Math. Proc. Camb. Phil. Soc. (1996), 120, 411 4 1 1

Printed in Great Britain

On the complement of a nef and big divisor on an algebraic variety

BY FRANCESCO RUSSO*

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764,70700 Bucharest, Romania

e-mail address: frusso@imar.ro

(Received 3 April 1995; revised 12 September 1995)

Introduction

Let X be an algebraic (complete) variety over a fixed algebraically closed field k.To every Cartier divisor D on X, we can associate the graded fc-algebraR(X,D) = ®%_0H°(X, 6x{nD)). As is known, for a semi-ample divisor D, R(X,D) is afinitely generated ^-algebra (see [21] or [9]), while this property is no longer true forarbitrary nef and big divisors (see [21]).

In this paper we will be interested in studying the scheme Proj (R(X,D)), e.g.conditions under which this scheme is an algebraic (quasi-projective) variety over k.The latter property is relevant especially when R(X,D) is not finitely generated overk. It turns out that such questions are closely related to the study of the complementsof divisors belonging to the linear systems \mD\, for m ^ 1.

It is well known that the complement of an effective ample divisor D on X is anopen affine subset of X. On the other hand, if D is an effective semi-ample divisor onX, then its complement is proper over an affine scheme, i.e. it is a semi-affine variety(see [11]). One shows that for a semi-afnne variety, it is not possible to extend globalfunctions in any non-trivial open immersion; varieties with this last property will bereferred to as (n— l)-valuation convex, where n = dim(X).

The main result of Section 2 is that the complement of an effective nef and bigdivisor D on X is (n— 1 valuation convex. This is proved by applying a result ofWilson concerning the boundedness of the multiplicities of the fixed components of\mD\ as m -> oo and by using the classical technique of constructing rational functionsas 'quotient of divisors'. Then we show that, if D is a nef and big divisor, everydivisor in \mD\ has semi-affine complement for every m ^ 1 if and only ifProj (R(X,D)) is an algebraic variety over k.

In the first section we fix our terminology and recall the definition of a mapintroduced by Grothendieck to define the notion of ample invertible sheaf. As anapplication, we observe that some properties (e.g. affineness, semi-affineness, notcontaining complete curves, finite generation of the ring of global functions) ofcomplements of divisors in a linear system on an arbitrary variety are linear, i.e. thedivisors whose complements satisfy one of the conditions above form themselves alinear (sub)system.

In Section 3 we treat the case of surfaces. We show that on a surface every nef andbig divisor has semi-affine complement. As a corollary we obtain that if a Cartierdivisor D on a normal complete variety X of arbitrary dimension has Kodaira-Iitaka

* Supported by grants of the Istituto di Alta Mathematica T. Severi' of Rome and of theItalian C.X.R.

412 FRANCESCO R U S S O

dimension K(X,D) = 2, then Proj (R(X,D)) is always a quasi-projective surface (ifK(X,D) =% 1, B(X,D) is finitely generated over k). The last result, originally provedby Constantinescu in [7] with algebraic methods coming from the theory ofalgebraization of schemes, refines a theorem of Zariski (see [20], [15] Section 3 andthe remarks in Section 3). To our knowledge, these are the first examples of gradedalgebras not finitely generated over k, for which Proj is of finite type over k. It wouldbe interesting to see to what extent the property of Proj (R(X,D)) being a quasi-projective surface is connected with periodicity of the function A(m), defined for msufficiently large by h°(X, d^(mD)) = P(m) + A(m), where P(m) is a quadraticpolynomial (see [8], the introduction and theorems 2 and 8). Then a geometricalrealization of Proj (R(X,D)) as an open subset of a complete algebraic space givessome interesting corollaries.

We end by showing that when K(X,D) ^ 3, Proj (R(X,D)) need not be a variety, i.e.is not of finite type over k, and by analyzing some examples. As an application of theresults, one can see that in the example of Wilson (see [19]) of a complete Gorenstein3-fold X, with canonical ring R(X) not finitely generated over k, Proj (R{X)) is a quasi-projective (canonical) variety.

1. Background material

We shall use the standard terminology of [E.G.A.]. By an algebraic variety wemean an algebraic scheme over the fixed algebraically closed field k, which is reducedand irreducible. To a pair (X, <£), consisting of an algebraic variety X and aninvertible sheaf Jz? on X, we can associate the graded ^-algebra

Denote by R(X, Jz?)+ the set of all homogeneous elements of positive degree, and set

u= U xf,feR(X,2') +

with Xf = {xeX\f(x) #= 0}. Then U is an open subset of X, and according to [E.G.A.II, 3"7-l], there is a canonical morphism

rse '•= r<e, t: u ^ P r o J (R(x>

associated with the identity i of R(X,J§?), defined by the following properties:

r2(D+(f)) = Xf for all feR(X, <£)+;the restriction of r^ toXy, r#:Xf-*D+(f), corresponds to the identity ofR(X, <(which is also identified with T(Xf,(9x )).

The morphism r^ will also be referred to as the Grothendieck map.One natural problem is to find conditions under which Proj {R(X, j£?)) is an

algebraic variety (in general it is only an integral ^-scheme).Goodman and Landman introduced the following:

Definition (see [11])- A variety X is semi-affine if there is an affine scheme Y anda proper morphism i/r:X->Y. Equivalently, X is a semi-affine if and only if thecanonical morphism n:X^> Spec (F(X, C^)) is proper.

Complement of a nef and big divisor 413

We observe that the Grothendieck map r^ is a proper morphism if and only if Xf

is semi-affine for every /ei?(X, Jz?)+.Let sf(&) {Sf{S£), <«(&), F<S(£e)) :={feH°(X,3'):Xf is affine (semi-affine, does

not contain complete curves, has coordinate ring r(Xf) finitely generated over&)} U {0} E H°{X,Se) with X an arbitrary variety.

As an application of the Grothendieck map, we shall prove:

PROPOSITION 1. The subsets s£(S£), 6^(£C), <&(£?), &&(&) are linear subspaces ofH\X,<£).

Proof. Recall that T{Xf) = R(X,&){f) = Y{D+(f)). If f,geH°(X,SC), thenf+geH°(X,£>) and D+(f+g)^D+(f)\jD+(g). So if Xf and Xg are affine, thenry.XfV Xg^D+(f) (jD+{g) is an isomorphism and so Xf+g is isomorphic to D+(f+g),whence affine.

If Xf and Xg are semi-affine, then r^ :Xf U Xg-+D+(f) U D+(g) is a proper morphismand since properness is local on the base ([E.G.A. II]), Xf+g is semi-affine.

Suppose that Xf and Xg do not contain complete curves. If Xf+g contains acomplete curve C, then r# maps C to a closed set C of D+{f+g). Then C is an affinescheme with F(C") = k, i.e. a point. This shows that C is contained in a fibre of rx andhence is in Xf or in Xg.

If F(Xf) and r(Xg) are finitely generated algebras over k, then D+(f) [)D+(g) is ascheme of finite type over k by definition and the conclusion follows from the factthat D+(f+g)zD+(f)UD+(g).

Remark. The fact that J^(£C) is linear was observed by Borelli (see [6], theorem1-1) with a completely different proof which makes use of a cohomological argumentdue to Hartshorne and Goodman. The linearity of £f{££) and (€{S£) was shown in[11], corollary 8-9, under the assumption that the variety is complete. The linearityof ^''S(^) is stated without proof in [20], section 4. We observe that the linearityof «P/(J2?), ^(JS?) and S'<3{^£) remains true on an arbitrary quasi-compact ^-scheme.

2. Nef and big divisors

We recall the following definition introduced by Wilson:

Definition (see [19], 2-1). For an invertible sheaf i f ona variety X, we say that thefixed locus of |J5?"| is numerically bounded if for every birational morphism <p:X'-+X,the fixed components of |^*(JS?")| have bounded multiplicities as n-> oo.

Wilson then showed that a nef and big invertible sheaf on a compact complexalgebraic variety has this property (see [19], 2-2) and so it is almost base point free inthe sense of Goodman (see [10]), i.e. for each point (not necessarily closed) xeX andeach e > 0, there exists n = n(x, e) > 0 and Dn e \nD\ such that vau\tx{Dn) < ne.Fujita extended this result to the class of complete normal varieties over analgebraically closed field k (see [9], 6-13). Here is an application of the result above.

PROPOSITION 2. Let X be a normal complete variety and let <£ be a nef and biginvertible sheaf onX. JffeR(X, S£ ) m , let (/)„ = 2 4 nt Yt be the Weil divisor correspondingto f and let x be the discrete valuation ring associated to the irreducible component Yt.Then for all i = 1,. . . , w there exists (f)te T(Xf) such that <j>t$&x x.

414 FRANCESCO R U S S O

Proof. Since i£ is almost base point free, for all i there exists lt and gieR(X>

such that multy(((^)0) < lt.njm. If fa :=g?/f'e T{Xf), then

multy (04) = m.multy ((gr4)0) —Zt.multy ((/)0) < m.li.nilm — li.ni < 0,

whence the assertion.

Definition (see [22], p. 88). A prime divisor of a field of functions X over k is adiscrete valuation ring (R.m) of K/k such that trdegk(R/m) = trdegk(K) — 1.

PROPOSITION 3. Le< (X, <£)beas in Proposition 2 awd letfeR(X, !£ )m. Let (R, m) bea prime divisor ofK(X), with no centre on Xf. Then there exists a complete normal varietyX' and a proper birational morphism n.X'^-X such that n is an isomorphism on Xf andsuch that R is of the first kind on X'', i.e. R ^ (3^ x. with x' eX'.

Proof. We essentially make an adaptation of an argument contained in [11],proposition 4-3. Let Y be a normal complete model of K(X) on which R is of the firstkind. Such a model exists because R m Ap, where A is a finitely generated domainover k and p is a minimal prime ideal of A of height 1 (see [22], p. 89) and henceR ~ (% y with Y the normalization of a projective closure of Y' = Spec(vl).

Then as in the proof of lemma 4-1 of [16], it is possible to patch together Xf andan open neighbourhood of y in Y in order to obtain an open immersion i :Xf c+ X, withR of the first kind on X (see also [11], proposition 4-3). By a result of Nagata (see[16]), we can suppose X complete and normal. By theorem 3-2 of [16], there existsa (normal) complete variety X' and a 'commutative' diagram

with nt isomorphisms on nJ1{Xf) (see also [13], p. 76-77). Since R is a discretevaluation ring and X a normal variety, R is still of the first kind on X' and we cantake n = n2.

Definition. A variety V of dimension n is (n— 1 valuation convex if for all openimmersion i:Vt+X with X a variety and for all peX\V of dimension n— 1, thereexists (j>eY{V) such that 4)$%^, i.e. F(V) <£ %x.

Remark. We have modified the original notion of algebraic convexity of [11],because in the proof of the implication (iii) => (iv) of proposition 43 there is an error(see also [11], 4-12iv). Indeed it is asserted (loc. cit.) that a discrete valuation ring(R,m) of a field of functions K is the local ring of a point on a normal completealgebraic 'model' of if. Unfortunately this is not true in general, because, accordingto [22] (see also [23]), one can easily construct an example of a discrete valuation ring(R,m) oiC(x,y) for which R/m m C, i.e. R is not the local ring of an algebraic 'model'of C(x,y). Take for example v to be the valuation of C((t)) associated to the discretevaluation ring C[[<]]; let x = t, y = el be an embedding of C(x, y) into C((t)) and thenlet (R,m) be the discrete valuation ring associated to the restriction of v to C(x, y).

The definition, inspired by [11], is sufficient for our applications; it is motivatedby Proposition 5 below.

Complement of a nef and big divisor 415

Definition (see [11], 4-2). Let K be a function field over k and si a &-subalgebra ofA'. Then

i^(jrf) := {varieties V:K(V) = K, T{V) = si)

with 1r{stf) partially ordered by inclusion, i.e. an open immersion.

PROPOSITION 4. Let V be a variety of dimension n. Then the following conditions areequivalent:

(i) V is (n-l)-valuation convex;(ii) V is maximal in "^"(F(F));

(iii) every prime divisor R of K(V) containing F(F) has a centre on V.

Proof, (i) => (ii). Let i: V c> X be an open immersion with X variety. Let X be theblowing-up of X along X\V. Then X\V contains (n— l)-dimensional points, so thatF(A') £ F(A') c F(V) and V is maximal in V(T(V)).

(ii) => (iii). Suppose there were a prime divisor R ofK(V), containing F(F) and withno centre on V. Let Y be a normal complete model of K(V) with respect to which Ris of the first kind. As in the proof of Proposition 3, we can construct an openimmersion r F c , X, with R still of the first kind on X. B37 eliminating some points inX\V, we can assume X\V irreducible with generic point x and 6 x cz R. SinceF(F) <=&Xix

a n d since X is normal outside V by construction, we have F(F) = T(X)whence V is not maximal in

(iii)=>(i). Suppose we have an open immersion i :Fc ,X , with X a variety andpeX\V of dimension n— 1 with T(V) c (^ x. Then (^ x is dominated by a primedivisor of K(X) with centre p outside V and containing T(V).

COROLLARY 1. Let V be a semi-affine variety, then V is (n— i)-valuation convex.Conversely, if V is (n— 1)-valuation convex and F(F) is a finitely generated k-algebra,then V is semi-affine.

Proof. The first assertion follows from the valuative criterion of properness (seealso [11], proposition 1-7). In particular in this case every valuation ring R of K(X),containing F(F), has a centre on V.

If F(F) is a finitely generated ^-algebra, by a theorem of Nagata (see [17], theorem1 or also [11], I ' l l) , there exists a commutative diagram

Spec(r(F))

with \]s a proper surjective morphism, i an open immersion and T{V) = F(X). Theconclusion follows by the maximality of V.

COROLLARY 2. / / V is (n-i)-valuation convex, then for every open immersioni: V c> X with X a variety, X\ V has pure codimension one in X.

Proof. We can suppose V and X normal. If V does not have pure codimension one,eliminating the codimension one part of X\V we would obtain an open immersioni'-.Vt+X' with codim (V,Xr) ^ 2 and hence F(F) = F(X') by the normality of X'. SoV is not (n— 1)-valuation convex.

416 FRANCESCO R U S S O

PROPOSITION 5. Let X be a normal complete variety and ££ a nef and big line bundleon X. IffeR(X,J£')m, then Xf is (n— i)-valuation convex.

Proof. We verify condition (iii) of Proposition 4. Let R be a prime divisor ofK(Xf) = K(X); we will show that, if it has no centre on Xf, then it does not containF(Xf). Since X is proper over k, R has a centre x on X. Let us suppose that x$Xf. IfR = (9,. x, then R cannot contain r(Xj) by Proposition 2. Otherwise let X' andn:X'->-X be as in Proposition 3. Then -n*{S£) is nef and big on X' andX;*(/) = TT~x{Xf) ~ Xf. We then apply Proposition 2 to (X', n*(f)) to obtain T(Xf) $ R.

COROLLARY 3. Under the same hypothesis as Proposition 5, if r(Xf) is a finitelygenerated k-algebra, then Xf is semi-affine.

REMARKS. (1) In [20], pp. 164-165, Zariski introduced the definition of 'inde-termination point' and ' base valuation' {place de base) for subrings of a field offunctions K/k of the form F(Xf), where X is a normal complete model of K/k andfeH°(X,^C), with JSfePic {X). Then the proof of Proposition 5 shows that if j£? is anef and big invertible sheaf on a complete normal variety X and/ei/0(X, j£f), thenV(Xf) has no ' base prime divisors' in K(X)/k.

(2) In Section 4 we will construct a class of examples of nef and big line bundlesfor which there exists feR(X, ^C)1, with Xf (n— l)-valuation convex and T(Xy) notfinitely generated over k, i.e. Xf is not semi-affine. These are inspired by Zariski'spaper [21] (see also [15]).

We can now prove the main result of this section:

THEOREM 1. Let X be a normal complete variety and S£ a nef and big line bundle onX. Then the following conditions are equivalent:

Proj (R(X,££)) is a quasi-projective variety over k.r# is a proper morphism.

Proof. If Proj (R(X, ££)) is a variety, then D+(f) is an affine variety and hencer(D+(/)) =R(X,!£)(/) = F(Xf) is a finitely generated ^-algebra; by Proposition 5and Corollary 2, Xf is semi-affine for every feR(X,^C)+ and then r^ is a propermorphism. If r^, is a proper morphism, then it is surjective and by Corollary 3"9 of[11] Proj (R(X, £?)) is a scheme of finite type over k and hence it is quasi-projectiveover k ([E.G.A. II], 5-3-1).

COROLLARY 4 (Wilson, [19], 1-2 and 2-3). Let X and <£ be as above. IfR(X,£?) isfinitely generated over k, then there exists n such that <£n has no base points.

Proof. Since Proj (R(X, Z£)) is a projective variety over k, by Theorem 1, r^ is aproper surjective morphism and hence U is proper over k, which implies U = X.

The converse is true in general and is due to Zariski (see [21]).

3. The case of surfaces

LEMMA 1. Let Xbea non-singular and complete variety and D an effective nef and bigdivisor onX. Then there exists a divisor D' with supp (D') = supp (D) and such that \D'\has no fixed components.

Complement of a nef and big divisor 417

Proof. Let V:= X\supp (D). I t is sufficient to construct a function 0eF(F), whichhas poles along each irreducible component Dt of supp(D) and take D' = (<p)m, thepolar divisor of <j). As in Proposition 2, for every i we can construct, ^ e F(F), with(p( having a pole along Dt, and then take <j> = 2 <f>f'e F(F) for suitable nt ^ 1 (see [10],proposition 2).

COROLLARY 5. Let X be a non-singular complete surface and let D be a nef and bigeffective divisor on X. Then X\supp(D) is semi-affine and Proj (R(X,D)) is a quasi-projective surface.

Proof. By the lemma and a well-known theorem of Zariski ([21], theorem 6-2) thereexists a divisor D' with supp(D') = supp (D) and such that \mD'\ has no base pointsfor ra sufficiently large. It follows easily that X\supp (D) is semi-affine; by Theorem 1Proj (R(X,D)) is quasi-projective.

Remarks. (1) This proof is essentially contained in [20] and [21]. In fact theorem10'1 of [21] assures that in our hypothesis \wD\ has bounded multiplicities and sothere exists a divisor D' without base points with supp (D) = supp (D') (see also [20],pp. 162-163).

(2) In [20] it is shown that for a divisor D on & surface X, R[D] = F(X\supp (D))is a finitely generated fc-algebra (algebro-geometric version of Hilbert's 14thproblem). This last observation can be translated into Proj (R(X,D)) is locally offinite type over k. Independently, Constantinescu proved that Proj (R(X,D)) is moreprecisely of finite type over k for an arbitrary divisor D on a surface X (see [7]). Infact this is a consequence of Corollary 5. Indeed, let D be a big divisor on a non-singular surface X (remember that if K(X,D) ^ 1, R(X,D) is finitely generated overk, see [21]). Let D = P+N be the Zariski decomposition of D into the nef part P andthe negative part N. If s is the least integer such that sP has integer coefficients wehave H°(&r(nsD)) = H°(C^(nsP)) for all ra 1. Hence

Proj (R(X,D)) ~ Proj (R(X,BYS)) =a Proj (R(X,sP)).

We have reduced the problem to the case in which D is nef and big; by Corollary 5we infer that Proj (R(X,D)) is a quasi-projective surface.

We shall now give a direct geometric proof of this last result in order to obtainsome interesting corollaries. Set

jrf :— {integral curves C on X: (P • C) = 0}.

By the Hodge index theorem and the Neron-Severi theorem, stf is a finite set and thematrix [(Ct • C})] is negative definite. The Artin—Grauert criterion of contraction (see[1], [2] and [12]) yields the existence of a birational proper morphism 7T:X-> Y, withY a normal complete algebraic space of dimension two, such that the connectedcomponents of JS/ are the only curves contracted to points of Y. Let

@ := {ys Y: n*(D) (= n*(P)) is not Q-Cartier}.

Then 38 is a finite set. We recall that Y is called the canonical model of (X,D) (see also[3], [4] or [18]).

PROPOSITION 6. Let X be a protective non-singular surface and let D be a big divisor

418 FRANCESCO RUSSO

on X. If Y is the canonical model of (X,D), then Proj (R(X,D)) ^ Y\&, whence it is aquasi protective normal surface with T (Proj (R(X,D)) 2; k. Moreover, Proj (R(X,D)) isprotective if and only if R(X,D) is finitely generated over k.

Proof. Let n:X-+ Y be as above, let V = Y\0& and Jzf' = (9Y{n *{sP))w for s suchthat n^(sP) is Cartier on U'. We have for all n ^ 0,

H°(U', £"n) ^ H°(Y, (9y(7T*(nsP)) ~ H°(X, (9x(nsP));

furthermore for all f'eR(U', ££'), U'f, = Yf because otherwise TT*(P) would beQ-Cartier in some point of 8ft. On the other hand by the projection formulafor Q-divisors on an algebraic normal space of dimension two (see [18]), thecomplement of Xn*(n = n'^^U'f) = n~1(Yf.) inX is the support of a nef and big divisorand by Corollary 4 it is semi-affine. Note that r(Xn*in) = V(U'f) since n^((9x) = (9Y.We obtain a commutative diagram:

n

—

where \fr and xj/' are the 'canonical' morphisms, n' is an isomorphism and n is asurjective proper morphism; then xjr' is a surjective proper Stein morphism becausexjr is proper by the semi-affineness ofX^y,,. Moreover rjr' has finite fibres since U'r, doesnot contain complete curves by construction. By Zariski's Main Theorem, ijr' is anisomorphism and so U'f- is affine for &\\ f eR(U',££")+. Theorem 4-5-1 of [E.G.A. II]shows that

Proj (R(X,D)) ca Proj (R(U',£")) ~ U U'r =

It also follows that r(Proj (R(X,D)) ^ T(Y\@) ~ F(Y) k. We can conclude theproof by observing that if Proj (R,X,D)) is projective, then 3ft is empty, n^P) is Q-Cartier and ample by definition (see [E.G.A. II], 4-5-2) and hence R(X,D) is a finitelygenerated ^-algebra.

COROLLARY 6. Let X, D and Y be as above and let Et = ~x{y^)red. with 2/i£Sing (Y).lfy^Sft, thenEt is not a fixed component of \rP\for r large and divisible. If y^Sft, thenEt is a fixed component of \rP\ for r large and divisible.

Proof. ££' is ample on V and so very ample for r sufficiently large. If y^SS we canfind a section seH0(U', j£?'r) = H°(X, &x(rsP)) such that s(yt) 4= 0. On the contrary ifyte3ft, for all r > 0 every seH°(U', £"r) = H°{Y,(9y(n*(rsP)) vanishes at yf.

COROLLARY 7 (Benveniste, see [5]). If F is a reduced fibre of n over a rationalsingularity of Y, then F is not a fixed component of \rP\ for r large and divisible.

Proof. Every rational singularity has finite divisor class group, so that n^P) isQ-Cartier near n(F).

COROLLARY 8 (Badescu, Cutkosky-Srinivas, see [4] or [8], theorem 3). If D is abig divisor on a normal complete surface defined over k = F p , then R(X,D) is a finitelygenerated k-algebra.

Complement of a nef and big divisor 419

Proof. We can suppose X non-singular. Now, every singularity of Y has torsiondivisor class group, hence TT*(D) is an ample Q-Cartier divisor on the normalprojective surface Y and R(X,D) ^ R(Y,n*(

COROLLARY 9 (Constantinescu, see [7]). Let X be a normal complete variety and jSfbe a line bundle on X ivith K(X,<£) ^ 2. Then Proj (R(X, £?)) is a quasi-projectivevariety over k, such that F(Proj (R(X, if))) ~ k. Further, Proj (R(X, J£)) is projective ifand only if R(X, i f) is a finitely generated k algebra.

Proof. We use a standard argument (see [14]) to reduce to the case when X is asmooth non-singular variety of dimension equal to K(X,£C). We let if c ; G^(D).Replacing X by the normalization X' of the blowing up of the base locus of \mD\ form large and D by its inverse image on X' we may assume that \mD\ = \D'm\ + Em, with\D'm\ a complete linear system without base points and Em an effective divisor whichis the base locus of \mD\. Since X is normal this procedure does not change the k-algebra R(X,D). Let n:X->~Pn be the morphism defined by \D'm\ and denote byi/r-.X^- W the morphism obtained from n, where W is the normalization in k(X) ofn(X) (the Iitaka fibration). Then W is normal variety of dimension K(X, ££). Let6: W -» W be the minimal resolution of singularities of W (dim (W) ^ 2). Consider therational map d'1 o\jf:X-+ W, which is defined (say) in the open subset UofX, and letFaUxW be the graph of the morphism 6~xoi]r:U^- W. Denote by X" thenormalization of the closure of F in XxW, and by f:X"-+W the morphismobtained from the projection XxW -y W'. Then/" is another representative for theIitaka fibration associated to (X, Z£). Replacing X by X" and ^£ by its pullback toX" we may assume W smooth in the original Iitaka fibration.

Set D = YiaiGi = Dvert+Dh0T for the decomposition of the (effective) divisor D,where Dvert:=Hi.ir{Ct) + waiCi and Dhor:='Lj.t/,(Cj)_wajCj. Let F be the largesteffective Q-divisor on W such that Dn^ — xJr^F) is an effective Q-divisor. Then,exactly as in [8] p. 544, one sees that for all n ^ 0

H°(X, (9x(nD)) * H°(W, &w([nF])).

It follows in particular that there exists a big divisor Dx on W such thatR(W,Dl) ~ R(X,D)(S) for some integer s ^ 1 and hence

Proj (R(X,D)) ^ Proj (R(X,D)(S)) ~ Proj (RiW^J).

If K(X, i f ) = 1, then Dl is ample and Proj (R(X, S£)) is a projective curve. If= 2, the conclusion follows from Proposition 6.

4. Examples

In this section we want to analyze some examples of schemes Proj (R(X, SC)) forpairs (X, $£), with X normal complete variety and S£ nef and big.

One may ask whether the conclusion of Corollary 9 remains true without theassumption K{X,<£) ^ 2. The first example shows that, if K(X, SC) = 3, it is no longertrue that Proj (R(X, ££)) is a quasi-projective variety.

Example 1. Let if be a nef and big line bundle on a smooth projective surfaceX, such that R(X, <£) is not finitely generated over k (see for example [21], p. 563).

420 FRANCESCO R U S S O

Let Y = Y((9X©&), JS?':=^(1) and Xo be the 'zero section' of Y. Then &' isa nef and big line bundle and R(Y,£P') ~R(X,3>)[T] with the grading givenby deg(/T") = deg(/) + n. We show that Proj (R(Y, g")) is not a variety.Indeed, if TeH°(Y,L') is the section which corresponds to Xo, thenT(YT) = r(Y\X0)~R(X,g')[T]iT)~B(X,g'). Hence D+{T) is an affine non-noetherian open subset of Proj (R(Y, <£')). It is not difficult to show that the 'cone'Proj (R(Y, $£')) is algebraic outside the vertex.

Exam/pit 2. According to Zariski (see [21], p. 562), fix a non-singular elliptic curveE in P2 (k =t= ¥p if char(k) > 0) and a set of 12 distinct points px, ...,p12 on E. Let

X = Blp p (P2)4-P2 be the blowing-up of P2 at px, ...,p12- I f / i s a line which does

not pass through any of the pt, set D = l + E, where overline means strict transform;

then D is a nef and big divisor on X, such that E is the only integral curve of X forwhich (D.C) = 0. Consider the following situations:

1. 0£(4) <g> G>B(-'2'iliPi) i s torsion in Vic°(E).2. 0£(4) <g) tfy-E^^) is not torsion in Pic°(jB).

In these two examples the ' canonical model' Y is obtained contracting the ellipticcurve E to a normal point yeY (see Proposition 6). Let n.X^-Y denote thecontraction morphism.

In the first case &?(D) ®&£ — n\k(^E(^) ® ®E( — ^«-IPi) is torsion, TT*(D) isaCartierdivisor in a neighbourhood of y and hence Y cz Proj (R(X,D)) is a projective normalsurface.

In case 2, one sees that E is a fixed component of \nD\ for all n ^ 1 and henceR(X,D) is not finitely generated over k (see [21], p. 563 or Corollary 4). Then Y is anormal algebraic space of dimension 2 proper over k and Y\y c~ Proj (R(X,D)) is aquasi-projective surface.

Example 3. According to [3], let X be a smooth projective ruled non-rationalsurface and let D = :—Kx be an anti-canonical divisor such that K(X, —KX) = 2.Then the canonical model Y of (X,D) has the following properties (see [3]):

Y is always a normal projective surface, having precisely one non rationalsingularity yeY and possibly finitely many rational singularities;n*(— Kx) = ~KY, where KY is a canonical Weil divisor on Y.

If we denote by R^^X) = R(X, —Kx) the anti-canonical ring of X, from Proposition6 it follows that Proj (R^iX)) is isomorphic to Y (if i2-1(X) is finitely generated overk), or to Y\y (otherwise). Moreover, there are simple examples of geometrically ruledsurface X = P(©B © 5£), where B is a non-singular curve of positive genus and $/? asuitable ample line bundle on B for which R~1(X) is not finitely generated over k (see[3] and [4]). In this case Y is the projective cone over the polarized curve (B, i f )obtained by blowing down the 'zero section' of X and Proj (R^iX)) is just the coneminus the vertex. We thus have an example in which Y is an algebraic'compactification' of Proj (R(X,D)).

A particular case of a result of Hartshorne allows us to construct some examplesof quasi-projective 3-fold Proj (R(X, <£?)). Let ^ n be the ideal sheaf of the base locusscheme of |JS?"|. Then we have:

Complement of a nef and big divisor 421

PROPOSITION 7. Let X be a complete variety and £C be a nef and big invertible sheafon X such that 3§n ~ 38^ for all n > 1. Then rx is a proper morphism and henceProj (R{X, ££)) is a quasi-projective variety.

Proof. Theorem 5-1 of [13] implies that in this case Xf is semi-affine for allfeR(X,£C)+ and so r^ is a proper morphism. Then Proj (R(X, S£)) is a quasi-projective variety by Theorem 1.

Example 4. In [19], Wilson constructs an example of a normal complete Gorenstein3-fold X with non-canonical singularities for which R(X) :=R(X,KX) is not finitelygenerated over k. In this example R(X) is isomorphic to R(X',D) for D a divisor ona 3-fold X' for which one has that 38n is constant. By Proposition 7 Proj (R{X)) is aquasi-projective (canonical) 3-fold.

Example 5. In [10] an example is constructed, originally due to Zarinski, of a primenef and big divisor D on a non-singular projective 3-fold X, such that X\D is affineand such that 8&n ^ Jc is constant, where C is a non-singular elliptic curve. By theProposition 7 Proj (R(X,D)) is a quasi-projective 3-fold.

I wish to thank Lucian Badescu for valuable suggestions, for his patience andencouragement and Adrian Constantinescu for interesting conversations. I wouldalso like to thank the Institute of Mathematics of the Romanian Academy for warmhospitality during the period spent in Bucharest as a fellow of the Istituto Nazionaledi Alta Matematica 'F . Severi' of Rome.

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