mella m. - #-minimal models of uniruled 3-folds (2002)(21s)

Digital Object Identifier (DOI) 10.1007/s002090100374Math. Z. 242, 687–707 (2002)

#-Minimal models of uniruled 3-folds

Massimiliano Mella

Dipartimento di Matematica, Universita di Ferrara, 44100 Ferrara, Italia(e-mail: [email protected])

Received: 3 March 2000 / in final form: 5 September 2001 /Published online: 1 February 2002 –c© Springer-Verlag 2002

Mathematics Subject Classification (1991):14J30, 14N05

1. Introduction

The aim of minimal model theory is to choose, inside of a birational class ofvarieties, a “simple” element. This program has been fulfilled in dimension2 by the Italian school of the beginning of the century and a decade agoin dimension 3. After its discovery, the theory of(−1)-curves has beenused thoroughly to study algebraic surfaces. Unfortunately in the threefoldcase the program and the “simple” output objects are not easily handled.It is difficult to use them as a tool to understand the geometry of threedimensional varieties. Here, after [Re], we rephrase the standard minimalmodel program for uniruled varieties, using a polarizing divisor. As in [Re],it will be called #-minimal model. We will be able, under strong assumptionon the variety studied, to govern the program and understand its output. Evenif quite restrictive, the assumption needed are very geometric in nature. Thisallows to apply the #-program in various concrete situations. Indeed onepurpose of this note is to give a generalized and unified treatment of variousresults on uniruled varieties, [Al] [CF] [Io].

We first state the #-program and observe some of its natural properties.This program is governed by a movable linear systemH. The crucial obser-vation is the following. If the generic elementH ∈ H is a smooth surfaceof negative Kodaira dimension, then the #-program is well understood in aneighborhood ofH. This allows to study both the steps of the program andthe final output for specific families of 3-folds. For Fano varieties with bad

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anticanonical sections we improve previous results of Alexeev, [Al], Theo-rem 4.1. We describe the birational type of 3-foldsT containing a big systemof uniruled surfaces Corollary 5.5. Finally we are able to list the possible #-models of small degree threefolds embedded in projective spaces. This is thehigher dimensional counterpart of the classical result for surfaces of reduceddegree≤ 1, Theorem 5.8. In the appendix we collect some observations andexamples about uniruled varieties embedded of small degree in projectivespaces. This part is based on Castelnuovo bound and its generalization dueto Harris, [Ha].

This note is a revised version of a preprint that circulated in 1996, inspiredby a long stay at Warwick University and by suggestions of M. Reid. I alsobenefited a lot from many conversations with A. Corti and I would like tothank V. Alexeev for pointing me out the papers of Campana–Flenner, [CF],and Sano, [Sa]. The referee’s comment on the first version of this papermade it clearer and, I hope, easier to read. I would like to thank both TrentoUniversity for supporting me along the first part of this research and CentroNazionale delle Ricerche (grant n. 203.01.66).

2. Notations and preliminaries

All the varieties are defined overC and our notations are compatible with[KMM]. Let X be a variety we will denote withNE(X) ⊂ N1(X) theclosure of the cone of effective cycles inside the vector space of 1-cyclesmodulo numerical equivalence. By means of the intersection product to anyQ-Cartier divisorDwe associate the hyperplaneD⊥ = {[Z] : D·Z = 0} ⊂N1(X). We are interested in the part of the cone where−KX is positive.If X is terminal this part is locally polyhedral and the rays delimiting it arecalledextremal rays.

A surjective morphismf : X → Y with connected fibers between nor-mal varieties is calledaMori fiber space if −KX isf -ample,rkP ic(X/Y )=1 anddimX > dimY . Given a projective morphismf : X → Y andA,B ∈ Div(X) ⊗ Q thenA is f -numerically equivalent toB (A ≡f B)if A · C = B · C for any curve contracted byf . A is f -linearly equivalenttoB ( A ∼f B ) if A−B ∼ f∗M , for some line bundleM ∈ Pic(Y ), wewill suppress the subscript when no confusion is likely to arise.

2.1 A varietyX is calleduniruled if it admits a generically finite dominantrational mapp : Y × P1−−>X. By means of Miyaoka–Mori characteriza-tion of uniruled varieties, [MM], the minimal model program and Miyaokacharacterization of minimal 3-folds, [Mi], a threefold is uniruled if and onlyif its Kodaira dimension is negative. A uniruled 3-fold is always birationalto a Mori fiber space.

# -Minimal models of uniruled 3-folds 689

LetH be a linear system of Weil divisors, not necessarily complete, on avarietyX.H ∈ H a general member of this linear system. We will say thatH is movable if h0(X,nH) > 1 for somen > 0. Observe that the pushforward of a movableH is again movable. In other words the elements ofH cannot be contracted by a birational projective morphism.

2.2 The following is a particular case of [BS, Prop. 1.4]. LetX be a terminal3-fold andf : X → W a Mori fiber space with generic fiberF � Pr.Assume there exists anf -ample line bundleH ∈ Pic(X)withH|F ∼ O(1).ThenX andW are smooth and(X,H) = (P(E),O(1)), with E = ϕ∗H ark(r + 1) vector bundle onW .

2.3 We want here to recall some basic properties of terminalQ-factorial 3-folds. We will use them throughout the paper without explicit mentioning,as a reference see for instance [Me]. LetX be aQ-factorial terminal 3-fold andx ∈ Sing(X) a singular point. ThenX has isolated singularitiesand any irreducible surfaceS troughx is singular atx. Moreover if we leti(X) = inf{t ∈ N : tKX ∈ Pic(X)} then for any integral divisorA ⊂ Xis i(X)A ∈ Pic(X).

3. #-Minimal models

Let T be a terminalQ-factorial uniruled 3-fold. As observed in 2.1, thereexists a birational modificationΦ : T−−>T 1−−> . . .−−>T k to a Mori spaceT k.Φ is obtained as a chain of birational modifications associated to extremalrays onT , T 1, · · · . We are looking for a way to choose these extremal raysin a “natural” way. To do this we will borrow an idea of Reid, [Re, (2.3)].Fix a movable linear systemH. We will choose an extremal ray, sayR+[z],having the smallest valueH · z.

Definition 3.1 Let T be a terminalQ-factorial uniruled 3-fold andH amovable linear system with generic elementH ∈ H onT . Assume thatHis nef, by abuse of language we will say thatH is nef, then

ρ=ρH = ρ(T,H)=: sup{m∈Q|H+mKT is an effectiveQ-divisor}≥0,

is the threshold of the pair(T,H), see [Re, (2.1)].

Theorem 3.2 There exists a pair(T#,H#), with generic elementH# ∈H# such that:

i) there is a birational mapϕ : T−−>T# withϕ∗H = H#;ii) π : T# → W is a Mori space;iii) ρ(T,H)KT# +H# ≡π OT# .

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Definition 3.3According to [Re],(T#,H#)will becalleda#-Minimal Modelof the pair(T,H).

Proof of the Theorem.We will proceed in an inductive way. Let(T0,H0) =(T,H). H0 is nef by hypothesis andT0 is uniruled. Therefore to(T0, H0)are naturally associated:

– the nef valuet0 = sup{m ∈ Q|mKT0 +H0 is nef},– a rational mapϕ0 : T0−−>T1, which is either an extremal contraction,

or a flip, of an extremal ray in the face spanned byt0KT0 + H0,– a movable linear systemH1 := ϕ0∗H0 onT1.

One inductively definesϕi : Ti−−>Ti+1 and(Ti+1,Hi+1) as follows. Firstnote thatti−1KTi +Hi is nef. Letδ = sup{d ∈ Q|dKTi +(ti−1KTi +Hi)is nef } and defineti := δ + ti−1. The second step is to prove that thereexists an extremal ray[Ci] ⊂ NE(Ti) in the face spanned bytiKTi +Hi. ByconstructiontiKTi +Hi = δKTi +(ti−1KTi +Hi). Furthermoreti−1KTi +Hi is nef and(ti + ε)KTi +Hi is not nef. Then there exists a curveCi, withKTi · Ci < 0, in the face spanned bytiKTi +Hi.

Let us defineϕi : Ti−−>Ti+1 the birational modification associated tothe extremal ray[Ci] ⊂ NE(Ti) andHi+1 := ϕi∗Hi.

The inductive process is therefore composed by divisorial contractionsand flips. SinceT0 is uniruled after finitely many steps we reach a Mori fiberspaceπ : Tk → W .

Claim 3.4. (Tk,Hk) satisfiesiii).

Proof of the claim. Let ϕi : Ti−−>Ti+1 a birational modification of theinductive procedure. By Kleiman’s criteria the cone of nef divisors is theclosure of the ample cone. The cone of nef divisors is therefore contained inthe closure of the cone of effective divisors. So that we always haveti ≤ ρHi .Let us observe that the threshold is preserved after any single birationalmodification. On one handρHi+1 ≥ ρHi because the push forward of aneffective divisor is effective. On the other hand for anyϕi

ϕ∗iKTi+1 = KTi − αE,

for someα ≥ 0 (if ϕi is a flip thenϕ∗i is the trace morphism andα = 0).

That is to say

ϕ∗i (rKTi+1 +Hi+1) = rKTi +Hi − (r − ti)αE,

thusρHi ≥ ρHi+1 . Let ρ = ρHi andπ := ϕk : Tk → W the Mori spaceending the process. Sinceπ is a fibration, then(tk + ε)KTk

+Hk cannot beeffective, thereforetk = ρ. ��Let (T#,H#) = (Tk,Hk) to conclude. ��


Remark 3.5Along the proof of Claim 3.4 we proved that the threshold ispreserved by #-MMP modifications. Note thatH# is relatively nef. Further-more if the rational map defined by|mH| is birational thenH# is relativelyample.

In general the #-Minimal Model is not uniquely determined. This prob-lem arise when two extremal rays, spanned by the same nef divisor havenot disjoint exceptional loci. For example letT = E × F1 andH = π∗A,whereE is a smooth curve of genusg > 0, A ∈ Pic(E) a movable di-visor andπ : T → E the natural projection. Then there are two extremalrays spanned byH itself, one of divisorial type and the other of fiber type.The order in which the rays are contracted determines the #-minimal model(either aP1-bundle or aP2-bundle).

As mentioned in the introduction we are not able to handle the whole #-program. We will restrict ourselves to study the following special situation.Let (T,H) a pair withρH < 1. If there is a smooth surfaceS ∈ H it ispossible to describe in detail the #-program in a neighborhood of the surfaceS, see also [CF,§2].

Proposition 3.6 Let ϕi : Ti−−>Ti+1 a birational modification in the #-program relative to(T,H), with ρH < 1. Assume thatS ∈ Hi is a smoothsurface. Thenϕi(S) = S is a smooth surface andϕi|S : S → S is eitheran isomorphism or the contraction of a disjoint union of (-1)-curves.

Proof. S is smooth andTi is terminalQ-factorial thereforeS∩Sing(Ti) =∅. In particularHi is a Cartier divisor.

Claim 3.7. If ϕi is a flip thenS is disjoint from the flipping curves

Proof of the claim. Let C be a curve flipped byϕi thenKTi · C > −1,[Mo2], andC is not on the smooth locus ofTi. So thatC �⊂ S andS · Cis an integer. By definition(ρKTi +Hi) · C ≤ 0 and by hypothesisρ < 1.ThereforeS · C = 0 and the claim is proved. ��Case 3.8 (ϕi contracts a divisorE onto a curve)The generic fiberF of ϕi

is out ofSing(Ti) andKTi · F = −1. Since(ρKTi + Hi) · F ≤ 0, thenHi · F = 0 andS ∩ E is the disjoint union of (-1)-curves.

Case 3.9 (ϕi is a divisorial contraction to a point)We can assume thatHi is ϕi ample, letE the exceptional divisor andB = ∪Bi = S ∩ E.MoreoverHi|E ample and therefore letE the exceptional divisor andB =∪Bi = S ∩ E. MoreoverHi|E ample and thereforeB is connected. LetϕS := ϕi|S : S → S, then by adjunction formulaKS ·Bi < 0. FurthermoreϕS is birational thereforeB is a (-1)-curve. ThenE is smooth alongB

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andE · B = −1. By construction−KT · B > 1, therefore−KE · B =−KT ·B − E ·B ≥ 3.

Letµ : E → E a relatively minimal resolution ofE andL = µ∗(−E|E).ThenL · µ∗B = 1. Let [µ∗B] ≡ ∑

i bi[Ci] for some[Ci] ∈ NE(E). Then

−3 ≥ KE · µ∗B =∑i

biKE · Ci

and1 =

∑i

biL · Ci.

L is nef andKE is µ-nef. MoreoverL · C = 0 if and only if µ(C) = pt.Therefore there is onei such thatKE · Ci ≤ −3, andE � P2, see forinstance [CKM]. FurthermoreL is ample andKE · B = KE · B = −3.That isKTi ·B = −2 and(Hi|E)2 = Hi ·B = 1. By relative spannedness,[AW], Hi|E is spanned and thush0(E,Hi|E) ≥ 3. This is enough to provethat∆(E,Hi|E) = 0 and(E,E|E) � (P2,O(−1)). ��

Using the Proposition 3.6 we can control #-minimal model and its output.

Corollary 3.10 Let T be a terminalQ-factorial uniruled 3-fold andH amovable and nef linear system. Set(T#,H#) a #-minimal model of(T,H).Assume thatρH < 1 andH is base point free. ThenH# ∈ Pic(T#), H#

has at most base points andH# is smooth.

Proof. By Bertini TheoremH is smooth therefore we can apply Proposition3.6 in an inductive way up to reach a model(T#,H#). ��

We will need the following relative version of Corollary 3.10.

Definition 3.11 Let T be a 3-fold andH a movable linear system, withdimH ≥ 1. Assume thatH = M+F , whereM is a movable linear systemwithout fixed component andF is the fixed component. A pair(T1,H1) willbe called a log minimal resolution of the pair(T,H) if there is a morphismµ : T1 → T , with the following properties:

– T1 is terminalQ-factorial– µ−1∗ M = H1, whereH1 is a Cartier divisor,dimBsl(H1) ≤ 0– a general elementH1 ∈ H1 is a minimal resolution of a general elementM ∈ M.

Corollary 3.12 For any pair(T,H) of an irreducibleQ-factorial 3-foldTand a movable linear systemH, withdimH ≥ 1, there exists a log minimalresolution.


Remark 3.13If T is an irreducible uniruled 3-fold andH a movable linearsystem of Weil divisors, we will call #-Minimal Model of the pair(T,H), a#-Minimal Model of a log minimal resolution. Note that this is well definedonly up to birational equivalence.

Using Corollary 3.12 we can describe a pair(T,H), with dimH ≥ 1, interms of a birational Mori fiber space. A #-Minimal Model of a log minimalresolution. It is useful to have a way to ensure that the resulting model iseffectively different from the starting one. The following Lemma is in thisdirection and will be used in the next section to studyQ-Fano 3-folds.

Lemma 3.14 Let (T0,H0) be a pair consisting of a uniruled 3-fold and amovable linear systemwithdimH0 ≥ 1. Letµ : (T1,H1) → (T0,H0) a logminimal resolution and(T2,H2) a #-minimal model. LetHi ∈ Hi genericelements. Assume thatH0 is Cartier and−KH0 , −KH2 are ample. ThenK2

H0≤ K2

H2. Furthermore ifH0 is normal and singular thenK2

H0< K2

H2.

Proof. −KH2 is ample. ThenρH1 < 1 and by construction there are twowell defined morphismsν : H1 → H2 andµ : H1 → H0. Moreoverµ isinduced by a log minimal resolutionµ : T1 → T0. If we define theai’s by

KH1 = (KT1 +H1)|H1 =(µ∗(KT0 +H0) −

∑aiEi

)|H1

= µ∗KH0 −∑

aiEi;

thenai ≥ 0.By Corollary 3.10ν is a morphism between two smooth surfaces, thus

KH1 = ν∗KH2 +∑

biEi,

with bi ≥ 0, whereEi are eitherµ or ν exceptional.This yields to

K2H0

= K2H2

+ 2ν∗KH2 ·(∑

(ai + bi)Ei

)+

(∑(ai + bi)Ei

)·(∑

(ai + bi)Ei

).

By construction(∑

(ai + bi)Ei

)· (aj + bj)Ej = (µ∗KH0 − ν∗KH2) · (aj + bj)Ej

If ν∗Ej = 0 then(∑

(ai + bi)Ei) · (aj + bj)Ej = (∑

(ai + bi)Ei) ·µ∗KH0 ≤ 0; while if µ∗Ej = 0 then (

∑(ai + bi)Ei) · (aj + bj)Ej =

−ν∗KH2 · (aj + bj)Ej . So in any case

K2H0

= K2H2

+(∑

(ai + bi)Ei

)· µ∗KH0 +

(∑(ai + bi)Ei

)· ν∗KH2 ,

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where in the right hand side the last two terms are non positive.To prove the last assertion, let us start with a simple but useful observa-

tion. There are not birational contractions, that is morphisms with connectedfibers, from a del Pezzo surface to a normal singular surface. In particu-lar H1 is not a del Pezzo surface. That is there exists the class of a cycle[C1] ∈ NE(H1) such thatKH1 · C1 ≥ 0. On the other handH2 is Fanothusν is not an isomorphism. There exists at least a (-1)-curve, sayE0,contracted byν. But µ is a log minimal resolution thereforeµ∗(E0) �= 0.Thus(

∑(ai + bi)Ei) · µ∗KH0 < 0, andK2

H2> K2

H0. ��

Let us derive another technical result, which will be useful in the ap-pendix.

Lemma 3.15 Letϕi : Ti−−>Ti+1 a step of the #-program of a pair(X,H).Assume thatHi is irreducible and letB ⊂ Ti any curve not contained inthe exceptional locus ofϕi. ThenHi ·B ≤ Hi+1 · ϕi∗(B).

Proof. If ϕi is a morphism projection formula

Hi+1 · ϕi∗B = ϕ∗iHi+1 ·B = (Hi + αE) ·B,

allows to conclude.Assume now thatϕi is the flip of the curveC. Let us consider the fol-

lowing diagramZ

p

��

�� q

��

��

�

Ti

f ��

��

�ϕi �� Ti+1

g��

��

��

W

wheref andg are the contractions of[C] and[C+], respectively, andp, qis a resolution ofϕi. SinceHi is f -nef, there exists anM ∈ Pic(W ) suchthatm(Hi + f∗M) is spanned form � 0. LetHZ := p∗(m(Hi + f∗M))thenHZ = q−1∗ (m(Hi+1 + g∗M)). Furthermoreq∗(m(Hi+1 + g∗M)) =HZ +

∑aiEi, with ai ≥ 0. Therefore again by projection formulaHi ·B ≤

Hi+1 · ϕi∗B. ��

4. General elephants of Q-Fano 3-folds

In this section we will apply the #-program to studyQ-Fano 3-folds with“bad” anticanonical class. LetT be aQ-Fano 3-fold and assume that thegeneral element of|−KT |, has worse than Du Val singularities. According to


the general elephant conjecture varieties of this kind shouldn’t exist. Alexeevproved that such aT , if one further assume thatdimφ|−KT |(T ) = 3, isbirational to aQ-Fano 3-fold on which a general anticanonical element hasonly Du Val singularities, [Al, Th 4.3]. Furthermore he proved that anyQ-Fano 3-fold admits a Gorenstein model, at the expense of introducingcanonical singularities, [Al, Th 4.8]. As observed by Corti, [Co2, 1.19], oneshould expect to have a Gorenstein terminal model for aQ-Fano 3-fold with“big” anticanonical system. Using #-Minimal Model Program we will showthat aQ-Fano 3-foldT with “big” anticanonical system and whose generalanticanonical element has worse than Du Val singularities, is birational toa smooth Fano 3-fold. In such a way we meet Corti’s expectations for thisparticular class, conjecturally empty, ofQ-Fano 3-folds.

Theorem 4.1 Let T be aQ-Fano such thatdimφ|−KT |(T ) = 3 and thegeneral element in| −KT | has worse than Du Val singularities. ThenT isbirational to a smooth Fano 3-foldT# of Fano index≥ 2.

Proof. Let H0 = | − KT | andµ : (T1,H1) → (T,H0) a log minimalresolution of(T,H0). A general element inH0 has worse than Du Valsingularities. Then the canonical class ofH1, KH1 = K + H1, is negativeon infinitely many curves dense inT1. In particularρH1 < 1.

Let ν : T1−−>T2 the result of the #-program applied to(T1,H1).π : T2 → W the fiber type contraction associated to an extremal ray, say[Z] ∈ NE(T2). By Corollary 3.10,H2 ∈ Pic(T2) and a general elementin H2 is smooth. Furthermore sincedimφH0(T ) = 3 thenH2 is relativelyample and

KT2 · Z = −1ρH · Z < −1.

There are the following possibilities:

– dimW = 1. The general fiberF of π is eitherP2 or Q2 andT2 isbirational toW × P2, see for instance the appendix in [Co1].T hasrational singularities andh1(T,OT ) = 0 thenW � P1. In particularT2is rational.

– dimW = 2. π : T2 → W is a conic bundle with a section. So thatTis birational toP1 × W . Again h1(T,OT ) = 0, and via Castelnuovorationality criteria,W and henceforthT are rational.

– dimW = 0. Thenρ(T2) = 1 andH2 is ample. The general elementH2 is a smooth del Pezzo surface andH2 has at most base points. Byassumptionsdim(H2) ≥ h0(T,−KX) ≥ 4 so thath0(H2, H2|H2) ≥ 3.By general properties of del Pezzo surfaces|H2| is spanned by globalsections.

If T2 is smooth ordimW > 0, we have finished. AssumeT2 is a singularQ-Fano. Letx ∈ Sing(T2) andL2 = |H2 ⊗Ix|. |H2| is ample and spanned

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thereforeL2 has only base points. In particular the general element is asingular normal irreducible surface. Let(T3,L3) a log minimal resolutionof (T2,L2), and(T4,L4) a #-model. Withπ4 : T4 → W2 the Mori fiberspace associated. Observe that also in this caseρL3 < 1.

We have now to analyze the morphismπ4 as before. Assume thatL4 isπ4-trivial. Sincedim(L2) ≥ 2 and the generic element is irreducible thenL2 is not composite with a pencil anddimW2 = 2. Let ϕ2 : T2−−>V2the map defined byL2 with dimV2 = 2. Let Φ = Φ|H2| : T2 ↪→ Pn, themorphism defined by sections of|H2| andl a curve contracted byϕ2. Anyhyperplane passing troughΦ(x) and any other point ofΦ(l) must containΦ(l). That isΦ(l) is a line troughΦ(x) ∈ Pn. Let us come back to themorphismπ4. The generic fiberf0 of π4, is contained inL4. The #-programis an isomorphism in a neighborhood off0. So that the strict transform ofH2is a birational section and we conclude as above that the 3-fold is rational.

Assume thatL4 is π4-ample. If dimW2 > 0 we conclude thatT4 isrational. IfdimW2 = 0 thenL4 is ample and a generic elementL4 ∈ |L4| isa smooth del Pezzo surface. Furthermore by Lemma 3.14K2

L4> K2

H2≥ 1.

In particular|L4| is spanned and since it is also ample thenh0(T4, L4) ≥ 4.We can therefore iterate our argument and, by Lemma 3.14, it must stopafter at most 7 steps, sinceK2

Lk≤ 9.

5. 3-folds with a big uniruled system

Definition 5.1 Let T be a terminalQ-factorial 3-fold andH a movablelinear system. We will say that(T,H) is a pair with a big uniruled systemif H ∈ H is nef and big andH is a smooth surface of negative Kodairadimension.

Lemma 5.2 Let (T,H) be a pair with a big uniruled system. ThenT isuniruled andρ(T,H) < 1.

Proof. Let f ⊂ H be a generic rational curve such thatKH · f < 0. H isnef and big thereforeKT · f < 0 and by [MM] T is uniruled. Form � 0letm(KT +H) = αH+B, with f �⊂ B.H is nef andm(KT +H) ·f < 0then eitherB is not effective orα < 0. Thus in any caseρH < 1. ��

Our first aim is to list the #-Minimal Models of pairs with a big uniruledsystem.

Theorem 5.3 Let (T,H) be a pair with a big uniruled system. Then(T#,H#) is one of the following:

i) a Q-Fano 3-fold of index1/ρ > 1, withKT# ∼ −1/ρH# andΦ|H#|birational, the complete classification is given in [CF] and [Sa]:


(P(1, 1, 2, 3),O(6))(X6 ⊂ P(1, 1, 2, 3, a), X6 ∩ {x4 = 0}), with 3 ≤ a ≤ 5(X6 ⊂ P(1, 1, 2, 2, 3), X6 ∩ {x3 = 0})(X6 ⊂ P(1, 1, 1, 2, 3), X6 ∩ {x0 = 0})(P(1, 1, 1, 2),O(4))(X4 ⊂ P(1, 1, 1, 1, 2), X4 ∩ {x0 = 0})(X4 ⊂ P(1, 1, 1, 2, a), X4 ∩ {x4 = 0}), with 2 ≤ a ≤ 3(P3,O(a)), with a ≤ 3, (Q3,O(b)), with b ≤ 2(X3 ⊂ P(1, 1, 1, 1, 2), X3 ∩ {x4 = 0}), (X3 ⊂ P4,O(1))(X2,2 ⊂ P5,O(1))a linear section of the Grassmann variety parametrising lines inP4, embedded inP9 by Plucker coordinates(P(1, 1, 1, 2),O(2)), the cone over the Veronese surface

ii) a bundle over a smooth curvewith generic fiber(F,H#|F ) � (P2,O(2))

and with at most finitely many fibers(G,H#|G) � (S4,O(1)), whereS4

is the cone over the normal quartic curve and the vertex sits over anhyper-quotient singularity of type1/2(1,−1, 1)with f = xy−z2 + tk,for k ≥ 1, [YPG],

iii) a quadric bundle with at mostcA1 singularities of typef = x2 + y2 +z2 + tk, for k ≥ 2, andH#

|F ∼ O(1),iv) (P(E),O(1)) whereE is a rk 3 vector bundle over a smooth curve,v) (P(E),O(1))whereE is a rk 2 vector bundle over a surface of negative

Kodaira dimension.

Before giving the proof let us make a few comments about the output.

Remark 5.4The singular varieties at the pointii) are canonically birational,to a smooth projective bundle. In particular there exist singular terminal Morifiber spaces with generic fiber(P2,O(2)) and containing a smooth ruledsurface. This corrects an error in [Me, Prop 3.7] and [CF, Prop 3.4], wheresuch varieties have not been detected. To have an example it is enough to takea smooth projective bundle over a curve, sayX = P2 ×P1, blow up a conicC which sits in a fiberG and contract the strict transform ofG. In this waywe produce a Veronese cone singularity. The smooth surface is a genericelement of the strict transform of divisors in|OX(2, 1) ⊗ IC |. Going onblowing up hyperplane sections and contracting down we get hyper-quotientindex two singularities. Indeed along the proof we will prove that reversingthis process we obtain a canonical desingularization of the singular pointsthat can appear in such a situation. The same is true also for singular quadricbundles at pointiii). The varieties at pointv) are birational to a projectivebundle over eitherP2 or a ruled surface, but not in a canonical way. Indeedthis is achieved following the #-program on the base surface by means oflinks on the 3-foldT#.

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Theorem 5.3 and Remark 5.4 allow to extend the result of [CF] to thefollowing class of 3-folds.

Corollary 5.5 LetT be an irreducible threefold andH a linear systemwithgeneric elementH. Assume that one of the following is satisfied:

– (T,H) is a pair with a big uniruled system– dimΦH(T ) = 3 and the Kodaira dimension of a resolution ofH isnegative.

ThenT is birational to one of the following:

– H × P1,– a terminal sestic in eitherP(1, 1, 1, 2, 3) or P(1, 1, 2, 2, 3),– a terminal quartic inP(1, 1, 1, 1, 2),– a terminal cubic inP4.

Proof of the Corollary.Let (T1,H1) a log resolution of(T,H). Under bothassumptions(T1H1) is a pair with a big uniruled system. Then a #-modelof (T1,H1) is in the list of Theorem 5.3. We can now argue exactly as in[CF, Rem. 1.6]. ��Proof of the Theorem.By Sect. 3 we have a Mori fiber spaceπ : T# → Wwith ρKT# +H# ≡π 0 andH# is smooth.

Case 5.6 (Fano case)Assume thatρKT# ≡ H# andrkP ic(T#) = 1, thenH# is ample and generated by global sections outside a finite set of smoothpoints.H# is smooth andKH# = (KT# + H#)|H# = (1 − 1/ρ)H#

|H#

thereforeH# is a del Pezzo surface. The classification of these 3-folds isknown [CF] and [Sa].

Case 5.7 (Fiber type case)LetF be a generic smooth fiber. Then(ρKT# +H#)|F ∼ OF thus1/ρH#

|F = −KF .

If dimW = 1 then eitherF � P2 and ρ = 1/3, 2/3 or F � Q2

andρ = 1/2. Furthermore all fibers are irreducible and reduced and themorphism is flat. Furthermore the delta genus of any fiber is defined andit is semi-continuous. This simple observation allows to give a biregulardescription of these varieties in the following way.

5.7.1 (F � P2 and ρ = 1/3) ThenT# � P(ϕ∗H#) by (2.2).

5.7.2 (F � P2 and ρ = 2/3) We will prove that the only possible fibersareP2 or S4. The latter with the vertex sitting on a terminal hyper-quotientsingularity of index 2 and type1/2(1,−1, 1) f = xy − z2 + tk, withk ≥ 1. Incidentally observe that−(KT# +H#)|F ∼ OF (1). ThusT# has


a birational section and it is birational to a projective bundle. By (2.2) all thefibers on the Gorenstein locus areP2. Assume that there is a singular nonGorenstein pointx and a singular fiberG � x. By Fujita classification andflatness, the only possible singular fiber is the cone over the normal quarticcurve. In particular|H#| is relatively very ample. Using this remark we willbe able to describe a birational map fromT# to a projective bundle overWfactored by elementary links in the Sarkisov category, [Co1].

Letϕ : T# ↪→ PN an embedding given by the sections ofL = |H# +π∗M |, for some very ampleM onW . Letν : Y → T# the blow up ofϕ(x)with exceptional divisorE =

∑eiEi. LetL1 andL2 two generic element

of |L ⊗ Ix|. Then

ν∗Li = LiY + E andν∗G = GY +∑

γiEi,

for integersγi ≥ ei. SinceL is very ample thenL1Y · L2Y · GY = 0 andL1Y · L2Y · Ei is the degree ofEi as subvariety ofPN−1. Therefore byprojection formula

(5.1) 4 = L1 · L2 ·G =∑

γideg(Ei).

GY � F4 andG is embedded inP5 byϕ. LetE0 an irreducible componentof the exceptional divisor that contains the rational normal curve of degree4. ThendegE0 > 2, γi = ei = 1 andGY |E ∼ LY |E is ample. So thatEis irreducible and reduced of degree 4. It is now immediate to observe thatE is either the Veronese surface or the cone over a rational normal quarticcurve. Letf be a fiber of the ruling onGY thenGY · f = −1. Moreover byadjunction formula

KY · f = −1.

This is enough to prove thatGY ⊂ Y is contractible to a 3-foldT1. MoreoverT1 is smooth along the exceptional locus of the contraction. Iterating thisprocess we get a resolution of the singularities ofT#. Keep in mind thatϕ(T#) ⊂ PN and we are making smooth blow ups of points in the ambientvariety. After finitely many steps we have the following picture

Yν

��

��

��

��

�. . .

��

��

�. . .

��

��

�� Y ′

��

��

η��

��

��

T# T1 Tk T ′

in particularT ′ is a smooth projective bundle overW andTk has a quotientsingularity of type1/2(1,−1, 1).

We have to describe an analytic neighborhood of the pointx ∈ T#.By results of Pinkham, [Pi] and Kollar–Shepherd-Barron, [KSB, pag 313-316], the versal deformation of the singularity inS4 the versal deformation

700 M. Mella

of the singularity inS4 has only one smooth component of dimension 1,with generic fiber a Veronese surface. Locally this family is the quotient ofC4/(xy − z2 + t) by a cyclic group of order 2 acting on(x, y, z, t) withweights(1,−1, 1, 0). The local equation ofT# can be obtained by this oneby a base changet → tk.

5.7.3 (F � Q2 and ρ = 1/2) Then by Fujita list the only possible singu-lar fiberG is a quadric cone. In particular|H#| is relatively very ample.Repeating the same argument of the quartic cone we exhibit a canonicalbirational map to a smooth quadric bundle overW , factored by elemen-tary links in the Sarkisov Category. Then argue by [KM, Prop 4.10.1] and[Cu]. To derive that the possible singularities ofT# arecA1 points of thetypek[x, y, z, t]/(x2 + y2 + z2 + tk). Another approach is to use the versaldeformation of the cone singularity. That is given byx2 + y2 + z2 + t.

If dimW = 2 the only possibility isF � P1 andρ = 1/2.

5.7.4 (F � P1 and ρ = 1/2) . Then H#|F = O(1) and by (2.2)W is

smooth andT# � P(ϕ∗H#). Let S in H# a smooth section thenϕ|S :S → W is a birational morphism, thereforeW is either ruled or the blowup of a ruled surface.

To have a better description of these conic bundles consider the #-minimal model of(W,G = (ϕ|S)∗(H|S)), [Re]. Let (W#, G#) the #-Minimal Model of (W,G) andH the push forward ofϕ∗H onW#. ThenW# is eitherP2 or a ruled surface and(T#, H#) is birational to(P(H),O(1))

T#

ϕ

��

�� P(H)

ψ

��W �� W#.

It is possible to factor the birational map(T#, H#)−−>(P(H),O(1)) byelementary links in the Sarkisov category. The links are described in thefollowing way. LetB ⊂ W a (-1)-curve thenD = ϕ∗(B) � Fa, forsomea ≥ 0. If a = 0 then the link is simply the blow down ofD. Ifa > 0 then letC0 the exceptional section of the rational scrollD. ThenNC0/T ∼ O(−1) ⊕ O(−a). It is possible to antiflip it (or flop ifa = 1)to get the following

T

ϕ

��

�� Tν �� T ′

ψ

��W �� W#.

whereν is the weighted blow up with weights(1, 1, a). ��


There exists a natural geometric interpretation of the conditions imposedin Theorem 5.3.

Theorem 5.8 LetTd ⊂ Pn be a degreed non degenerate 3-fold. Supposethatd < 2n − 4, then any #-Minimal Model(T#,H#) of (Td,O(1)) is inthe list of Theorem 5.3.

Remark 5.9This result is an answer to a question of Mumford, [Mu, 2.15pg 66], about reduced degree varieties. The reduced degree of a varietyTd ⊂ Pn of dimensionk and degreed is by definition

rd :=d

n+ 1 − k.

We could rephrase Theorem 5.8 saying that 3-folds of reduced degree< 2have a #-minimal model in the list of Theorem 5.3. If we forget about thepolarization and consider only the 3-foldT# then the birational type of thisclass of 3-folds is described in Corollary 5.5. Observe that the 3-folds listedadmit an embedding satisfying the numerical criteria.

Theorem 5.8 can be interpreted as the three dimensional generalizationof the following classical result for surfaces, [GH, Proposition at page 525].

LetS ⊂ Pn be a non degenerate irreducible surface of degreed ≤ n−1thenS is either a rational scroll or the Veronese surface.

In the threefold case we loose the biregular character of the classificationand smoothness. Furthermore the varieties listed can have arbitrarily highirregularity. Consider for exampleP2 × C, with C a curve of genusg ≥ 2embedded by(O(1),L). WhereL is a non special very ample divisor ofdegreeg+3. Even the irregularity of the base of conic bundles is unbounded.As the following example shows. LetT = F1×C, whereF1 is the blow up ofP2 andC is a genusg curve. Consider the linear systemH = |(C0+2f,L)|,with degL > 10g − 4. Then

12KT + H

is nef and defines a conic bundle structure ofT ontoP1 ×C. It is a simplecalculation then to observe that the embedding defined byH satisfies thedegree assumption. This is not surprising indeed. The surface classificationis essentially based on the fact that the unique uniruled curve of degreed ≤ nis the rational normal curve. While uniruled surfaces of degreed < 2n− 2have arbitrary irregularity.

Proof of the Theorem.Let ν : X → T a resolution of singularities andH = ν∗O(1). By Lemma A.2 in the appendix(KX + H) · H2 < 0. Sothat by adjunction formula,H ∈ H is a smooth surface of negative Kodairadimension and(X,H) is a pair with a big uniruled system. ��

702 M. Mella

Consider the proof of Proposition 3.6. If the nef value of the pair(Ti,Hi)is positive the birational modification associated is the blow down of(P2,O(−1)). If we restrict to study ample linear systemsH it is possible togeneralize all #-Minimal Model Theory to arbitrary dimensions.

Corollary 5.10 LetXd ⊂ Pn a non degeneratek-fold withk > 3 and onlyQ-factorial terminal singularities. Assume thatd < 2(n − k) − 2 then a#-minimal model(X#, H#) of (X,O(1)), in adjunction theory language(X#, H#) is the first reduction, is one of the following:

i) a Q-Fano n-fold of Fano index1/ρ > k − 2, withKT# ∼ −1/ρH#

andΦ|H#| birational, the complete classification is given in [Fu] ifX#

is Gorenstein and in [CF] and [Sa] in the non-Gorenstein case.ii) a projective bundle over a smooth curve with fibers(F,H#

|F ) � (Pk−1,

O(1)), or a quadric bundle with at mostcA1 singularities, withH#|F ∼

O(1);iii) (P(E),O(1)) whereE is a rk(k − 1) ample vector bundle either on

P2 or on a ruled surface.

Proof. Let us argue as in Theorem 5.8. By Lemma A.2 the threshold is<1/(k − 2). By adjunction theory on terminal varieties, [Me], after finitelymany blow downs of(Pk−1,O(−1)), we find a Mori space structure. IfX# is Fano and Gorenstein thenrkP ic(X#) = 1. Moreover the index isan integer≥ dimX−1, so that Fujita, [Fu], classification applies. All othercases are well described in [CF], [Sa] and [Me]. ��Remark 5.11An equivalent statement was proved by Ionescu for smoothvarieties in [Io]. As in the threefold case let me stress that the irregularity isunbounded.Xd is terminalQ-factorial. Under the minimal model conjec-ture, Corollary 5.10 furnishes the classification of #-models of irreduciblenon degenerate k-folds embedded inPn of degreed < 2(n− k) − 2.

Appendix

In this appendix we sum up some observations about small degree varieties.Let us considerT k

d ⊂ Pn, a non degenerate irreducible k-fold. It is quitenatural to predict that for “small” degreed, with respect ton, T must beuniruled. Our first aim is to find the region of the(n, d) plane banned to nonuniruled varieties.

Example A.1LetE be an elliptic curve andFk+1 ⊂ Pk an hypersurface ofdegreek+1. ConsiderT = E×Fk+1 together with the natural projectionsp1 : T → E andp2 : T → Fk+1. Let OT (Le) = p∗

1OE(e) ⊗ p∗2OFk+1(1),

thenOT (Le) is very ample for anye ≥ 3; letϕe : T ↪→ Pn, the embedding


associated to|OT (Le)|, thenn = h0(E,OE(e)) · h0(F4,OF4(1)) − 1 =(k + 1)e − 1 while the degreed = OT (Le)k = k(k + 1)e. In this caseκ(T ) = 0 andd = k(n+ 1).

The main message hidden in this example is the following. It is notpossible to take arbitrarily high Veronese embeddings of a fixed immersionto construct varieties sitting in the region in which we are interested in. Weneed to evaluate the negativity of the canonical class on hyperplane sections.

Lemma A.2 Let T kd ⊂ Pn an irreducible non degenerate k-fold. Leta =

min{t ∈ N|d < t(n − k) + 2}, and degenerate k-fold. Leta = min{t ∈N|d < t(n− k) + 2}, and assume thata ≤ k. Letν : X → T a resolutionof singularities andHi ∈ |ν∗O(1)|, for i = 1, . . . , k−1 generic hyperplanesections. LetC = ∩k−1

1 Hi, then(KX + (k − a)H) · C < 0.

Proof. Let S := ∩k−21 Hi a smooth surface containingC. From genus for-

mula on the surfaceS

(A.1) g(C) = 1 +12[d(k − 1 − (k − a)) + ((k − a)H +KX) · C].

We want to bound the genus from above. To do this we will will use Castel-nuovo inequality, [GH, pag 251]. Leth = h0(C,H) = n− k + 2, then thegenus ofC is bounded by the following sum

g(C) ≤ (d− h+ 1) + (d− 2h+ 3) + (d− 3h+ 5) + . . . ,

where we sum only positive terms. Therefore in our hypothesis we haveeither

g(C) ≤ (a− 1)d− a(a− 1)2

(n− k) − a+ 1,

or a(n− k) − (2a− 1) < d < a(n− k) + 2 and

g(C) ≤ ad− a(a+ 1)2

(n− k) − a.

So that combining with equation (A.1) we obtain the desired inequalities

(A.2) ((k − a)H +KX)) · C ≤ (a− 1)(d− a(n− k) − 2),

respectively

((k − a)H +KX)) · C ≤ (a+ 1)(d− a(n− k) − 2). ��

Theorem A.3 Let T kd ⊂ Pn. Assume thatd < k(n − k) + 2, thenT is

uniruled.

704 M. Mella

Proof. Let ν : X → T a resolution of singularities. FixHi ∈ |ν∗O(1)|, fori = 1, . . . , k− 1, andC = ∩Hi. By Miyaoka–Mori criterion, [MM], it willbe enough to prove thatKX ·C < 0. This is the content of Lemma A.2 fora ≤ k. ��

The following example settles completely the question. There exist k-foldsX of degreed with κ(X) = 0 andd = k(n− k) + 2.

Example A.4([Ha]) Let us consider a (k+1)-foldV , scroll over a rationalnormal, curve embedded of degreen − k in Pn and a generic elementT ∈ |(k+ 1)O(1) ⊗ In−k−2

F |, whereF is a fiber of the scroll. Then for the“generic” V , [Ha], T is spanned, thus for the generic elementKT ∼ OT

anddegT = k(n− k) + 2.

This example is particularly interesting. It is the example of a non unir-uled variety for which the reduced degree is less then the dimension. In thecurve and surface case this is impossible but starting from 3-folds on thereare plenty of non uniruled variety, even of general type, which have thisproperty.

Let us now come back to 3-folds. the next example gives a family ofuniruled 3-folds for whichd = 3n − 8 andn is arbitrarily large. Keep inmind that by Lemma A.2 this family is just on the edge of the uniruled sector.

Example A.5Let T4 ⊂ P4, Z = H1 ∩ H2, with Hi ∈ |OT (1)|, a smoothhyperplane curve andMa = |OT (a) ⊗ Ia−1

Z |. Let us considerπ : X =BlZ(T ) → T andHa ∼ π∗OT (a) − (a − 1)E the strict transform ofMa.ThenrkP ic(X) = 2 andX admits a fibrationf : X → P1 in K3 surfaces(the fibers are the strict transform of surfaces in|OT (1) − Z|). Thereforethe only extremal ray is the one of the blowing up.Ha ∼ f∗OP1(a− 1) +π∗OT (1), thus it is very ample, letϕa : X → Ya ⊂ Pn the morphismdefined by the sections of|Ha|.

Let d = H3a the degree ofYa ⊂ Pn then

Claim A.6 d = 3n− 8.

Proof of the claim.By constructionE3 = −8 andE2 · π∗OT (1) = −4.The degree ofYa is

d = H3a = 4a3 + 8(a− 1)3 − 12a(a− 1)2 = 12a− 8.

To calculaten let Ha = f∗O(a) + E, tensoring withHa the structuresequence off∗O(a) we get

0 → OX(E) → OX(Ha) → Of∗O(a)(E) → 0,

E −KX is nef and big therefore

0 → H0(X,E) → H0(X,Ha) → ⊕aH0(F,OF (1)) → 0.

Finally this yieldsn = h0(X,Ha) − 1 = ah0(F,OF (1)) = 4a. ��


Let us further note that there is only one extremal ray onTa. Then the#-minimal program of the pair(Ya,O(1)) consists just in the contractionof E. The #-minimal model of(Ya,O(1)) is (T4,O(a)). In such a way wehave constructed varieties with Fano #-minimal model, embedded insideour area withn arbitrarily large.

Remark A.7It is important to stress the following. There are only finitelymany deformation types ofQ-Fano 3-folds, [Ka2]. Hence only finitely manycomplete linear systems of bounded degree on aQ-Fano 3-fold. The aboveexample shows that, choosing a non complete linear system, one can getQ-Fano #-models with arbitrarily high codimension.

Definition A.8 ([Ko]) Let T be a uniruled variety andH an ample linebundle. We say thatT is uniruled ofH-degree at mostd if there is a coveringfamily of rational curves{Cλ} such thatCλ ·H ≤ d.

Let us consider a 3-foldTd ⊂ Pn, with d < 3(n − 3) + 2. By LemmaA.2 T is uniruled. Our aim is to bound itsO(1)-degree of uniruling.

Example A.9Let T = P1 × F4, whereF4 ⊂ P3 is a quartic surface.Consider the line bundleLa of bidegree(a, 1) on the productT . ThenL3 = 12a andh0(T,OT (L)) = 4(a + 1). La is very ample for anya ≥ 1and embedsT ⊂ Pn as a 3-fold of degreed = 3n − 9. Furthermore theLa-degree of uniruling isa.

Example A.9 shows that there is no hope to bound the uniruled degreeof 3-folds embedded in our area. On the other hand the behavior of uniruleddegree is not completely uncontrolled.

Theorem A.10 LetTd ⊂ Pn and assume thatd < 3n − 7. If d < 2n − 4thenTd is uniruled ofO(1)-degree at most 5. Assume that2n − 4 ≤ d <3(n− 3) + 2. Let

δ =d

3(n− 3) + 2,

ThenT is uniruled ofO(1)-degree at most

3δ(1 − δ)

.

Proof. Let (T#, H#) a #-model of(T,O(1)) andρ the threshold. By aresult of Kawamata, [Ka1], a dense subset ofT# can be covered by rationalcurvesB such that−KT# · B ≤ 6. ThereforeH# · B ≤ 6ρ. LetBT thestrict transform ofB onT . Then by Lemma 3.15 we get

H ·BT ≤ H# ·B ≤ 6ρ.

706 M. Mella

It is therefore enough to get a bound on the threshold depending ond.If d < 2n − 4 we already know thatρ < 1 thusT is covered by rationalcurves of degree at most 5.

Assume that2n − 4 ≤ d < 3n − 8. Then by equation (A.2) witha = k = 3 in Lemma A.2

(A.3) KT ·H2 ≤ 2(δ − 1)(3(n− 3) + 2).

By definition of threshold if(rKT + H) · H2 < 0 thenr > ρ, thereforesubstituting Equation (A.3) we obtain the required upper bound for thethreshold

ρ ≤ δ(3(n− 3) + 2)2(1 − δ)(3(n− 3) + 2)

,

which allows to conclude.

Remark A.11The bound in Lemma A.10 is not sharp. In fact we are estimat-ing the threshold onKT ·H2 and we are using Kawamata result. Nonethelessit is interesting to observe that the degree of the ruling is bounded by a func-tion depending only onδ, that is on the ratiod/n. Similar results are truefor irreducible varieties of arbitrary dimension, under the Minimal Modelconjecture.

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mella m. - #-minimal models of uniruled 3-folds (2002)(21s)

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