(de Gruyter 2002
Extending extremal contractions from an ample section
Marco Andreatta and Gianluca Occhetta
(Communicated by A. Sommese)
Abstract.LetEbe an ample vector bundle of rankron a complex projective manifoldXsuch that there exists a sectionsAGðEÞwhose zero locus,Z¼ ðs¼0Þ, is a smooth submanifold of the expected dimension dimXr. We study the problem of extending birational contractions of Zto the ambient variety proving an extension property for blow-ups and we apply our results to classify Xas above whenZis aP-bundle on a surface with nonnegative Kodaira dimension.
Key words.Vector bundle, extremal ray, Fano–Mori contraction.
2000 Mathematics Subject Classification. Primary 14E30, 14J40; Secondary 14F05
1 Introduction All through the paper we will work in the following
Setup 1.1. Let Xbe a smooth complex projective variety of dimension n andE an ample vector bundle of rank ronXsuch that there exists a section sAGðEÞwhose zero locus,Z¼ ðs¼0Þ, is a smooth submanifold of the expected dimension dimZ¼ dimXr¼nr.
A classical and natural problem is to exploit the geometric properties ofZto get information on the geometry of X; for an account of the results in caser¼1, i.e.
whenZis an ample divisor, see [6, Chapter 5]. In [3] we considered the problem from the point of view of Mori theory, posing the following question: assume thatZis not minimal, i.e. Zhas at least one extremal ray in the negative part of the Mori cone;
does this ray (or the associated extremal contraction) determine a ray (or a contrac- tion) inX, and if so, does this new ray determine the structure ofX?
Through the paper we will assume that Z is not minimal, we assume also that dimZd2; if dimZ¼1 thenZFP1and this case is treated in [13], where the prob- lem of special sections of an ample vector bundle was studied first.
We showed in [3] that there is always an extremal faceFZofNEðZÞthat determines an extremal faceFX ofNEðXÞ; now we slightly improve our results, proving that, if N1ðZÞFN1ðXÞ(which is always true if dimZd3, see 2.8) then there is always an
extremal faceFZ ofNEðZÞwhich coincides with an extremal face ofNEðXÞand in this case we say that the face isliftable toX; this is the context of Theorem 3.2 and Corollary 3.4.
Note that, a priori, the liftability of a face does not imply the extendability of the associated contraction; namely the contraction associated toFX onXrestricted toZ is not necessarily the contraction associated toFZ(see Remark 3.5); if this is the case we say that the face isextendable. IfFZcorresponds to a fiber type contraction onZ, thenFZ is extendable and the contraction of the face FX in Xis again a fiber type contraction [3, 3.12 and 3.13].
In this paper we study the extendability problem for birational contractions: a gen- eral result is given in 3.8. Then we prove an extension property for blow-ups, namely:
Theorem 1.2.Let RZbe an extremal ray on Z,whose associated contractionj:Z!Z0 is the blow-up of a smooth subvariety C of codimension md3.
If RZis liftable to RX then it is extendable;moreover,iff:X !X0 is the contrac- tion associated to RX,then X0is smooth,Z0is isomorphic to a subvariety of X0,andf is the blow-up of C.
A straightforward corollary of this theorem is the following
Corollary 1.3.Assume thatj:Z!Z0is the blow-up of a smooth minimal variety(KZ0
is nef)along a smooth subvariety of codimension md3.Then X is the blow-up of a smooth variety X0along a smooth subvariety of codimension mþr.
Another application of the above theorem concerns the problem to determine whether a blow-up of a projective spacePnralong a linear subspace can be the zero locus of a section of an ample vector bundle in a projective manifold. We discuss this problem in the second part of Section 4.
We finally apply our results to prove the following
Theorem 1.4. Assume that Z is a Pd-bundle on a surface S of nonnegative Kodaira dimension.Then X is aPrþd-bundle on S.
In the case r¼1 this theorem, together with previous existing results, completes the proof of a long lasting general conjecture by A. Sommese forn¼4, see [6, Con- jecture 5.5.1]; the same theorem was proved in a di¤erent way (only for the caser¼1 andn¼4) recently in [16].
We would like to thank the referee for remarks and suggestions which improved the exposition of the results in the paper. We are both partially supported by grants of the MURST.
2 Notations and preliminaries
We use the standard notation from algebraic geometry, in particular it is compatible with that of [11] and of [12]. This paper is a sequel of [3] to which we refer constantly.
In the paperXwill always stand for a smooth complex projective variety of dimen- sionn andKX will be its canonical divisor; the famousCone Theoremof Mori says that the closure of the cone of e¤ective 1-cycles into the real vector space of 1-cyles modulo numerical equivalence,NEðXÞHN1ðXÞ, is polyhedral in the part contained in the setfZAN1ðXÞ:KX:Z<0g. An extremal faceF(orFX) ofXis a face of this polyhedral part; an extremal ray is a face of dimension 1. To every extremal face is associated a morphism to a normal variety; namely we have the followingBase point free theoremof Kawamata and Shokurov.
Theorem 2.1. Let X and F be as above. Then there exists a projective morphism j:X!W from X onto a normal variety W which is characterized by the following properties:
i) For an irreducible curve C in X,jðCÞis a point if and only if the class of C is in F.
ii) jhas only connected fibers.
Definition 2.2. The map j of the above theorem is usually called the Fano–Mori contraction(or theextremal contraction) associated to the faceF. A Cartier divisorH such thatH ¼jðAÞfor an ample divisor onWis called agood supporting divisorof the mapj(or of the faceF).
The contraction is of fiber type if dimW<dimX, otherwise it is birational. We usually denote with E¼EðjÞ:¼ fxAX :dimðj1jðxÞÞ>0g the exceptional locus ofj; ifjis of fiber type then of courseE:¼X.
Remark 2.3.Note that a good supporting divisor for a Fano–Mori contraction is of the form H¼KXþrL, whereris a positive integer andLis an ample line bundle.
In fact if H is a good supporting divisor then HKX is an ample line bundle by Kleiman’s criterion.
On the other hand note also that any nef but not ample line bundleHof the form H ¼KXþrL, with r a positive integer and L an ample line bundle, defines (or is associated to) an extremal faceF :¼ fZANEðXÞ:H:Z¼0g.
Example 2.4.Letj:X :¼BlYðX0Þ !X0be the blow-up of a projective manifoldX0 along a submanifold YHX0 of codimension m; let also EHX be the exceptional divisor. This is a Fano–Mori contraction and a good supporting divisor for this con- traction isH¼KXþ ðm1ÞL, whereL¼ EþjðAÞfor an ample divisorAonX0. Let Wbe a projective manifold and letGbe a rankmvector bundle onW; then j:X:¼PðGÞ !W is a Fano–Mori contraction, called a P-bundle contraction or a (classical) scroll over W; to avoid possible confusion, we will use always the first denomination. If xG denotes the tautological bundle onX then a good supporting divisor for the contraction jisH ¼KXþmL, whereL¼xGþjðAÞfor an ample divisorAonW.
More generally a fiber type contraction j:X!W of a projective manifold X onto a normal projective variety W supported by a divisor of the type H¼KXþ ðdimXdimWþ1ÞL, withL an ample line bundle, is a Fano–Mori contraction
and it is called an adjunction theoretic scroll. In a neighborhood of a generic fiber an adjunction theoretic scroll is a P-bundle (this is a theorem of Fujita); however there can be special fibers of dimension greater thanðdimXdimWÞ.
Another important result of Mori, see [15], is the existence of rational curves in the extremal rays. Namely ifXhas an extremal rayRthen there exists a rational curve Con Xsuch thatR¼R½Cand 0<KXCcnþ1. Such a curveCis called an extremal curve.
Definition 2.5.LetRbe an extremal ray onX. We define the positive integerlas l¼lðRÞ:¼minfKX:C:Cis a rational curve inRg:
l is called the length of the ray while a rational curve C in the ray R such that l¼ KX:Cis called aminimal extremal curve.
The importance of this integer comes from the following proposition, proved by Ionescu and Wisniewski.
Proposition 2.6([19]).Letjbe an extremal contraction associated to the extremal ray R;let S be an irreducible component of a non-trivial fiber ofj.The following formula holds
dimSþdimEðjÞddimXþlðRÞ 1:
The zero locus of a section of an ample vector bundle has a lot of good properties, we will frequently use the following two:
Proposition 2.7([3, 2.18]).Let X and Z be as in1.1and let Y be a subvariety of X of dimensiondr.ThendimZVYddimYr.
The other is a very strong result, a Weak Lefschetz type theorem for ample vector bundles, proved by Sommese in [17] and subsequently with slightly weaker assump- tions in [13].
Theorem 2.8.Let X,Eand Z be as in1.1and let i:Z,!X be the embedding.Then (2.8.1) HiðiÞ:HiðX;ZÞ !HiðZ;ZÞis an isomorphism for icdimZ1.
(2.8.2) HiðiÞis injective and its cokernel is torsion free for i¼dimZ.
(2.8.3) PicðiÞ:PicðXÞ !PicðZÞis an isomorphism fordimZd3.
(2.8.4) PicðiÞis injective and its cokernel is torsion free fordimZ¼2.
The following lemma is probably well known but we will provide anyway a proof for the interested reader, since we did not find a good reference for it.
Lemma 2.9. Letj:X !X0 be the blow-up of a smooth variety along a smooth sub- variety YHX0with exceptional divisor EðjÞandEarankr vector bundle on X.Assume thatEFFlrOPð1Þfor every fiber ofj.Then there exists arankr vector bundleE0on X0such that
EnEðjÞ ¼jE0:
Proof. The vector bundle EE~:¼EnEðjÞis trivial along any fiber of j. We have to prove that fðEEÞ ¼:~ E0is a locally free sheaf of rankr. This is a local question at any point yAY, and we can apply the Theorem on Formal Functions, see [10, Theorem III.11.1], which says that
fðEEÞ~y ¼lim H0ðFn;EE~nÞ
(with Fn¼XYSpecðOy=mynÞ). SinceEE~is trivial on F¼j1ðzÞ there arerlinearly independent non-zero sections ofEE~jF ¼FCr; the same proof of the Castelnuovo criterion for blow-up, as for instance in [5, Proposition 2.4] or [10, Theorem V.5.7], gives that the sections actually extend tornon-zero and linearly independent sections of the limH0ðFn;EE~nÞ(i.e. they extend in a formal neighbourhood ofF). This gives that fðEEÞ~yis locally free of rankr. r
In the setup of the previous lemma, it is useful to find conditions which ensure the ampleness ofE0:
Lemma 2.10.In the situation of the previous lemma ifEis ample and Y is a point then also E0 is ample; the same is true if E is ample and there exists a surjection NY ! L!0with L a nef line bundle on Y.
Proof. The first part has been proved in [14, Lemma 5.1]. To prove the second part we will apply [7, Lemma 5.7]; let E¼PðNYÞ be the exceptional divisor of j and consider the following diagram:
PðE0Þ jj~ PðjE0Þ
p1
??
?y
??
?yp
X0 j X
wherejj~is the blow-up ofPðE0Þalongp11 ðYÞ. LetxE0be the tautological line bundle ofPðE0Þ; we have
j~
jxE0 ¼xjE0¼xEnOðEÞ¼xEþpE so thatjj~xE0pE¼xE is ample onPðjE0Þ ¼PðEÞ.
MoreoverxE0 is ample on p11 ðYÞ. In fact, letY0HE be the section correspond-
ing to the surjectionNY ! L!0;Y0 is mapped isomorphically ontoYbyj and EY0 ¼LY0, hence we have
ðEnLÞjY0¼ ðEnEÞjY0 ¼jEjY0 0 ¼EjY0
andE0is ample onY. We can thus apply the quoted lemma to get the ampleness of xE0. r
3 Lifting of birational contractions
LetX;EandZbe as in 1.1; in this section we will improve some general results in [3];
we start with a definition which was not stated there.
Definition 3.1.Assume thatN1ðZÞFN1ðXÞ, which is always the case if dimZd3 b y Theorem 2.8, and letFZ be an extremal face inNEðZÞ. If under the above identifi- cationN1ðZÞFN1ðXÞthe faceFZis an extremal faceFXinNEðXÞ, then we will say that the faceFZisliftabletoFX.
The following is a refinement of [3, Theorem 3.4]:
Theorem 3.2(Lifting of extremal faces).Assume that Z is not minimal in the sense of Mori theory,i.e. KZ is not nef. Let FZ be an extremal face of Z, DZ¼ ðKZþtHZÞ a good supportingQ-Cartier divisor of FZ and H the line bundle on X which restricts to HZ.If H is ample on X,theQ-divisor D¼KXþdetEþtH is nef,not ample and defines an extremal face FX of X.
Moreover,if N1ðZÞFN1ðXÞ,FZis liftable to FX.
Proof.The first part of the theorem has been proved in [3, Theorem 3.4], so we have to prove only the last assertion. Since under the identification N1ðZÞFN1ðXÞ we have NEðZÞHNEðXÞ andNEðZÞK<0HNEðXÞK<0 it is enough to show that, for every extremal rayRX in the faceFX, there is a curve inRX lying onZ.
Let RX be an extremal ray of FX and jR:X!T the associated extremal con- traction; since the contraction ofFX is supported byKXþdetEþtH, this divisor is zero on the curves inRX, yieldinglðRXÞdrþt; if the contraction is birational, then, using 2.6, for a non-trivial fiberFofjR, dimFdrþt, hence dimFdrþ1 and we are done by 2.7.
In the same way we get our result if the contraction is of fiber type and has a fiber of dimension rþ1; so we are left with the case of an equidimensional fiber type contraction withr-dimensional fibers; note that in this case, thejR-ample line bundle Hhas intersection number one with the extremal rational curve generating the ray by [3, Proposition 2.7], so that, letting H0¼HþjRA, withAample onT, the divisor KXþ ðrþ1ÞH0 is a good supporting divisor forjR, which, by [8, 2.12] is thus a P- bundle contraction.
As in the proof of the first part of [3, Theorem 3.4], we get thatZis a regular sec- tion of this bundle, a contradiction with Theorem 2.8. r
Proposition 3.3.Assume that N1ðZÞFN1ðXÞand that the extremal face FZis liftable to FX.Denote byjandfthe contractions associated to FZand FX;let KZþtHZbe a good supporting divisor for FZand let H be the divisor on X which restricts to HZ.
Then,up to replacing H with H0¼HþfA,with Aa su‰ciently ample line bundle, we can assume that that H0is ample on X and thatjis supported by KZþtHZ0. Proof. The line bundle H isf-ample and thus H0 is ample. Moreover KZþtHZ0 ¼ KZþtHZþtðfAÞZis a good supporting divisor ofFZsincetðfAÞZis nef and it is zero on the curves ofFZ.
Proposition 3.4.If Z is not minimal there exists at least one extremal face FZwhich is liftable to X.
Proof. Let Lbe an ample line bundle onX; the restriction of this line bundle toZ, LZ, is ample onZ, so, ifKZ is not nef there exist a rational numbers>0 such that the divisorKZþsLZ is nef but not ample and it defines an extremal faceGZ. This face satisfies the assumptions of Theorem 3.2 and so it is liftable to an extremal face GX. r
Remark 3.5. Let us note that, a priori, the fact that an extremal face of NEðZÞ is liftable to an extremal face of NEðXÞdoes not imply that the restrictionfZ of the extremal contraction f associated to FX coincides with the extremal contraction j associated toFZ; as explained in [3], we have a commutative diagram
X i Z
f
??
?y fZ
??
?yj
Y p W
ð3:6Þ
wherep:W!fZðZÞis a finite morphism.
To complete the lifting process, we introduce the following definition:
Definition 3.7. In the above notation, ifp is a isomorphism onto its image, that is if the restrictionfZcoincides with the extremal contractionjofFZ, then we will say that the faceFZ, or the associated contractionj, isextendable.
In [3] we proved that if FZ is a liftable face associated to a fiber type contraction then it is extendable and moreoverpis the identity. Now we will deal with birational contractions.
Proposition 3.8. Assume that there exists an extremal ray RZ, whose associated con- traction, j:Z!W, is birational, which is liftable to an extremal ray RX. We can assume thatjis supported by KZþtHZwith td1 (Remark2.3)and that HZ is the restriction of an ample line bundle H on X(Proposition3.3).
If t>1 then f is birational, fZ has connected fibers and p:W !fZðZÞ is the normalization morphism.Ift¼1thenfcan be either birational or of fiber type;in the first casefZ has connected fibers and p:W!fZðZÞis the normalization morphism while in the secondfis an adjunction theoretic scroll contraction onto W(see2.4)and RZis extendable.
Proof.Ift>1 the proof is as in [3, Propositions 3.13, 3.14], observing that the inter- section ofZwith any non-trivial fiberFoffhas dimension dimðZVFÞd1.
Ift¼1, by Theorem 3.2 the contractionfis supported byKXþdetEþH, and so its lengthlðfÞisdrþ1.
If f is birational then again the proof of [3, Propositions 3.13] applies since dimðZVFÞd1.
Iffis of fiber type, by Inequality 2.6 we have that all its fibers have dimensiondr;
if the generic fiber has dimensiondrþ1 then it has non-trivial intersection withZ, and this is impossible as in the proof of [3, Proposition 3.14]. So the generic fiber off isr-dimensional,lðfÞ ¼rþ1 and, ifCis a minimal extremal rational curve in a fiber offwithKX:C¼lðfÞ, thenH:C¼1 and detE:C¼r. In particularKXþ ðrþ1ÞH is a good supporting divisor forf, which thus is an adjunction theoretic scroll.
We have to prove now thatRZis extendable; for this we will first prove thatfZhas connected fibers and then thatfZðZÞis normal. SincefZ contracts only the curves whose numerical class is inRZ, outside of the exceptional locusEðjÞfZis finite-to- one; in particular, if f is a fiber offwhich does not contain curves ofEðjÞthen f is r-dimensional, and thus is a projective spacePr.
Since detE:C¼r for a minimal extremal rational curve, for every line in f, ðdetEÞlFOP1ðrÞ, Ef ¼lrOð1Þ, andZVf is one point, thus it is connected (note that we have proved that, outside off1ðfðEðjÞÞÞfis a projective bundle andZis a regular section).
On the other hand, the non-trivial fibers offZare connected since they are inter- sections of Z with fibers of f and [3, 3.13] applies again. Thus fZ has connected fibers andfZðZÞ ¼fðXÞ, which is normal; sop:W !fZðZÞis the identity andRZ is extendable. r
Example 3.9.Let us note that the last case of the above proposition is e¤ective: let X ¼P2P1andZb e aF1-surface in the linear systemOPPð1;1Þ; the contraction of theð1Þcurve ofZlifts to theP-bundle contraction ontoP2.
Proposition 3.10. In the setup of the above proposition ifjand fare both birational, then EðjÞ ¼EðfÞVZ.
Proof.IfxAEðjÞ, then there exists a curveCHZwhich containsxand is contracted byj; but, onZ,jandfZ contract the same curves, thereforexACis contained in EðfÞ.
On the other hand, if xAEðfÞVZ we consider the unsplit family Vof deforma- tions of a minimal extremal rational curve contracted by f (see [12, IV.2]). If LocusðV;0!xÞdenotes the locus of the curves inVwhich pass throughx, by [12,
IV.2.6], dim LocusðV;0!xÞdrþt, hence dimðLocusðV;0!xÞVZÞd1, so that xlies in a curve contracted byf, and so byj. r
Remark 3.11.Actually, the proof of the last proposition shows that the fibers ofjare exactly the intersections of the fibers offwithZ.
4 Blow-ups
Proof of Theorem 1.2. Let DZ¼KZþ ðm1ÞHZ be a good supporting divisor of j:Z!Z0, whereHZis an ample line bundle onZwhich restricts toOPð1Þon every non-trivial fiber ofj. By the Proposition 3.3 we can assume that the extension ofHZ
toX, namelyH, is ample.
Let as usualfbe the contraction associated to the rayRXto whichRZis liftable; it is supported byD¼KXþdetEþ ðm1ÞH and, by 3.8, it is birational.
The non-trivial fibers offhave dimensioncrþm1 by Proposition 2.7 (see also Remark 3.11); on the other hand, by Proposition 2.6 the dimension of any non-trivial fiber is exactlyrþm1 andlðfÞ ¼rþm1.
We can apply [2, Theorem 5.2] to deduce that f:X !X0 is the blow-up of a smooth subvariety of codimension rþm1. Let us point out also that the restric- tion of detEto every line in a fiber offisOP1ðrÞ, and soEsplits on the fibers off aslrOPð1Þ; thus by Lemma 2.9, we have thatEn½EðfÞ ¼fE0.
We want to prove now thatZ0!X0is a closed embedding, that is thatRZis ex- tendable. For this, in the spirit described in the introduction of the paper [4] we now consider a local situation: choose a pointzAfZðEðjÞÞ, an a‰ne neighbourhoodUof zinX0and consider the restrictions offandjto the inverse images ofU; to simplify the notation denote again byX;X0andZthe new spaces and byf;fZandjthe re- stricted maps.
In this a‰ne situation and in the notation of the Lemma 2.9 we have thatE0is trivial and in particularEsplits aslrL, whereL¼ EðfÞ; note thatKXþ ðrþm1ÞLis a good supporting divisor forf. We will now use the horizontal slicing procedure ([4, Lemma 2.6]): let Li, withi¼1;. . .;r, be general smooth sections ofLand let Xi¼ 7j¼1;...;iLj; note thatX0¼X and thatXrFZ; we have a chain of surjections
H0ðX;DÞ !!H0ðX1;DX1Þ !! !!H0ðZ;DZÞ
and this implies (see the proof of [4, Lemma 2.6]) that p:Z0!X0 is a closed em- bedding. r
As mentioned in the introduction we have also the following application.
Proposition 4.1.Assume that Z is the blow-up of a projective spacePnralong a linear space Y of codimension md3.Then X is a projective bundle onPm1;namely g:X ¼ PðGÞ !Pm1for some vector bundleGonPm1with
0!V!G!lnrm1OPlOPð1Þ !0 andE¼xGngV,wherexGis the tautological line bundle ofG.
Proof.The varietyZhas two extremal rays: the blow-down contraction toPnrand a P-bundle contraction onPm1; by Proposition 3.4 one of these rays is liftable toX; if the fiber type ray is liftable, then we are done by [3, Corollary 4.2]; as stated in that corollary the existence of the sequence is a known fact about vector bundles (see for instance [9, B.5.6]).
We will now show that the birational ray cannot be liftable. Suppose, by contra- diction, that this is the case: by Theorem 1.2 we have that the associated contraction is extendable to a contractionf:X !X0which gives a commutative diagram
X i BlYðPnrÞ
f
??
?y j
??
?y
X0 j Pnr
We also know thatEE¼lrOð1Þ; hence there exists a vector bundleE0 onX0 such thatE¼fE0nðEÞ; this vector bundle is ample, by Lemma 2.10, andPnris the zero locus of a section of it. This implies, by [13, Theorem A], thatX0is a projective space Pn and E0 decomposes as lrOPnð1Þ, but this contradicts the ampleness of E. r
Remark 4.2. Let us note that a blow-up of a projective space Pnr along a linear space Yof codimensionmd3 cannot be an ample section of a line bundle or of a vector bundle which is a direct sum of line bundles; this follows from [7, Proposition 5.8]. Therefore there exists no example for the above proposition ifEis a line bundle (that is ifVis a line bundle), or a direct sum of line bundles.
For general vector bundlesE however this can happen, as the following example will show.
Example 4.3. On X¼PkP2¼Pðlkþ1OP2Þ, with kd3, consider the line bun- dles p1ðOðaÞÞnp2ðOðbÞÞ ¼:ða;bÞ, where pi are the projections. For mg0, let D1Ajð1;mÞjandD2Ajð1;mþ1Þjbe su‰ciently general divisors. They correspond to sections oflkþ1OP2ðmÞandlkþ1OP2ðmþ1ÞonP2 and therefore they will give an injective morphism of vector bundles, with cokernelV:
0!OP2ðm1ÞlOP2ðmÞ !lkþ1OP2!V !0:
We notice thatVis actually a vector bundle since, ifkd3, the two sections can be taken linearly independent at each point ofP2.
Moreover we have thatVis ample; in fact the tautological bundle ofPðVÞis the restriction ofx¼p1ðOð1ÞÞ, the tautological bundle ofX, toPðVÞand therefore our claim follows if we show that the restriction ofp1toPðVÞis a finite-to-one map onto Pk, by a general choice of the sections. SincePðVÞ ¼D1VD2, this can be proved applying twice the next lemma; the first time to p1:X !Pk and L¼ jð1;mÞj, the second time to p1:D1!Pk and L¼ jð1;mþ1ÞjD1. Note in fact that, for mg0, p1ðð1;mÞÞ ¼Smðl3OPkÞð1Þis a spanned vector bundle onPk of rank>k.
Lemma 4.4.Let p:X !Y be a flat morphism of projective manifolds and let L be an ample and spanned line bundle on X.Suppose moreover that pL is spanned by global sections on Y and that rankðpLÞis bigger than dimY.Then the restriction of p to a general DAjLjis equidimensional,hence flat.
Proof.It is enough to show thatDmeets any fiber of the mappproperly. In factDis ample and therefore it meets any fiber; if a fiber p1ðxÞ is contained in Dthen it means that the section corresponding toDin pLwill vanish at the pointx, but this is impossible since the assumptions imply that a general section ofpLdoes not vanish anywhere. r
Now, dualizing the sequence we have constructed onP2 and twisting it byOðmÞ we get
0!VðmÞ !lkþ1OP2ðmÞ !OP2ð1ÞlOP2!0:
If we setV:¼VðmÞandG:¼lkþ1OP2ðmÞthen this is a sequence as in Prop- osition 4.1: in factxGnp2V¼p1Oð1Þnp2V is an ample vector bundle.
Let therefore X ¼PðGÞ ¼Pðlkþ1OP2ðmÞÞ, gð¼p2Þ:X!P2 and E¼ xGngV. ThenEis an ample vector bundle onXwith a sections, which corresponds to the composite of the duals of the canonical mapgðGÞ !xGand ofgV!gG, whose zero locus isZ:¼PðOP2ð1ÞlOP2Þ, which is the blow-up ofP3 at a point.
5 Pd-bundles on surfaces withkI0 and their Mori cone
Proposition 5.1. Let p:Z!S be a Pd-bundle over a smooth surface such that kðSÞd0;assume that Z has an extremal ray R di¤erent from the one associated to the P-bundle contraction.Then the associated contractionjRis a blow-downjR:Z!Z1
of a divisor E¼p1ðCÞ,with C an exceptionalð1Þ-curve on S,such that EFP1Pd and EEFOð0;1Þ.
Moreover,Z1has aPd-bundle structure on S1,where S1is the surface obtained con- tracting the exceptional curve C on S,andjRðEÞis a fiber of p1:Z1!S1.
Proof.Suppose thatZhas an extremal ray,R, di¤erent from the bundle contraction;
there exists a rational curveC0 (½C0AR) such thatKZ:C0 >0 and pðC0Þis not a point. LetC¼pðC0Þ, letn:P1 !Cbe the normalization ofCand consider the fiber product
ZSP1 n! Z
p
??
?y
??
?yp P1 n! S
ð5:2Þ
The map p:ZC:¼ZSP1!P1 is aP-bundle onP1; letC0 be a minimal section ofp; the proof of [18, Lemma 1.5] applies and we getKS:C<0.
Since onSthere is only a finite number of curves which have negative intersection withKS (theð1Þ-curves) we deduce that the image of the exceptional locus ofjRis C, andCis að1Þ-curve.
Moreover, since the fibers of di¤erent extremal contractions can meet only in points, we have that all the fibers ofjRhave dimension one; combining these facts we get that jRis a divisorial contraction. By [1]jRis a smooth blow-down contraction.
The exceptional locus ofjR,E, is thus p1ðCÞand carries two di¤erent P-bundle structures, it is so forced to beP1Pd; the description ofEEis clear observing that the lines in one ruling are extremal curves for the blow-up, while those in the other ruling are contained in fibers of the bundle projection.
LetFbe a rankdþ1 vector bundle onSsuch thatZ¼PSðFÞ; the restriction of FtoCis, up to twist, ldþ1OP1; therefore if we denote bys:S!S1 the contrac- tion ofC, by Lemma 2.9 there exists a rankdþ1 vector bundleF1 onS1 such that F¼sF1. Consider the commutative diagram
Z¼PðFÞ s! Z1¼PðF1Þ
p
??
?y p1
??
?y
S s! S1
The map s is a good contraction which contracts exactly the curves in R, so it coincides withjR. r
6 Pd-bundles on surfaces as ample sections
Proof of Theorem1.4. By Proposition 5.1 the extremal rays of Zare the ray corre- sponding to thePd-bundle fibration and, possibly, other rays of birational type; such a ray corresponds to a blow-downb:Z!Z1 which contractsPdP1 toYFPd.
By Proposition 3.4 we have that at least one extremal ray ofZis liftable toX; if this ray is the fiber type one, then, by [3, Corollary 4.2]Xis aPrþd-bundle onSand we are done.
Suppose now that the ray that is liftable is a birational one, corresponding to a blow- down b:Z!Z1; by Proposition 5.1Z1 has a Pd-bundle structure over a smooth surfaceS1, p1:Z1!S1, obtained contracting að1Þ-curve ofSto a points1.
Nowb is supported by KZþHZ(e.g. takingHZ¼ EðbÞ, whereEðbÞis the ex- ceptional divisor ofb) and, by Proposition 3.3 we can assume that the line bundleH which restricts toHZis ample onX. By Proposition 3.8 ifbis liftable to a fiber type ray, f, then f is a scroll contractionf:X !Z1; the proof of Proposition 3.8 also shows that, outside of f1ðfðEðbÞÞÞ fis a projective bundle andZis a regular sec- tion.
Choose a smooth non-rational curveBinS1which does not contains1;T¼p11 ðBÞ is aP-bundle onB, and it is not contained infðEðbÞÞ. Denote byUthe inverse image ofTviaf,U¼f1ðTÞ;Uis aP-bundle onTandZVUis a regular section. Therefore ZVUis isomorphic toTand thusrðZVUÞ ¼2; on the other handZVUis the zero
locus of a section of the ample vector bundleEU, thus, by Theorem 2.8rðZVUÞd3, a contradiction.
Sob is liftable to a birational ray, corresponding to a contractionb:X !X1; b y the Theorem (1.2)bis extendable andbis a smooth blow-up ofY1HX1, such that the restriction ofEðbÞtoZisEðbÞ; thus by Lemma 2.9 there exists a vector bundleE1on X1such that
EnOXðEÞ ¼PE1
and moreover, by Lemma 2.10,E1is ample.
Summing up, we have replaced the starting triple ðX;E;ZÞ with a new triple ðX1;E1;Z1Þsatisfying the assumptions of the theorem and such thatrðZ1Þ ¼rðZÞ 1.
Thus we can repeat the above procedure, i.e. one of the extremal contractions ofZ1is liftable. Since rðZÞis finite, at some point of this process we must find some triple ðXk;Ek;ZkÞsuch that thePd-bundle contraction ofZk is extendable to aPrþd-bundle contraction ofXk andEkjFFlrOð1Þfor every fiber of the bundle contraction.
Let bk:Xk1!Xk be the last blow-down contraction and let F be the fiber of the Prþd-bundle contraction of Xk which contains Yk, the center of bk (which is FVZkFPd). Letl0be a line inFwhich meetsYk transversally and letlbe its strict transform inXk1. We have
ðEk1ÞlFðbkEkÞlnOðEkÞlFðlrOP1ð1ÞÞnOP1ð1ÞFlrOP1;
contradicting the ampleness ofEk1. r
References
[1] T. Ando, On extremal rays of the higher-dimensional varieties.Invent. Math.81(1985), 347–357. MR 87g:14045 Zbl 0554.14001
[2] M. Andreatta, G. Occhetta, Special rays in the Mori cone of a projective variety. Preprint AG/0010101.
[3] M. Andreatta, G. Occhetta, Ample vector bundles with sections vanishing on special varieties.Internat. J. Math.10(1999), 677–696. MR 2000g:14024 Zbl 01629311 [4] M. Andreatta, J. A. Wis´niewski, A note on nonvanishing and applications.Duke Math. J.
72(1993), 739–755. MR 95c:14007 Zbl 0853.14003
[5] M. Andreatta, J. A. Wis´niewski, A view on contractions of higher-dimensional varieties.
In:Algebraic geometry—Santa Cruz 1995, 153–183, Amer. Math. Soc., Providence, RI 1997. MR 99f:14020 Zbl 0948.14014
[6] M. C. Beltrametti, A. J. Sommese,The adjunction theory of complex projective varieties.
Walter de Gruyter & Co., Berlin 1995. MR 96f:14004 Zbl 0845.14003
[7] T. Fujita, On the hyperplane section principle of Lefschetz.J. Math. Soc. Japan32(1980), 153–169. MR 81c:14005 Zbl 0422.14004
[8] T. Fujita, On polarized manifolds whose adjoint bundles are not semipositive. In:Algebraic geometry, Sendai, 1985, 167–178, North-Holland 1987. MR 89d:14006 Zbl 0659.14002 [9] W. Fulton,Intersection theory. Springer 1984. MR 85k:14004 Zbl 0541.14005
[10] R. Hartshorne,Algebraic geometry. Springer 1977. MR 57a3116 Zbl 0367.14001
[11] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem.
In: Algebraic geometry, Sendai, 1985, 283–360, North-Holland 1987. MR 89e:14015 Zbl 0672.14006
[12] J. Kolla´r,Rational curves on algebraic varieties.Springer 1996. MR 98c:14001 Zbl 0877.14012
[13] A. Lanteri, H. Maeda, Ample vector bundles with sections vanishing on projective spaces or quadrics.Internat. J. Math.6(1995), 587–600. MR 96d:14039 Zbl 0876.14027 [14] A. Lanteri, H. Maeda, Geometrically ruled surfaces as zero loci of ample vector bundles.
Forum Math.9(1997), 1–15. MR 97i:14027 Zbl 0876.14026
[15] S. Mori, Threefolds whose canonical bundles are not numerically e¤ective.Ann. of Math.
(2)116(1982), 133–176. MR 84e:14032 Zbl 0557.14021
[16] E. Sato, Z. Yicai, Smooth 4-folds which contain aP1-bundle as an ample divisor.Manu- scripta Math.101(2000), 313–323. MR 2001a:14043 Zbl 01441728
[17] A. J. Sommese, Submanifolds of Abelian varieties. Math. Ann. 233 (1978), 229–256.
MR 57a6524 Zbl 0381.14007
[18] M. Szurek, J. A. Wis´niewski, Fano bundles overP3andQ3.Pacific J. Math.141(1990), 197–208. MR 91g:14036 Zbl 0705.14016
[19] J. A. Wis´niewski, On contractions of extremal rays of Fano manifolds.J. Reine Angew.
Math.417(1991), 141–157. MR 92d:14032 Zbl 0721.14023
Received February 13, 2001; revised May 21, 2001 and June 25, 2001
M. Andreatta, G. Occhetta, Dipartimento di Matematica, Universita` degli Studi di Trento, Via Sommarive 14, I-38100 Povo (TN), Italy
Email:fandreatt,[email protected]
