Examples of threefolds with Kodaira dimension <span class="mathjax-formula">$1$</span> or <span class="mathjax-formula">$2$</span>

Examples of Threefolds with Kodaira Dimension 1 or 2

ALBERTOCALABRI(*) - MASAAKIMURAKAMI(**) - EZIOSTAGNARO(***)

ABSTRACT- We construct three nonsingular threefoldsX,X⁰andX⁰⁰with vanishing irregularities.Xhas Kodaira dimensionˆk(X)ˆ1and itsm-canonical trans- formationW_jmKXjhas the following property: the minimum integer numberm0, such that the dimension of the image dimW_jmKXj(X)ˆk(X)ˆ1formm0, is given by m0ˆ32. X⁰and X⁰⁰ have Kodaira dimensionk(X⁰)ˆk(X⁰⁰)ˆ2and their m-canonical transformations have the properties: dimW_jmKX0j(X⁰)ˆ

ˆk(X⁰)ˆ2 if and only if m12, dimW_jmKX00j(X⁰⁰)ˆk(X⁰⁰)ˆ2if and only if mˆ9;10orm12.

Introduction.

One of the problems regarding the projective, algebraic, nonsingular variety X, of dimension dim Xˆd and of general type, is to establish the finiteness and also the birationality of the m-canonical transfor- mation (improperly called a map) W_jmKXj:XTT333444P^{Pm 1}, where KX is a canonical divisor onXand Pmis them-genus ofX. In other words, the problem is to establish when the dimension of the image of X under W_jmKXj is d and, in addition, when X is birationally equivalent to its image W_jmKXj(X).

We want to generalize the above problem to any varietyXwithKodaira dimensionˆk(X)>0. Since ``of general type'' is equivalent to ``k(X)ˆdˆ dimX'', the new problem is to establish when dimW_jmKXj(X)ˆk(X).

We indeed consider the following two problems:

(1) what is the minimum integerm₀such that dimW_jmKXj(X)ˆk(X) for eachmm₀ and for each threefoldXwith Kodaira dimensionk(X)?

(*) Indirizzo dell'A.: Department of Mathematics, University of Ferrara, Italy.

(**) Invited at DMMMSA with a biennial fellowship of the University of Padua.

(***) Indirizzo dell'A.: Dept. of Mathematical Methods and Models for Scien- tific Applications (DMMMSA), University of Padua, Italy.

(2) what is the maximum integerM0such that there exists a threefoldX with Kodaira dimensionk(X) and dimW_jmKXj(X)5k(X) for eachm5M0?

In particular, we are interested in these problems when X moreover has vanishing irregularities.

First we recall the known answers to the analogous problems for sur- faces.

WhenXis a surface of general type, one hasm₀ˆM0ˆ5 (cf. F. Enriques [E], Cap. VIII, § 21; E. Bombieri [B]).

WhenXis a surface withk(S)ˆ1, it is known thatM₀ˆ8, as recently communicated by I. Dolgachev and proved by F. Catanese via email, and thatm₀ˆ14 (cf. F. Enriques [E], pp. 410-412, T. Katsura - K. Ueno [KU], F. Catanese - I. Bauer [CB]).

Consider now varieties of dimensiond3.

C. Hacon - J. McKernan [HM], S. Takayama [Ta] and H. Tsuji [Ts] proved the existence ofm₀(depending only ond) for varieties of general type, but the value ofm₀ is still unknown, even fordˆ3. M. Chen [Che₁, Che₂, Che₃] obtained upper bounds ofm₀for threefoldsXof general type, under some hypotheses on the geometric genuspgˆP1, or on them-genusPm,m>1, of X. In some cases, such limitations are optimal, thanks to examples due to Chen himself [Che₂], S. Chiaruttini - R. Gattazzo [CG], S. Chiaruttini [Chi]

and C. Hacon considering an example of Reid (cf. [Che₃, Re]).

For threefoldsXwithk(X)ˆ1, it is known that there exists an effec- tively computable integer m1 such that the m1-canonical linear system jm1K_Xjinduces the Iitaka fibration for every threefoldX (cf. [FM], Cor- ollary 6.2).

When k(X)ˆ2, J. KollaÂr conjectures the existence of an integer m₂, independent ofX, such that them-canonical transformationW_jmKjgives the Iitaka fibration formm2[K, Remark 3.4]; concerning this conjecture cf.

[P]. Moreover, KollaÂr conjectures that them-genus ofXisPm(X)>0 for somem524492 and for everyX[K, Corollary-Conjecture 3.3].

In the present paper we give three examples of threefolds,Xin Chapter 1, X⁰ andX⁰⁰ in Chapter 2, with vanishing irregularities.X has Kodaira dimension 1 and its m-canonical transformation W_jmKXj has the following property: the minimum integerm0, such that dimW_jmKXj(X)ˆk(X)ˆ1 for mm₀, is given bym₀ˆ32. This implies thatm₀32 in problem (1) for threefolds withkˆ1 and vanishing irregularities.

The properties ofX(cf. Section 1.8) imply also thatM₀20 in problem (2) for threefolds withkˆ1 and vanishing irregularities.

X⁰ andX⁰⁰ have Kodaira dimension 2 and their m-canonical transfor- mations have the properties: dimW_jmKX0j(X⁰)ˆk(X⁰)ˆ2 if and only if

m12; dimW_jmKX00j(X⁰⁰)ˆk(X⁰⁰)ˆ2 if and only ifmˆ9;10 orm12. It follows thatm₀M012 in problems (1) and (2) for threefolds withkˆ2 and vanishing irregularities.

The irregularities and the first plurigenera ofX,X⁰,X⁰⁰are as follows:

TABLE1. Irregularities and the first plurigenera ofX,X⁰andX⁰⁰.

q1 q2 pg P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14

X 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1

X⁰ 0 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3 X⁰⁰ 0 0 0 0 1 2 1 1 2 3 3 3 3 5 5 5 In our constructions, the ground fieldkis an algebraically closed field of characteristic 0, which we may assume to be the field of complex num- bersC.

1. Construction ofX.

1.1 ±Imposing singularities on a degree six hypersurface V inP⁴. Let us indicate as f₆(X₀;X₁;X₂;X₃;X₄) a form (a homogeneous poly- nomial) defining a hypersurface of degree six V P⁴ with a triple point at each of the five vertices A₀ˆ(1;0;0;0;0), A₁ˆ(0;1;0;0;0), A2ˆ(0;0;1;0;0), A3ˆ(0;0;0;1;0), A4ˆ(0;0;0;0;1) of the funda- mental pentahedron.

V :f6(X0;X1;X2;X3;X4)ˆ

X₀³(a₃₃₀₀₀X₁³‡ )‡X³₁(a₂₃₁₀₀X₀²X₂‡ )‡X₂³( )‡X₃³( )‡X₄³( )‡

‡a22200X₀²X₁²X₂²‡a22110X²₀X₁²X2X3‡ ‡a00222X₂²X₃²X₄²; whereaijkhl2kdenotes the coefficient of the monomialX₀ⁱX₁^jX₂^kX₃^hX₄^l.

We want to impose an infinitely near double surfaceS_iat the pointA_i, iˆ0;1;2;3;4, in the first neighbourhood. The surface S_i is locally iso- morphic to a plane, according to our hypothesis on the singularities in [S₁], Introduction and section 1.

We impose here a double surfaceS4infinitely nearA4and after this, by means of a permutation of indices and variables, we impose the same singularity at the otherA_j,j54.

The permutations of the indices ijkhl of the coefficient aijkhl and of variablesX0;. . .;X4, which appear inaijkhlX₀ⁱX₁^jX₂^kX₃^hX₄^l, passing fromA4to A3, fromA3toA2, fromA2toA1 and fromA1toA0, are as follows.

Permutations of indices and variables A₄^jjj !A₃^jjj !A₂ ^jjj !A₁ ^jjj !A₀ ijkhl^jjj !jiklh^jjj !jhlki^jjj !klhji ^jjj !lkhij Let us considerA₄.

Letp₁:P₁ !P⁴be the blow-up ofP⁴atA₄. LetU₄be the affine open set fX₄6ˆ0g with coordinates xˆX₀

X4;yˆX₁

X4;zˆX₂

X4 and tˆX₃ X4. The polynomial definingV\U₀ is

V\U₄:f₆(x;y;z;t;1)ˆa₃₃₀₀₀x³y³‡ ‡a₀₀₂₂₂z²t²: Locally the blow-upp1is given by the formulas:

Bx1 :

xˆx1

yˆx1y1

zˆx1z1

tˆx1t1

8>

><

>>

: ; By2 :

xˆx2y2

yˆy2

zˆy2z2

tˆy2t2

8>

><

>>

: ; Bz3:

xˆx3z3

yˆy3z3

zˆz3

tˆz3t3

8>

><

>>

: ; Bt4 :

xˆx4t4

yˆy4t4

zˆz4t4

tˆt4

8>

><

>>

:

and we consider, for example,B_x1. The strict (or proper) transformV_x1 of V\U4with respect toBx1is given by

V_x1: 1

x³₁ f₆(x₁;x₁y₁;x₁z₁;x₁t₁)ˆa₃₃₀₀₀x³₁y³₁‡ ‡a₀₀₂₂₂x₁z²₁t²₁: We impose on Vx1 the double surface, better the double plane, fx1ˆy1ˆ0g(i.e. we impose that such a plane is a locus of double points on Vx1). Since the coefficientsa_ijklh are arbitrary, according to Bertini, this means that the plane is (at least) double on every monomiala_ijklhx^m₄yⁿ₄z^p₄t^q₄. It is consequently very easy to compute the conditions on the coefficients, in order thatV_x1has the double planefx₁ˆy₁ˆ0g.

Let us denote byV1the strict transform ofVwith respect to the blow- up p1. The above conditions on the coefficients impose on V1 the double surface we called S₄, i.e. S₄ is a double surface on V₁ infinitely near A₄ˆ(0;0;0;0;1) in its first neighbourhood.

Now, we impose a double surfaceS_iinfinitely nearA_i, foriˆ0;1;2;3 by means of the above permutations without repeating the above calcula- tions.

By imposing all the conditions, we obtain for V an equation de- pending on 26 free coefficients (also called parameters) because some conditions are a duplicate of previous ones. Several of the 26 parameters can be chosen as equal to zero, because they are inessential in the computation of the birational invariants of a desingularization sj_X :X !V ofV, as well as in the computation of the dimensions of the images under pluricanonical transformations. The shortest form with the essential coefficients and defining our hypersurface with the above- said singularities is given by

f6ˆa31002X₀³X1X₄²‡a13020X0X₁³X₃²‡a20301X₀²X₂³X4‡a20031X₀²X³₃X4

‡a02013X₁²X3X₄³‡a12210X0X₁²X²₂X3‡a12120X0X₁²X2X₃²

‡a₁₁₂₂₀X₀X₁X₂²X₃²‡a₁₁₂₀₂X₀X₁X₃²X₄²‡a₁₀₂₂₁X₀X₂²X₃²X₄.

The hypersurfaceV, obtained for a generic choice of the parametersaijkhl, will be called ageneric V. In the sequel, when we shall consider ourV, it is understood our genericV.

1.2 ±Imposed and unimposed singularities on V.

We consider the hypersurfaceV at the end of section 1.1.

Close to the singularities imposed onV (the triple pointA_ihaving an infinitely near double surfaceS_i,iˆ0;1;2;3;4), new singularities appear on the genericV, either actual or infinitely near. We callactual singula- ritiesthe singularities on V that are not infinitely near. Let us find the unimposed actual singularities onV in the present section.

According to Bertini's theorem (characteristic zero), the actual singu- larities on the generic V belong to the base points of the linear system definingV. It is not difficult to find that the unimposed actual singularities are given by the following five double (straight) lines

fX0ˆX1ˆX_iˆ0g; iˆ2;3;4; fX_jˆX3ˆX4ˆ0g; jˆ0;2;

and a double plane cubic

fX0ˆa13020X₁²X3‡a12210X1X₂²‡a12120X1X2X3‡a11220X₂²X3ˆX4ˆ0g:

In the following picture the five double lines are drawn in bold type.

In particular the unimposed actual singularities have codimension 2, thereforeV it is reduced, irreducible and normal.

REMARK 1.1. We note that the double singular curves, actual or in- finitely near, do not give conditions of adjointness, that is, they do not give conditions to the hypersurfaces for them to be (any kind of) adjoints (cf.

[S1], Remark 17, section 11). In particular, such curves do not affect the birational invariant of a desingularizationXsuch asq₁;q₂;p_g;P_m, as well as the computation of dim W_jmKXj(X). Moreover, we note explicitly that the singularity given by the double plane cubic can be resolved blowing up the plane containing it.

RESOLUTION OF SINGULARITIES OFV

The main purpose of this resolution of singularities ofV is to find the infinitely near unimposed singularities and to check that they do not give conditions to any kind of adjoints toV. More precisely we find that the infinitely near unimposed singularities are given by double singular lines and isolated double singular points.

1.3 ±Blowing up the triple point A₄2U₄.

According to section 1.1, we consider the affine open setU4 ˆ fX46ˆ0g of affine coordinates (x;y;z;t). The actual singularities onV belonging to U4are given by the two double lines

A₂A₄\U₄; A₃A₄\U₄: The equation ofV\U₄ is given by

f₆(x;y;z;t;1)ˆa₃₁₀₀₂x³y‡ ‡a₁₀₂₂₁xz²t²ˆ0:

By using the notations of section 1.1, and denoting byVx1,Vy2,Vz3,Vt4

the strict (or proper) transform ofV\U4with respect toBx1,By2,Bz3,Bt4, respectively, we obtain the following results

Vx1is given by V_x1 : 1

x³₁ f₆(x₁;x₁y₁;x₁z₁;x₁t₁)ˆa₃₁₀₀₂x₁y₁‡ ‡a₁₀₂₂₁x²₁z²₁t²₁ˆ0:

On Vx1 there is a unique singularity given by the double plane fx1ˆy1ˆ0g; this means that the double plane is infinitely nearA4in the first neighbourhood.

V_y2 is nonsingular.

OnV_z3there are two singularities: the double planefy₃ˆz₃ˆ0gon the exceptional divisorz3ˆ0 and the double linefx3ˆy3ˆt3ˆ0gout- side the exceptional divisor; it is the strict transform of the actual double lineA₂A₄\U₄.

On V_t4 there are again two singularities: the double plane fy4ˆt4ˆ0g on the exceptional divisor t4ˆ0 and the double line fx4ˆy4ˆz4ˆ0g outside the exceptional divisor; it is the strict trans- form of the actual double lineA₃A₄\U₄.

1.4 ±The blow-upp2:P2 !P1ofP1along the surfaceS4

1.4.1 Let us considerVx1. OnVx1the surfaceS4is given by the double plane fx1ˆy1ˆ0g.

Locally the blow-up along this plane is given by the formulas

B_x11 :

x₁ˆx₁₁ y₁ˆx₁₁y₁₁ z₁ˆz₁₁ t₁ˆt₁₁ 8>

><

>>

: ; B_y12 :

x₁ˆx₁₂y₁₂ y₁ˆy₁₂ z₁ˆz₁₂ t₁ˆt₁₂ 8>

><

>>

: :

Let us denote byV_x11, the strict transform ofV_x1with respect toB_x11and by Vy12 the strict transform ofVx1with respect toBy12.

V_x11 is nonsingular; its equation is given by Vx11 : 1

x⁵₁₁ f6(x11;x²₁₁y11;x11z11;x11t11;1)ˆa31002y11‡ ‡a10221z²₁₁t²₁₁ ˆ0:

V_y12 is nonsingular.

1.4.2 Let us considerVz3. OnVz3the surfaceS4is given by the double planefy3ˆz3ˆ0g:Blowing up this plane, we find that there are no sin- gularities infinitely near it.

1.4.3 Let us considerV_t4. OnV_t4 the surfaceS₄is given by the double planefy₄ˆt₄ˆ0g:Blowing up this plane, again we find that there are no singularities infinitely near it.

1.5 ±The blow-ups along the double lines that are strict transforms of the double lines A₂A₄\U₄and A₃A₄\U₄.

Blowing up the strict transforms ofA2A4\U4andA3A4\U4, it is not difficult to see that infinitely near each of these double lines there is another double line and infinitely near the last double line there are no singularities.

The tree of the blow-ups resolving the singularities on V\U₄ is de- scribed below.

Where the nonsingular threefolds are drawn in bold type.

In the above sections 1.3-1.5, we gave, as an example, the blow-ups resolving the singularities of V\U₄ and we wrote the equations of the strict transforms that we need in the sequel. We calculated also the other similar desingularizations but we do not reproduce them here: we consider them as being achieved, as well as the complete desingularization ofV.

1.6 ±The m-canonical adjoints to V P⁴.

Let P_r ^p!^r !^p3 P₂ ^p!² P₁ ^p!¹ P₀ ˆP⁴ be a sequence of blow-ups resolving the singularities ofV.

If we callV_iP_ithestrict transformofV_{i 1}with respect top_i, then we obtain from the above sequence

V_r ^p!^0r !^p03 V₂ ^p!⁰² V₁ ^p!⁰¹ V₀ˆV;

where p⁰_iˆp_ijVi:V_i !V_{i 1} and s_jX :X !V, sˆp_r p₁, is a de- singularization ofV P⁴.

Let us assume thatp_iis a blow-up along a subvarietyY_{i 1} ofP_{i 1}, of dimensionj_{i 1}, which can be either a singular or a nonsingular subvariety ofV_{i 1}P_{i 1}(i.e.Y_{i 1}is a locus of singular or simple points ofV_{i 1}). Let m_{i 1}be the multiplicity of the varietyY_{i 1}onV_{i 1}.

Let us setni 1ˆ 3‡ji 1‡mi 1, foriˆ1;. . .;rand deg(V)ˆd.

A hypersurface F_{m(d 5)} of degree m(d 5) in P⁴ is an m-canonical adjoint toV (with respect to the sequence of blow-ups p₁;. . .;p_r) if the restriction toXof the divisor

D_m ˆp_rfp_{r 1}[ p₁(F_{m(d 5)}) mn₀E₁ ] mn_{r 2}E_{r 1}g mn_{r 1}E_r is effective, i.e.Dmj_X 0, whereEiˆp ¹(Yi 1) is the exceptional divisor of p_i andp_i :Div(P_{i 1}) !Div(P_i) is the homomorphism of the Cartier (or locally principal) divisor groups (cf. [S₁], sections 1,2).

An m-canonical adjoint F_{m(d 5)} is a global m-canonical adjoint to V (with respect top₁;. . .;p_r) if the divisorD_mis effective onP_r, i.e.D_m0 (loc. cit.).

Note that, if F_{m(d 5)} is an m-canonical adjoint toV, thenDmj_X mK, where `' denotes linear equivalence andKdenotes a canonical divisor onX.

In our above example, an order can be established in the sequence of blow-ups, e.g. let us assume that the blow-upp₁is the blow-up at the 3-ple pointA4,p2 is the blow-up along the double surfaceS4 infinitely nearA4

(see also section 1.1, 1.3 and 1.4),p3is the blow-up at the triple pointA3,p4

is the blow-up along the double surfaceS₃infinitely nearA₃,p₅is the blow- up at the triple point A₂, p₆ is the blow-up along the double surface S₂ infinitely nearA₂,p₇is the blow-up at the triple pointA₁,p₈is the blow-up along the double surfaceS4infinitely nearA1, andp9is the blow-up at the triple point A0, p10 is the blow-up along the double surfaceS0 infinitely nearA₀.

The hypersurfaceV has degreedˆ6 andDmis given by:

(^}) Dmˆp_r. . .fp₂[p₁(Fm) mE2] mE4g mE6 mE8 mE10‡P

mE;

whereE_iis the exceptional divisor of the blow-upp_iand, to be more specific, E₂is the exceptional divisor of the blow-upp₂alongS₄;. . .;E₁₀is the ex- ceptional divisor of the blow-upp₁₀alongS₀.

No other exceptional divisors are subtracted inD_mbecause, as we said, the unimposed singularities are either actual or infinitely near double singular curves or isolated double points on our (generic)V. We note that the exceptional divisors of the blow-ups at double isolated points appear with coefficientn_hˆ 1, we have indicated these divisors asP

mE. From here on, we omit writing P

mE, because they are not essential in the computation of the birational invariants, as well as in the computation of dimW_jmKXj(X), that we shall consider.

1.7 ±The global and non-global m-canonical adjoints to V P⁴.

PROPOSITION1. If we consider a non-globalm-canonical adjoint toV Fm:Fm(X0;X1;X2;X3;X4)ˆ X

i‡j‡k‡h‡lˆm

bijkhlX₀ⁱX₁^jX₂^kX₃^hX^l₄ˆ0;

where b_ijkhl2k, then a form AˆA(X₀;X₁;X₂;X₃;X₄) exists such that F^?_m:F_m Af₆ˆ0is a global m-canonical adjoint to V. In other words, the following equality holds

F_mjV ˆF^?_mj

V:

Proposition 1 holds for the three constructions in the present paper (see Proposition 2 in the Appendix). The three proofs are similar, so, to avoid unnecessary repetitions, we produce only one proof in the Appen- dix. There are two ways to prove these Propositions: the first way is contained in [S₂], cf. the proof of Lemma 5, section 18, p. 1177; the second way is due to Maria Cristina Ronconi. We reproduce in the Appendix the proof of Mrs. Ronconi.

LEMMA1. The global m-canonical adjoints to V are given by C_m: X

i‡j‡k‡h‡lˆm

c_ijkhlXⁱ₀X₁^jX^k₂X₃^hX₄^l ˆ0;

where cijkhl 2k and where lihjl, ik, i.e. lˆiˆhˆj and ik (for all monomials).

PROOF OFLEMMA1. Let us consider a globalm-canonical adjoint toV C_m: X

i‡j‡k‡h‡lˆm

c_ijkhlX₀ⁱX₁^jX₂^kX₃^hX₄^l ˆ0:

The total transformC ofCm\U4 with respect toBx1 (section 1.1) is given by

Cˆ B_x1(C_m\U₄): X

i‡j‡k‡h‡lˆm

c_ijkhl(x₁)ⁱ(x₁y₁)^j(x₁z₁)^k(x₁t₁)^hˆ0;

The double surfaceS4infinitely nearA4in affine coordinates (x1;y1;z1;t1) is given byfx1ˆy1ˆ0gand the blow-upp2alongS4is given locally by the formulasBx11 andBy12 (see section 1.4.1).

The total transform C of Cˆ B_x1(C_m\U₄) with respect to B_x11 is given by

C ˆ B_x11(C): X

i‡j‡k‡h‡lˆm

c_ijkhlxⁱ₁₁(x²₁₁y11)^j(x11z11)^k(x11t11)^hˆ X

i‡j‡k‡h‡lˆm

cijkhlx^i‡2j‡k‡h₁₁ y^j₁₁z^k₁₁t^h₁₁ˆ0:

SinceCmis a globalm-canonical adjoint toV, by definition in (^}), section 1.6, we haveDm0.

We note that B_x11 B_x1 coincides, up to isomorphisms, with the de- singularizations_jX on the affine open setV_x11. In fact,V_x11 is nonsingular (see the tree of blow-ups, section 1.5) and then it is isomorphic to a Zariski open set on the desingularizationXof V. The above coincidence and the inequality Dm0 imply the inequality p₂[p₁(Fm) mE2]0 and the following inequality between divisors (of rational functions) on the affine open setU_x11 of (affine) coordinates (x₁₁;y₁₁;z₁₁;t₁₁)

C x^m₁₁

ˆ 1 x^m₁₁

X

i‡j‡k‡h‡lˆm

cijkhlx^i‡2j‡k‡h₁₁ y^j₁₁z^k₁₁t^h₁₁ˆ0 0:

Since, on the affine open setUx11,x11ˆ0 is the local equation, of the ex- ceptional divisorE2of the blow-upp2, this last inequality is equivalent to

i‡2j‡k‡h m0; i:e:jl:

We have proved the above inequality for a particular sequence of open sets

appearing in the blow-ups. If we consider all the other sequences of open sets, we find the same inequality.

This proves the last inequality in the sequence of inequalities lihjlin the statement of Lemma 1.

If we consider the singular pointsA₃;A₂;A₁andA₀ and we go on with the blow-ups p₃;. . .;p₁₀, then all the other inequalities follow in a similar way. More precisely, if we consider the point A3, then we obtain ih;

consideringA2we obtainik;A1leads tohjandA0 toli.

We note that, up to isomorphisms, we can start by blowing upA_ifirst, withi6ˆ4, and in this case, we repeat forA_iwhat we did forA₄obtaining in the same way all the inequalities.

So, Lemma 1 is proved.

1.8 ±Computing the plurigenera of X.

Now, we consider [S1], Corollary 8, section 3: ifV is normal, there is an isomorphism of projective spaces for anym1

linear system of m canonical adjoints to V

jV

! jmKXj

F_mjV ^jjj ! D_mjX: Dmis defined in (^}), section 1.6.

Bearing in mind that our purpose is to compute them-canonical genusP_m

ˆdimjmK_Xj ‡1ˆdim linear system of m canonical adjoints

j_V ‡1, we can substituteF_mwithF⁰_mifF⁰_mj

V ˆF_mjV.

Next, the Proposition 1 in section 1.7 tells us that in order to compute them-genusPm, we can restrict ourselves to consider globalm-canonical adjoints toVand Lemma 1 in the same section tells us that the globalm- canonical adjoints are given by

C_m: X

4s‡vˆm

c_4s‡vX₀^sX₁^sX₂^vX₃^sX₄^sˆ0;

wherec4s‡v2kandvs; s>0.

By doing the easy calculations, we obtain pgˆP1ˆP2ˆP3ˆ0;

P4ˆ1, the global 4-canonical adjoint is defined byX0X1X3X4; P₅ˆ1, the global 5-canonical adjoint is defined byX₀X₁X₂X₃X₄;

P6ˆP7ˆ0;

P8ˆ1, the global 8-canonical adjoint is defined byX₀²X₁²X₃²X₄²; P9ˆ1, the global 9-canonical adjoint is defined byX₀²X₁²X2X₃²X₄²; P₁₀ˆ1, the global 10-canonical adjoint is defined byX₀²X₁²X₂²X₃²X₄²; P₁₁ˆ0;

P₁₂ˆP₁₃ˆP₁₄ˆP₁₅ˆP₁₆ˆP₁₇ˆP₁₈ ˆP₁₉ˆ1;

P20ˆ2 the global 20-canonical adjoints are defined byl1X₀⁵X₁⁵X₃⁵X₄⁵‡ l2X₀⁴X⁴₁X₂⁴X₃⁴X₄⁴ˆX₀⁴X₁⁴X₃⁴X₄⁴(l1X0X1X3X4‡l2X₂⁴),l_i2k; and so on.

REMARK 1.2. 1) We have seen that the first integer m such that Pmˆ2 is mˆ20 and the 20-canonical adjoints are given, up to fixed components, by the pencill1X₂⁴‡l2X0X1X3X4ˆ0.

Continuing the above list, we obtain

2) the minimum integer m₀ such thatP_m 2 for mm₀ is given by m0ˆ32;

3) the first integer m such that P_mˆ3 is mˆ40 and the global 40-canonical adjoints are defined by m₁X₀⁸X₁⁸X₂⁸X⁸₃X₄⁸‡m₂X₀⁹X₁⁹X₂⁴X₃⁹X₄⁹‡ m3X₀¹⁰X₁¹⁰X₃¹⁰X₄¹⁰ˆX₀⁸X₁⁸X₃⁸X₄⁸(m1X₂⁸‡m2X0X1X₂⁴X3X4‡m3X₀²X₁²X₃²X₄²),mi2k;

1.9 ±The m-canonical transformationW_jmKXj. Let us consider the following triangle

PP

wheres_jX :X !V, withsˆpr p1, denotes our desingularization of V, whereLmdenotes the (incomplete) linear system ofm-canonical adjoints toV restricted to V andW_Lm the rational transformation defined by the linear systemL_m.

The above triangle is commutative. This follows from the fact that the divisors of the linear system jmKXj on X are the divisors Dmj_X, where Dmˆp_rfp_{r 1}[ p₁(Fm) mn0E1 ] mnr 2Er 1g mnr 1Er, withFm

varying in the linear system ofm-canonical adjoints toV.

In the present section, we want to find the minimum integerm0such that the m-canonical transformation W_jmKXj enjoys the property: dim W_jmKXj(X)ˆ1, formm0. The values ofmsuch that dimW_jmKXj(X)ˆ1 are exactly given by the values for whichP_m2. This is a consequence of the following facts:

a) restricting the globalm-canonical adjoints toV there is no identifi- cations;

b) the commutativity of the above triangle;

c) 4), 5) in Remark 3.

So, thanks to 2) in Remark 3, the above value ofm₀is given bym₀ˆ32.

Therefore, the Kodaira dimension of X isk(X)ˆ1and the minimum integer m₀such that dimW_jmKXj(X)ˆ1, for mm₀, is given by m₀ˆ32.

Moreover, we note that the generic fiber of the rational transformation W_jmKXjis irreducible (by BeÂzout theorem) for those values ofmfor which Pmˆ2, whereas such generic fiber is reducible for those values ofmfor whichPm3.

1.10 ±Computing the irregularities of X.

It remains to prove thatq_iˆdim_kHⁱ(X;O_X)ˆ0, foriˆ1;2. We know thatq₁ˆdim_kH¹(X;O_X)ˆq(S_r)ˆdim_kH¹(S_r;O_Sr), whereS_rXis the strict transform of a generic hyperplane sectionSofV (cf. [S₁], section 4, for instance).Shas several isolated (actual or infinitely near) double points and no other singularities. This follows from the fact that the hypersurface V, outside the points A0;A1;A2;A3 and A4, only has actual or infinitely near double curves or isolated double points. So,q₁ˆ0.

To prove thatq₂ˆ0, we use the formula (36), section 4 in [S₁], which states that

q2ˆpg(X)‡pg(Sr) dimk(W2);

whereW2is the vector space of the degree 2 forms defining global adjoints F2toV, i.e. defining hyperquadricsF2 such that

p_r p₂[p₁(F₂)] E₂ E₄ E₆ E₈ E₁₀0;

(cf. the expression ofDm in (^}), section 1.6). So the above hyperquadrics F₂ are those passing through the pointsA₀;A₁;A₂;A₃ andA₄. Thus, we

have dimk(W2)ˆ15 5ˆ10. It follows from pg(Sr)ˆ10 and pg(X)ˆ0 (cf. section 1.8), thatq2ˆ0.

2. Construction ofX⁰and ofX⁰⁰.

In this chapter we construct two threefoldsX⁰andX⁰⁰, with the prop- erties described in the Introduction, as desingularizations of two hy- persurfaces V⁰ and V⁰⁰ of degree six in P⁴. Using the same method of Chapter 1, we impose sextic hypersurfaces to have triple points at the coordinate points A_l, lˆ0;. . .4, each one with infinitely near a double plane, namely a double surface whose local equations are linear, obtained with slight modifications of the permutations described in Chapter 1, § 1.1.

Indeed, the singularities ofV⁰;V⁰⁰ are of the same type as the singu- larities ofVin Chapter 1. The difference withV(and betweenV⁰andV⁰⁰) is mainly given by the position of the double surface infinitely near to A₄, which will imply the difference in the birational equivalence classes ofV,V⁰ andV⁰⁰.

The explicit equation ofV⁰is

f₆⁰ˆa30102X₀³X2X²₄‡a13020X0X₁³X₃²‡a20301X₀²X₂³X4

‡a₂₀₀₃₁X₀²X₃³X₄‡a₁₀₂₀₃X₀X₂²X₄³‡a₁₂₂₁₀X₀X₁²X₂²X₃

‡a₁₁₂₂₀X₀X₁X₂²X₃²‡a₀₂₁₁₂X₁²X₂X₃X₄²ˆ0;

while the equation ofV⁰⁰is

f₆⁰⁰ˆa30012X₀³X3X₄²‡a13020X0X₁³X₃²‡a20301X₀²X₂³X4‡a20031X²₀X₃³X4

‡a₁₀₀₂₃X₀X₃²X₄³‡a₂₂₀₁₁X₀²X₁²X₃X₄‡a₁₂₂₁₀X₀X₁²X₂²X₃

‡a₁₂₀₁₂X₀X₁²X₃X₄²‡a₁₁₂₂₀X₀X₁X₂²X₃²‡a₁₀₂₂₁X₀X₂²X₃²X₄

‡a10212X0X₂²X3X₄²‡a02112X₁²X2X3X₄²ˆ0;

where the coefficientsa_ijkhl 2kare sufficiently general. (Actually, one can construct such a threefoldX⁰⁰from a hypersurfaceV⁰⁰depending on 28 Ð instead of 12 Ð parameters, but we used only 12 of them for brevity.)

Reasoning like in Chapter 1, one sees thatV⁰;V⁰⁰are normal, they have triple points atAl, 0l4, they are double along the linesA0A1,A1A2, A1A4,A2A3and along the rational cubic plane curve:

X₀ˆX₄ˆa₁₃₀₂₀X₁²X₃‡a₁₂₂₁₀X₁X₂²‡a₁₁₂₂₀X²₂X₃ˆ0:

The further actual singularity of V⁰ is the double line A₃A₄, while the

further actual singularities ofV⁰⁰are the double lineA2A4and two double rational cubic plane curves:

X₀ˆX₁ˆa₁₀₂₂₁X₂²X₃‡a₁₀₂₁₂X₂²X₄‡a₁₀₀₂₃X₃X₄²ˆ0;

X₂ˆX₃ˆa₃₀₀₁₂X₀²X₄‡a₂₂₀₁₁X₀X₁²‡a₁₂₀₁₂X₁²X₄ˆ0:

SettingP0ˆP⁴, we perform the blow-upsp_i:P_i !P_{i 1},iˆ1;. . .;9, wherep_2l‡1,lˆ0;. . .;4, is the blow-up atA_l2U_lˆ fX_l6ˆ0gandp_2l‡2,

lˆ0;. . .;3, is the blow-up along the double surfaceS_l infinitely near to

A_l, which is the same forV,V⁰andV⁰⁰.

With the usual affine coordinatesx;y;z;tinUl,lˆ0;. . .;4, each blow upp2l‡1 is given locally by the formulasBx1,By2,Bz3,Bt4 written in § 1.1.

With respect toB_y2, we see that the local equation of the double surfaceS₀, infinitely near toA₀, isy₂ˆt₂ˆ0; the local equation ofS₁isy₂ˆz₂ˆ0;

those ofS₂and ofS₃arex₂ˆy₂ˆ0.

Finally, forV⁰, letp⁰₁₀:P⁰₁₀ !P9be the blow up along the surfaceS⁰₄ infinitely near toA4, with local equation y2ˆz2ˆ0 with respect toBy2. ForV⁰⁰, let p⁰⁰₁₀:P⁰⁰₁₀ !P₉be the blow up along the surfaceS⁰⁰₄ infinitely near toA₄, with local equationy₂ˆt₂ˆ0 with respect toB_y2.

We then checked that the strict transforms ofV⁰inP⁰₁₀and ofV⁰⁰inP⁰⁰₁₀ are singular along double curves only, and that no further essential sin- gularity appears in the resolution process. In other words, the double surfaces infinitely near to the coordinate triple points are the only essential singularities ofV⁰and of V⁰⁰. Therefore we may compute globalm-cano- nical adjoints toV⁰and toV⁰⁰in the same way we did in Chapter 1.

LEMMA2. Let FˆP

b_ijkhlX₀ⁱX₁^jX₂^kX₃^hX₄^l;be a homogeneous polynomial of degree m, i.e. with i‡j‡k‡h‡lˆm and b_ijkhl2k. Then F_mˆ fFˆ0g P⁴ is a global m-canonical adjoint to V⁰ [resp. to V⁰⁰] if and only if jhiˆkˆl [resp. jiˆhˆlk] for each monomial in F.

PROOF. Following the proof of Lemma 1 in Chapter 1, we compute the conditions imposed by the double surfaces infinitely near toA0[A1,A2,A3, resp.] and we find out thatli[hj,ik,ih, resp.]. ConcerningA₄, we find thatklforV⁰and thathlforV⁰⁰. It follows thatiˆkˆlforV⁰ and thatiˆhˆlforV⁰⁰, which conclude the proof.

By Proposition 2 in the Appendix, globalm-canonical adjoints toV⁰and toV⁰⁰are enough to compute the plurigeneraPm(X⁰);Pm(X⁰⁰) ofX⁰andX⁰⁰, and their pluricanonical transformations, which we study now.

2.1 ±Canonical adjoints to V⁰, pluricanonical transformations of X⁰. By Lemma 2, the globalm-canonical adjoints toV⁰are given by

F_m: X

5j‡4u‡3vˆm;j0;uj;vk

b_juvX₁^jX^j‡u₃ (X₀X₂X₄)^j‡u‡vˆ0;

where b_juv2k. Denote by A⁰_m thek-vector space of the polynomials de- fining globalm-canonical adjoints toV⁰. SettingY ˆX0X2X4, it follows that

A⁰₁ˆ A⁰₂ˆ f0g; A⁰₃ˆ hYi; A⁰₄ˆ hX₃Yi;

A⁰₅ˆ hX₁X₃Yi; A⁰₆ˆ hY²i; A⁰₇ ˆ hX₃Y²i;

A⁰₈ˆ hX1;X3iX3Y²; A⁰₉ˆ hX1X₃²;YiY²; A⁰₁₀ˆ hX₁²X3;YiX3Y²; A⁰₁₁ ˆ hX1;X3iX3Y³; A⁰₁₂ˆ hX1X₃²;X³₃;YiY³;

A⁰₁₃ ˆ hX₁²X₃;X₁X²₃;YiX₃Y³; A⁰₁₄ˆ hX₁²X₃²;X₁Y;X₃YiX₃Y³; and so on, which give theP_m(X⁰)'s written in Table 1 in the Introduction.

We now study them-canonical transformationsW⁰_mˆW_jmKX0jofX⁰. W e will show thatX⁰has Kodaira dimension 2 and thatW⁰_m(X⁰) has dimension 2 if and only ifm12. Clearly, dimW⁰_m(X⁰)52 ifm512.

Then, it is easy to check that W⁰_m(X⁰)ˆP² for mˆ12;13;14. Since P3(X⁰)ˆ1, it follows that dimW⁰_m(X⁰)2 also for eachm15.

An upper bound toP_m(X⁰) is given by the functionn:N!N, n(m)ˆ]f(j;u;v)2N³j5j‡4u‡3vˆmg P_m(X⁰);

whereNis the set of non-negative integers. We then see that P_m(X⁰)n(m) m

3 ‡1

m

4 ‡1

;

thus the Kodaira dimension ofX⁰is 2 and hence dimW⁰_m(X⁰)ˆ2 form12.

2.2 ±Canonical adjoints to V⁰⁰, pluricanonical transformations of X⁰⁰. By Lemma 2, the globalm-canonical adjoints toV⁰⁰are given by

F_m: X

3i‡j‡kˆm;i0;ij;ik

b_ijkX₁^jX₂^k(X₀X₃X₄)ⁱˆ0;

where b_ijk2k. Denote by A⁰⁰_m the k-vector space of the polynomials de- fining global m-canonical adjoints of X⁰⁰. Setting YˆX0X3X4, it follows