lemma 2.3 impliesOD(H) is very ample and |H | separates in D. By the same argument |H | separatesin E9. Finally, let Z be a 2–cluster in D + E9 that is not contained entirely in eitherD or E9. Then the trace of Z on D is a point p and its residual is a point q in E9.Clearly O(H − D) = O(C9 + E9) has no base points in E9 and we conclude that |H |separates Z as in the proof of Lemma 5.5.

The rest of the section is devoted to proving Proposition 6.2. Note that unlikeTheorem 5.1, there is no specialization of the points in this proposition.


6.1. Break down of the proof of Proposition 6.2

The notation used in the rest of the section is that of Proposition 6.2. We letD ≡ H0 −C9 and we show that the generic pencil P in |D| on X8 (l9 = 1) and on X9

otherwise, is a Lefschetz pencil and then prove Proposition 6.2 by making a dimensioncount on the variety of pairs (D, ∆′), where D is an irreducible curve in |D| with atmost nodes as singularities and ∆′ is a smooth divisor on D of degree n.

Put in precise terms, we obtain Proposition 6.2 from the following proposition andtwo lemmas as we will establish immediately after their statement.

Lemma 6.3. The generic pencil P in |D| is a Lefschetz pencil (i. e. the base issmooth, the generic member is smooth and every curve in the pencil is integral with atmost one node as singular locus).

Proposition 6.4. Let C be an integral curve with at most one node as singularlocus. Let δ ≥ p(C) and let OC(H) be an invertible sheaf on C of degree p(C) +4 + δ.Then in the open subset of Divδ(C), locus of divisors ∆ such that OC(H − ∆) andOC(∆) are non–special, the closed locus of divisors ∆ such that OC(H − ∆) is notvery ample, is in codimension ≥ 2.

Lemma 6.5. In the notation of Proposition 6.2, OD(H0 − ∆) and OD(∆) arenon–special for all curves D in P .

The following is an elementary calculation.Calculation:

D . H0 = 2p(D) + 2H0 . C9 − 2 and n = p(D) + 2H0 . C9 − 6

so that n = p(D) if H0 . C9 = 3 and n = p(D) + 2 if H0 . C9 = 4. In all casesH0 . D = n + p(D) + 4.

Proof of Proposition 6.2 Let D ≡ H0 − C9. Let a = 0 if (H0 − C9) . C9 = 4 anda = 1 if (H0 −C9) . C9 = 3, so that dim |D| = n + a + 1 and the curves of |D| throughn + a generic points of Xr form the generic pencil of |D|. Now, consider the followingdiagram

U

p0

Divn(D) × Diva(D)

p

qHilbn+a(X)

V P(H0(X, O(D))

)

where1. D is the tautological curve over P

(H0(X, O(D))

)= |D|.

2. Div is the scheme of smooth, relative Cartier divisors. (Note that points ofDivn(D) × Diva(D) are triples (C, ∆′, Q), where C is a curve in |D|, ∆′ is a smootheffective divisor of degree n on C and Q is a smooth divisor of degree a.)

3. V is the open set of integral curves having at most a node as singular locus.

62 Math. Nachr. 245 (2002)

4. U is the open subset of p−1(V ) formed by triples (C, ∆′, Q) such that OC(∆′)and OC(H0 − ∆′) are non–special.

5. q sends a divisor on a curve in |D| to the corresponding subscheme of X by thenatural map on Hilbert schemes.

Using Lemmas 6.5 and 6.3, the fiber of U → Hilbn+a(X) over the generic pointis formed by curves in the generic pencil P of |D| with n + a marked points in itsbase locus, hence is one dimensional. Let G be the closed set of U formed by thosetriples (C, ∆′, Q) for which OC(H0 − ∆′) is not very ample. By Proposition 6.4 andLemma 6.5, G has codimension ≥ 2 in all fibers over points of V , hence G has codi-mension ≥ 2 in U and does not meet the generic fiber over Hilbn+a(X). This meansthat OC(H0 − ∆) is very ample and non–special on every curve C in the pencil P .

6.2. The codimension argument

Proof of the Proposition 6.4. Let P m = Picm(C) and Pm

be respectively theJacobian and the generalized Jacobian of C in degree m and let Bp(C)+2 = P

p(C)+2 \P p(C)+2.

The closed set G of Pp(C)+2

formed by torsion free rank one sheaves that are special,is the image of Hilbp(C)−4(C) by the map that translates an ideal I by the dualizingsheaf ω of C; I → I ⊗ ω, so that G is in codimension 4.

Moreover, this map sends Hilbp(C)−4(C) \ Divp(C)−4(C) into Bp(C)+2 , so that G ∩Bp(C)+2 also has codimension 4 in Bp(C)+2.

Consider the following canonical diagram

P p(C)+4

P p(C)+4 × Hilb2(C)

p1

q

p2

Pp(C)+2 G

Hilb2(C) Hilbp(C)−4(C)

The subset of P p(C)+4 of non–special, not very ample sheaves is just p1

(q−1(G)

)which

is closed in the non–special locus. We will show that this has codimension ≥ 2, byshowing that q−1(G) has codimension four in all the fibers of p2.

Over each point of Div2(C) ⊂ Hilb2(C), q induces an isomorphism of P p(C)+4 withP p(C)+2, so that q−1(G) has codimension four in the fibers of p2 over Div2(C).

Let T 2 = Hilb2(C) \ Div2(C). The restriction of q to T 2 induces a surjection

T 2 −→−→ Bp(C)+2 .

Now by [9, Chap. III, Corollary 1.5], Bp(C)+2 = B is the only orbit of dimension≤ p − 1 under the action of P 0(C), so that the fibers of q restrained to T 2 haveconstant dimension 1. Since G∩B has codimension 4 in B, one concludes that q−1(G)has codimension 4 in fibers of p2 over points of T 2. This shows that q−1(G) hascodimension 4 in P p(C)+4 × Hilb2(C) and that its image in P p(C)+4 has codimension≥ 2 in the open, non–special locus.


Now consider the map

Divns(C) −→ P p(C)+4 ; ∆ −→ OC(H − ∆)

where Divns(C) is the open locus of non–special divisors in Divδ(C). Over the opennon–special locus of P p(C)+4, the fibers are smooth of dimension

dim (|∆|) = dim(H0(O(∆))

) − 1 = χ(O(∆)) − 1 = δ − p(C)

so that the inverse image of p1

(q−1(G)

)has codimension two in the open locus of

non–special divisors ∆ such that OC(H0 − ∆) is non–special.

6.3. The non–speciality condition

Proof of Proposition 6.5. Note that with x1, . . . , xn generic on X9 and ∆ the unionof the xi, the closed subscheme ∆ of D is a smooth divisor on all curves D in P , byLemma 6.3. For each such D we thus have the exact sequence

0 −→ ID −→ I∆ −→ I∆/ID −→ 0

which translates as

0 −→ OX9(−D) −→ I∆ −→ OD(−∆) −→ 0 .

Tensoring respectively with O(H0) and O(D−C9), we get the following exact sequences

0 −→ O(C9) −→ I∆(H0) −→ OD(H0 − ∆) −→ 0 ,(6.1)

0 −→ O(−C9) −→ I∆(D − C9) −→ OD(D − C9 − ∆) −→ 0 .(6.2)

From the exact sequence (6.1), one deduces that OD(H0−∆) is non–special becauseboth I∆(H0) and O(C9) are.

From the exact sequence (6.2), one deduces that OD(∆) is non–special. In factby Serre duality, H1(OD(∆)) = H0(OD(D − C9 − ∆)) and it suffices to show thath0(OD(D − C9 − ∆)) = 0.

Clearly h1(O(−C9)) = 0, so it suffices to show that h0(I∆(D − C9)) = 0.Now D ∈ |H0 − C9|, where H0 = mL − l1E1 − . . . − l9E9 with m ≥ l1 + l2 + l3,

l1 ≥ . . . ≥ l8 ≥ 2, l9 ≥ 1 and H0 . C9 ≥ 3. As such, if l9 ≥ 2, H0−2C9 is still in standardform, and is non–special. If m9 = 1, then H0−2C9 = m′L− l′1E1− . . .− l′8E8 +E9 andH1(O(H0 − 2C9)) = H1(O(m′L− l′1E1 − . . .− l′8E8)). Since m′L− l′1E1 − . . .− l′8E8 isin standard form H0 − 2C9 is again non–special. As such, p(D) = χ(O(H0 − 2C9)) =h0(O(H0 − 2C9)) and since n ≥ p(D), we have h0(I∆(H0 − 2C9) = 0, because ∆ is ageneric set of points on X9.

6.4. P is a Lefschetz pencil

Proof of Lemma 6.3. Firstly, |D| = |H0 −C9| is very ample on X8 if l9 = 1 and onX9 otherwise by Corollary 2.6, so the base of the generic pencil in |D| is smooth.

Let r = 8 if l9 = 1 and r = 9 otherwise. Let L = OXr (H0−C9) and F = H0(Xr , L).Let p1, p2 be the projections of Xr × Xr to the first and second factors and let I be

64 Math. Nachr. 245 (2002)

the ideal of the diagonal. Let E = p1

(p2L ⊗ I2

). Note that E is locally free of rank

dimF − 3 because L is very ample.One then has the following diagram

P(E) D

X X × P(F )

P(F )

where D is the universal divisor and P(E) is the relative singular locus of D as definedby the relative Jacobian ideal defined in Lemma 2.4. This locus is thus smooth andirreducible, and parameterizes pairs formed by a divisor of the linear system P(F ) anda singular point of this divisor. Let S be the (integral) image of P(E) in P(F ).

Let C = Cr and let D ≡ H0 − C9. Then D − C is standard = mC9 so that thegeneric curve of |D − C| is smooth and C meets the generic member of |D − C| ina smooth, irreducible locus as in the proof of 2.5. In particular the generic curve inC + |D − C| is nodal and the same is true for the generic singular curve of |D|. Thismeans that P(E) → S is generically finite and S is a hypersurface in P(F ).

Now, consider the universal family of singular curves in |D| furnished with a markedsingular point

P(E)

Cπ C = D ×P(F ) P(E)

σ

∩

P(E)

where σ is the universal singular point and π is the blowing–up of the universal curvein this point. Let V ⊂ S be the locus of nodal curves and let U ⊂ P(E) be the inverseimage of V .

Let U0 be the closed subset of U dominating the nodal curves in the fixed componentlinear system C + |D − C| ⊂ |D|. Over the generic point ξ of U0, the singular locusof C is smooth and irreducible. In fact, taking one generic point p on C, this singularlocus is just the generic divisor in the effective linear system

∣∣H0(C, OC(D −C − p))∣∣

on the elliptic curve C, where p is a marked singular point of the generic curve inC + |D − C|.

As well, if η is the generic point of U , then Cη is geometrically irreducible becauseas we saw in the proof of 2.6, the reducible curves in |D| are in codimension ≥ 2 whilethe singular ones are in codimension one. Then applying Lemma 2.4 we conclude thatthe generic fiber of C over U is geometrically smooth (i. e. the generic singular curveof |D| has a single node).


Finally, the generic line in P(F ) meets S transversally in an irreducible locus, so thatall the singular curves in the generic pencil in |H0 − C9| are generic singular curves,hence geometrically integral curves with a single node as singular locus.

Acknowledgements

Both authors would like to thank J. Alexander for his continuous support and the two

referees for their useful comments. The second author would also like to thank M. Coppens

for suggesting this problem and for the many helpful conversations.

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[6] Chauvin, S.: Systemes Lineaires de Courbes sur les Surface Rationelles Generiques, PhD The-sis, Universite d’Angers, 2000

[7] D’Almeida, J., and Hirschowitz, A.: Quelques Plongements Projectifs non Speciaux deSurfaces Rationelles, Mathematische Zeitschrift 211 (1992), 479 – 483

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[11] Harbourne, B.: Complete Linear Systems on Rational Surfaces Transactions of the AmericanMathematical Society, 289, No. 1 (1985), 213– 226

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66 Math. Nachr. 245 (2002)

[16] Manin, Y.: Cubic Forms: Algebra, Geometry, Arithmetic, North–Holland, Amsterdam, 1974

Cindy De Volder

Postdoctoral Fellow of the Fundfor Scientific Research–Flanders (Belgium)Department of Pure Mathematics and ComputeralgebraGalglaan 2B–9000 GhentBelgiumE–mail:[email protected]

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Some Very Ample and Base Point Free Linear Systems on Generic Rational Surfaces