A threefold with <span class="mathjax-formula">$p_g=0$</span> and <span class="mathjax-formula">$P_2=2$</span>

A Threefold with p

ˆ 0 and P

ˆ 2

EZIOSTAGNARO(*)

ABSTRACT - We construct a nonsingular threefold X with q1ˆq2ˆpgˆ0 and P2ˆ2whosem-canonical transformationW_jmKXjhas the following properties

i) W_jmKXjhas the genericfiber of dimension1, for 2m5;

ii) it is generically a tranformation 2:1, for 6m8 andmˆ10;

iii) it is birational formˆ9 andm11.

So, we have a gap formˆ10 in the birationality ofW_jmKXj.

Introduction.

In the classification of nonsingular varietiesXof general type, them- canonical tranformationW_jmKXj, whereK_Xis a canonical divisor onX, plays an important part. The main problem concerning W_jmKXj regards its bi- rationality. The property of W_jmKXj to have the genericfiber given by a finite set of points is important too.

In the case whereXis a threefold, Meng Chen has given several lim- itations for the birationality ofW_jmKXj. In the particular case whereXhas the geometricgenusp_g2, Chen ([Che₂], [Che₃] ) proved that:

ifpg4, thenW_jmKXjis birational form5;

ifpgˆ3, thenW_jmKXjis birational form6;

ifp_gˆ2, thenW_jmKXjis birational form8.

Such limitations are optimal, as demonstrated by examples costructed by Chen himself [Che₂] ifp_g4, by S. Chiaruttini - R. Gattazzo ([CG]) if p_gˆ3, by S. Chiaruttini ([Chi]) and by C. Hacon, considering an example of M. Reid [Re], ifpgˆ2 (see [Che3]).

In the case ofpgˆ1 and pgˆ0, we have only partial results and the

(*) Indirizzo dell'A.: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via Trieste, 63, UniversitaÁ di Padova, 35121 Padova - Italy.

E-mail: [email protected]

problem of finding an optimal limitation for the birationality ofW_jmKXjremains ([Che1]). Ifpgˆ1 and the bigenus ofXisP2ˆ2, then a Chen-Zuo's lim- itation ([CZ]) states thatW_jmKXjis birational form11. We costructed ([S₄]) a threefoldXwithq₁ˆq₂ˆ0 (whereq₁andq₂are the first and second ir- regularities ofX)p_gˆ1 andP₂ˆ2 suc h thatW_jmKXjis birational if and only if m11, (cf. alsoX22in [Re], p. 359, and [F]); so the above limitation is optimal.

As for threefolds with pgˆ0, we tried to find examples of X with q1ˆq2ˆ0, P2ˆ2 and with the birationality of W_jmKXj form large. The results obtained were worse than expected as regards the birationality of W_jmKXj, while an interesting result emerged for the gaps in the birationality ofW_jmKXj. Having obtained the birationality ofW_jmKXjif and only ifm11 in the case ofpgˆ1 andP2ˆ2, the expected result in the new case ofpgˆ0 and P2ˆ2 is birationality if and only if m>11. Instead, all our con- structions of threefolds X with q₁ˆq₂ˆp_gˆ0 and P₂ˆ2 have the 9- canonical transformationW_j9KXj, which is birational, but some of them also haveW_j10KXj, which is not birational, andW_jmKXj, which is birational if and only ifmˆ9 andm11.

So, the threefolds with this property have a gap in the birationality of W_jmKXjformˆ10. This came as a surprise because the only cases of gaps in the birationality of W_jmKXj that we found were in threefolds with q₁ˆq₂ˆp_gˆP₂ˆP₃ˆ0 orq₁ˆq₂ ˆp_g ˆP₂ˆ0. Such examples with gaps are in [S3], where an example is constructed with the same properties as the exampleX46in Reid's list ([Re]), and in [Ro2].

In the present paper, we construct a threefold Xwith the properties described ± i.e. W_jmKXj is birational if and only if mˆ9 and m11, q₁ˆq₂ˆ0 and p_gˆ0, P₂ˆ2 ± and with further plurigenera P₃ˆ2, P4ˆP5ˆ4,P6ˆP7ˆ8,P8ˆ13,P9ˆ15,P10 ˆ19,P11 ˆ22.

We note thatXis birationally distinct from the threefolds appearing in the lists of [Re], pp. 358-359 and [F], pp. 151-154, 169-170, becauseX has different plurigenera from those of the threefolds in said lists.

The example X is constructed as a desingularization of a degree six hypersurfaceV P⁴ endowed with a singularity at each of the five ver- tices A0;A1;A2;A3 and A4 of the fundamental pentahedron. The con- struction is similar to those in [S4]. Precisely, we put a triple point with an infinitely-near double surface at A₀ on V, we put a triple point with an infinitely-near triple curve atA₁;A₂;A₃, and an ordinary 4-ple point atA₄. Other unimposed singularities appear on V, but they do not affect the birational invariants ofX.

The ground fieldkis an algebraically closed field of characteristic zero, which we can assume to be the field of complex numbers.

1. Imposing singularities on a degree six hypersurfaceV inP⁴. Let (x₀;x₁;x₂;x₃;x₄) be homogeneous coordinates inP⁴and let us in- dicate asf₆(X₀;X₁;X₂;X₃;X₄) a form (homogeneous polynomial) of degree 6, in the variables X₀;X₁;X₂;X₃;X₄, defining a hypersurface VP⁴ of degree six. We impose a triple point on V at each of the four vertices A0ˆ(1;0;0;0;0), A1ˆ(0;1;0;0;0), A2ˆ(0;1;0;0;0), A3ˆ(0;0;0;1;0) and an ordinary 4-ple (quadruple) point atA4ˆ(0;0;0;0;1) of the funda- mental pentahedronX₀X₁X₂X₃X₄ˆ0.

The equation forV, with the imposed singularities, is of the following type

V :f6(X0;X1;X2;X3;X4)

ˆX₀³(a33000X₁³‡ )‡X₁³(a23100X₀²X2‡ )‡X₂³( )‡X₃³( )‡X₄²( )

‡a₂₂₂₀₀X₀²X₁²X₂²‡a₂₂₁₁₀X₀²X²₁X₂X₃‡ ‡a₀₀₂₂₂X₂²X₃²X₄²ˆ0;

wherea_ijkhl2kdenotes the coefficient of the monomialX₀ⁱX₁^jX₂^kX₃^hX₄^l. Moreover, we impose a double surfaceS0infinitely nearA0in the first neighbourhood. We impose the same double surface S₀, which is locally isomorphicto a plane as in [S₂]. In addition, we impose a triple curveC_i infinitely near Ai, iˆ1;2;3 in the first neighbourhood. Ci is locally iso- morphicto a straight line as in [S1].

As an example, we provide a few details on the realization of the sin- gularity at A₀ on V. This will also enable a better understanding in the sequel of the computation of the m-canonical adjoints toV and of them- genusPmof a desingularizationXofV,s:X !V (cf. section 5). Let us consider the affine open setU03A0inP⁴given byX06ˆ0 of affine coordi- nates xˆX₁

X0 ;yˆX₂

X0 ;zˆX₃

X0 ;tˆX₄ X0

. The affine equation ofV\U0is given byf₆(1;x;y;z;t)ˆ0.

The affine coordinates ofA₀ are (0;0;0;0), so the blow-up ofP⁴ at the pointA0is locally given by the formulas:

B_x1:

xˆx₁ yˆx₁y₁ zˆx₁z₁ tˆx₁t₁ 8>

><

>>

: ; B_y2:

xˆx₂y₂ yˆy₂ zˆy₂z₂ tˆy₂t₂ 8>

><

>>

: ; B_z3 :

xˆx₃z₃ yˆy₃z₃ zˆz₃ tˆz₃t₃ 8>

><

>>

: ; B_t4:

xˆx₄t₄ yˆy₄t₄ zˆz₄t₄ tˆt₄ 8>

><

>>

:

and we considerBt4. The strict (or proper) transformV⁰ofVwith respect to

the local blow-upBt4has an affine equation given by V⁰: 1

t³₄ f6(1;x₄t₄;y₄t₄;z₄t₄;t₄)ˆa₃₁₂₀₀x₄y²₄‡ ‡a₀₀₂₂₂y²₄z²₄t³₄ˆ0:

On this threefold V⁰ we impose the plane S0\U0 given affinely by x₄ˆ0

t₄ˆ0

as a singular plane of multiplicity two (i.e. as a double plane). The conditions on the coefficients aijkhl, such that V has the double plane S0\U0infinitely nearA0, are given by

a31200ˆ0 a30210ˆ0 a30012ˆ0 a20220ˆ0 a₃₁₁₁₀ˆ0 a₃₀₂₀₁ˆ0 a₃₀₀₀₃ˆ0 a₂₀₂₁₁ˆ0 a₃₁₁₀₁ˆ0 a₃₀₁₂₀ˆ0 a₂₀₃₁₀ˆ0 a₂₀₂₀₂ˆ0 a₃₁₀₂₀ˆ0 a₃₀₁₁₁ˆ0 a₂₀₃₀₁ˆ0 a₂₀₁₂₁ˆ0 a₃₁₀₁₁ˆ0 a₃₀₁₀₂ˆ0 a₂₀₁₃₀ˆ0 a₂₀₁₁₂ˆ0 a₃₁₀₀₂ˆ0 a₃₀₀₃₀ˆ0 a₂₀₀₃₁ˆ0 a₂₀₀₂₂ˆ0:

a₃₀₃₀₀ˆ0 a₃₀₀₂₁ˆ0

In much the same way as above and precisely as in [S₁], we impose a triple curveC_i infinitely nearA_iand in the first neighbourhood, which is locally isomorphic to a straight line, foriˆ1;2;3. Further information on the above singularities can be found in [S4].

We give the final equation for our hypersurfaceVafter imposing all the above-mentioned singularities. We have chosen several coefficients as equal to zero because they are inessential for the computation of the birational invariants of a desingularizations:X !V ofV. The shortest equation with the essential coefficients is

V :f6(X0;X1;X2;X3;X4)

ˆa33000X₀³X₁³‡a32100X₀³X₁²X2‡a32001X³₀X₁²X4‡a23010X₀²X₁³X3‡a13020X0X³₁X₃²

‡a10302X0X³₂X₄²‡a03030X₁³X₃³‡a02031X₁²X₃³X4‡a01032X1X₃³X₄²‡a22200X²₀X₁²X₂²

‡a₂₂₀₂₀X₀²X₁²X₃²‡a₂₂₀₀₂X₀²X²₁X₄²‡a₂₁₂₁₀X₀²X₁X₂²X₃‡a₂₁₂₀₁X₀²X₁X₂²X₄‡

‡a21102X₀²X1X2X₄²‡a21021X₀²X1X₃²X4‡a21012X²₀X2X3X₄²‡a12012X0X₁²X3X₄²

‡a02022X₁²X₃²X₄²‡a00222X₂²X²₃X₄²ˆ0:

From here on,V denotes this last hypersurface defined by the above formf₆(X₀;X₁;X₂;X₃;X₄) for a generic choice of the parametersa_ijkhl. As a reminder of this generic choice, we sometimes call V: the generic V.

2. Imposed and unimposed singularities ofV :the actual singularities.

We consider the hypersurfaceV given at the end of section 1.

New unimposed singularities appear on the (generic) V close to the singularities imposed onV; they are actual or infinitely-near singularities.

We call a singularity on V actualto distinguish it from those which are infinitely near. We call a singularity ofV unimposedif it does not appear in the list of singulatities in section 1.

There are six unimposed actual double (straight) lines on V given by A₀A₂, A₀A₃, A₀A₄, A₁A₂, A₁A₄, A₂A₃ and the unimposed double plane cubic X1 ˆ0

X2 ˆ0

a01032X₃²X4‡a21021X₀²X3‡a21012X₀²X4ˆ0:

8<

:

The generic V has no other actual singularities. It follows that the genericV is reduced, irreducible and normal.

The cubic lies on the plane X1ˆ0 X2ˆ0

;which is simple onV. The picture of the six double lines is as follows, where the double lines are drawn in bold type.

3. The infinitely-near singularities ofV.

In section 2, we described the actual singularities onV; in the present section, we briefly describe the infinitely-near singularities. Here again, new infinitely-near singularities appear on the generic V alongside the infinitely-near singularities imposed onV. They are only double singular curves and isolated double points, so none of the unimposed singularities (be they actual or otherwise) affect the birational invariants of a desin- gularizations:X !VofV, such as the irregularities and the plurigenera

ofX. This means that, in calculating these invariants, we can assume that there are only the imposed singularities onV.

We compute said birational invariants ofXusing the theory of adjoints and pluricanonical adjoints developed in [S₁]. We can apply this theory because the singularities on the hypersurfaceV satisfy the hypotheses of [S1], i.e. it must be possible to resolve the singularities onVwith local blow- ups along linear affine subspaces; moreover, the degree six hypersurfaces in P⁴ must have singularities of codimension2 (i.e. the hypersurfaces must be normal).

Such hypotheses on the singularities are satisfied by either actual or infinitely-near singularities ofV. In particular,V is normal (section 2). To be precise, all the singularities ofVare resolved with local blow-ups either along straight lines, that are double onV and on strict transforms ofV, or along planes containing double curves and points. These planes are simple onV and on strict transforms ofV, e.g. the simple plane X₁ˆ0

X2ˆ0

;con- taining the cubic curve onV in section 2.

Having said as much, we only give details on the imposed infinitely-near singularities ofV that are needed in the sequel.

From section 1, we already have the information that we need about the triple pointA0and the double surfaceS0 infinitely nearA0.

Next, we consider the triple pointA₁onVand the blow-up atA₁. Let us consider the affine open set U₁3A₁ in P⁴ given by X₁6ˆ0 of affine co- ordinates xˆX₀

X₁; yˆX₂

X₁; zˆX₃

X₁; tˆX₄ X₁

. The affine equations of V\U₁ are given by f₆(x;1;y;z;t)ˆ0. The affine coordinates of A₁ are (0;0;0;0).

We can assume that the blow-up atA₁is the first to be performed, so we can use the local blows-upBx1;By2;Bz3;Bt4 in section 1.

The strict transform ofV\U1, with respect toBt4, is given by V_t⁰₄ :1

t³₄ f6(x4t4;1;y4t4;z4t4;t4)

ˆa₃₃₀₀₀x³₄‡ ‡a₀₃₀₃₀z³₄‡ ‡a₁₂₀₁₂x₄z₄t₄‡ ˆ0:

We are interested in the triple curve infinitely near A₁. So, we focus locally on the triple line onV_t⁰₄belonging to the exceptional divisort₄ˆ0 of the local blow-upB_t4. This triple line is given by x₄ˆ0

z₄ˆ0 t4ˆ0 8<

: :

Let us go on to consider the triple pointA₂onV, the blow-up atA₂and the affine open set U23A2 in P⁴ given by X26ˆ0 of affine coordinates

xˆX₀

X2;yˆX₁

X2;zˆX₃

X2; tˆX₄ X2

. The affine equations of V\U₂ are given byf₆(x;y;1;z;t)ˆ0. The affine coordinates ofA₂are (0;0;0;0).

Here again, we can assume that the blow-up at A2 is the first to be performed, so we can use the local blow-upsBx1;By2;Bz3;Bt4 in section 1.

The strict transform ofV\U₂, with respect toB_y2, is given by V_y⁰₂: 1

y³₂ f6(x2y2;y2;1;y2z2;y2t2)

ˆa₁₀₃₀₂x₂t²₂‡ ‡a₂₂₂₀₀x²₂y₂‡ ‡a₀₀₂₂₂y₂z²₂t²₂ˆ0:

We are interested in the triple curve infinitely near A₂, so we focus locally on the triple line onV_y⁰₂belonging to the exceptional divisory₂ˆ0 of the local blow-upBy2. This triple line is given by

x2ˆ0 y₂ˆ0 t₂ˆ0 8<

: :

Finally, let us consider the triple pointA₃onV, the blow-up atA₃and the affine open set U₃3A₃ in P⁴ given by X₃6ˆ0 of affine coordinates

xˆX0

X₃;yˆX1

X₃;zˆX2

X₃; tˆX4

X₃

. The affine equations of V\U₃ are given byf6(x;y;z;1;t)ˆ0. The affine coordinates ofA3are (0;0;0;0).

We can again assume that the blow-up atA3is the first to be performed, so we can use the local blow-upsBx1;By2;Bz3;Bt4 in section 1.

The strict transform ofV\U₃, with respect toB_x1, is given by V_x⁰₁: 1

x³₁ f6(x1;x1y1;x1z1;1;x1t1)

ˆa03030y³₁‡ ‡a22020x1y²₁‡ ‡a21021x1y1t1‡ ˆ0:

We are interested in the triple curve infinitely near A3, so we focus locally on the triple line onV_x⁰₁belonging to the exceptional divisorx1ˆ0 of the local blow-upBx1. This triple line is given by x1ˆ0

y1ˆ0 t1ˆ0 8<

: :

To end this section, we add one more item of information, drawing the picture of the tree of local blow-ups resolving the singularity at A0 and those infinitely near.

where ``ns'' means ```nonsingular''.

4. The m-canonical adjoints toVP⁴. Let

P_r ^p!^r !^p3 P₂ ^p!² P₁ ^p!¹ P₀ˆP⁴ be a sequence of blow-ups solving the singularities ofV.

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

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

where p⁰_iˆpijV_i:Vi !Vi 1 and s_jX :X !V, sˆpr p1, is a de- singularization ofVP⁴.

Let us assume thatpiis a blow-up along a subvarietyYi 1 ofPi 1, of dimensionji 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 setn_{i 1}ˆ 3‡j_{i 1}‡m_{i 1}, foriˆ1; :::;rand deg(V)ˆd.

A hypersurface Fm(d 5) of degree m(d 5), m1, inP⁴ is anm-ca- nonical adjointtoV (with respect to the sequence of blow-upsp1; :::;pr) 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, whereE_iˆp ¹(Y_{i 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 adjointF_{m(d 5)} is an global m-canonical adjoint to V (with respect top1; :::;pr) if the divisorDm is effective onPr, i.e.Dm0 (loc. cit.).

Note that, ifF_{m(d 5)} is anm-canonical adjoint toV, thenD_mjX mK, where `' denotes linear equivalence andK denotes a canonical divisor onX.

In our above example, an order can be established in the sequence of blow-ups, e.g. let us assume thatp1is the blow-up at the triple pointA0,p2

is the blow-up along the double surfaceS0infinitely nearA0,p3is the blow- up at the triple point A₁, p₄ is the blow-up along the triple curve C₁ in- finitely nearA₁,p₅is the blow-up at the triple pointA₂,p₆is the blow-up along the triple curveC2 infinitely nearA2,p7is the blow-up at the triple pointA3,p8is the blow-up along the triple curveC3infinitely nearA3and the blow-upp9 is the one at the 4-ple pointA₄.

The exampleVhas degreedˆ6 andD_m, relative to ourX, is given by:

D_mˆp_r p₃fp₂[p₁(F_m)] mE₂g mE₄ mE₆ mE₈ mE₉; …†

where E_i is the exceptional divisor of the blow-up p_i and, to be more specific,E₁is the exceptional divisor of the blow-upp₁at the triple point A0,E2is the exceptional divisor of the blow-upp2alongC1, ... andE9is the exceptional divisor of the blow-upp9 at the 4-ple pointA4.

No other exceptional divisors are subtracted inDmbecause, as we said before, the unimposed singularities are either actual or infinitely-near double singular curves or isolated double points on our (generic) V. Put more precisely, the exceptional divisors of the blow-ups along the double curves appear with coefficientniˆ0 in the above expression ofDmand the exeptional divisors of the blow-ups along simple planes appear again with

coefficientnjˆ0. Since we have resolved all the unimposed singularities with blow-ups either along double curves or along simple planes, only the exeptional divisors E₂;E₄;E6;E₈ and E9 appear inDm. Note, moreover, that the exceptional divisor of a blow-up at a triple point also appears with coefficientn_hˆ0 inD_m.

5. The plurigenera of a desingularizationXofV.

Let us consider the equation ofV:f₆(X₀;X₁;X₂;X₃;X₄)ˆ0 at the end of section 1 and arrange the formf₆according to the powers ofX₄.

f₆ˆW₄(X₀;X₁;X₂;X₃)X₄²‡W₅(X₀;X₁;X₂;X₃)X₄‡W₆(X₀;X₁;X₂;X₃);

…†

whereW_i(X0;X1;X2;X3) is a form of degreeiinX0;X1;X2;X3and precisely W₄(X0;X1;X2;X3)ˆa10302X0X₂³‡a01032X1X₃³‡a22002X₀²X₁²‡ ‡a00222X₂²X₃²: Next, let us consider the hypersurfaceFm, appearing in (*) section 4 and assume that its equation is Fm(X0;X1;X2;X3;X4)ˆ0, of degree m.

Arranging the formFm according to the powers ofX4, we can write F_m(X₀;X₁;X₂;X₃;X₄)

…†

ˆc_s(X0;X1;X2;X3)X₄^{m s}‡c_s‡1(X0;X1;X2;X3)X^{m s}₄ ¹‡ ‡

‡c_m(X0;X1;X2;X3);

wherec_j(X₀;X₁;X₂;X₃) is a form of degree jin X₀;X₁;X₂;X₃ ands is an integer satisfying 0sm.

Under the sole hypothesis thatV has a 4-ple point atA4the following lemma holds.

LEMMA1. With the above notations, ifF_m is an m-canonical adjoint (be it global or not),then, modulo V :f₆ˆ0, we canassume that sm 1 in(***); i.e.ifFm:Fmˆ0is an m-canonical adjoint, then we can assume that

F_mˆc_{m 1}(X₀;X₁;X₂;X₃)X₄‡c_m(X₀;X₁;X₂;X₃):

Moreover, we have the equality

c_{m 1}(X0;X1;X2;X3)ˆAm 5(X0;X1;X2;X3)W₄(X0;X1;X2;X3);

where Am 5(X0;X1;X2;X3)is a form of degree m 5inX0;X1;X2;X3and W₄(X₀;X₁;X₂;X₃)is defined above in(**).

The idea for the proof of the above lemma came from M. C. Ronconi [CR], [Ro1]. A detailed proof can be found in [S4] (Lemma 1, section 5).

REMARK 1. In Lemma 1, we have F_mˆA_{m 5}W₄X₄‡c_m. We see that, ifA_{m 5}ˆ0, then F_mdefines a globalm-canonical adjointF_mtoV, whereas if Am 56ˆ0, then Fm is a ``non-global'' m-canonical adjoint to V. The non-global m-canonical adjoints to V are important for estab- lishing the birationality of the m-canonical transformation W_jmKXj (see next section).

The following lemma is proved in [S₄], Lemma 2, section 12, where the singularities at three fundamental points on a degree six hypersurfaceV⁰ differ from those on V in the present case. More precisely, V⁰ has three triple points with an infinitely-near double plane, whereas V has three triple points with an infinitely-near triple curve. But the proof remains the same in both cases.

LEMMA2. The m-canonical adjoint to V given by

F_m:A_{m 5}(X₀;X₁;X₂;X₃)W₄(X₀;X₁;X₂;X₃)X₄‡c_m(X₀;X₁;X₃;X₄)ˆ0;

has the following property

Dmj_X 0()Dm‡E90;

where Dmˆp_r p₃fp₂[p₁(Fm)] mE2g mE4 mE6 mE8 mE9, is defined in(*),section4.

REMARK 2. Roughly speaking, the result in Lemmas 1 and 2, that permits us an easy computation of the m-genus Pm (8m) of a desingu- larizations:X !V ofV, is the following. Our degree six hypersurface V has a 4-ple point, so from Lemma 1 we c an assume that them-cano- nical adjointF_m is defined by a form of the typeF_mˆA_{m 5}W₄X₄‡c_m, where the variableX₄ appears to the power 1. In order to compute the linear conditions given by the other singularities to the hypersurfaces Fm so that they arem-canonical adjoints to V, i.e. to obtain Dmj_X 0, we find that we do not need to restrict D_m to X and, after imposing D_mjX 0, we only need to haveD_m‡E₉0. This follows from the fact that F_m contains the variable X₄ to the power 1, whereas the form f₆ defining V contains the variable X4 to the power 2, and also from the particular singularities obtained in our examples. We note that E9 has to be added toD_m, otherwiseD_m may not be effective (whenA_{m 5}6ˆ0,

see Remark 1). So it is very easy to compute the conditions onFm such that Dm‡E9 is effective and, since Pm ˆ number of linearly in- dependent forms contained inFm(cf. [S₁]), the computation ofPm,8m, is very easy too.

Now, we are ready to compute the plurigenera of a desingularization s:X !V ofV. Let us write

Am 5(X0;X1;X2;X3)X4ˆ X

i‡j‡k‡hˆm 5

aijkhX₀ⁱX₁^jX₂^kX₃^h

! X4;

c_m(X0;X1;X2;X3)ˆ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

b_ijkhX₀ⁱ⁰X₁^j0X₂^k0X₃^h0;

wherea_ijkh;b_ijkh2k.

First let us consider the two blows-upp₁andp₂. We know that the blow-up p1 ofP⁴ atA0 is given byBx1;By2;Bz3;Bt4 (cf. section 1). Let us consider the affine open setU0ˆ fX06ˆ0gas in section 1.

The total transform ofF_m\U₀with respect toB_t4 is given by B_t4(F_m\U₀):A_{m 5}(1;x₄t₄;y₄t₄;z₄t₄)W₄(1;x₄t₄;y₄t₄;z₄t₄)t₄‡

c_m(1;x4t4;y4t4;z4t4;t4)ˆ0:

The double surface S₀ infinitely near A₀ in affine coordinates (x4;y4;z4;t4) is given by x4ˆ0

t4ˆ0

(cf. section 1).

The blow-upp2 alongS0is locally given by the formulas:

B_x41 :

x₄ˆx₄₁ y₄ˆy₄₁ z₄ˆz₄₁ t4ˆx41t41

8>

><

>>

: ; B_t42 :

x₄ˆx₄₂t₄₂ y₄ˆy₄₂ z₄ˆz₄₂ t4ˆt42

8>

><

>>

: :

The total transform ofB_t4(F_m\U₀) with respect toB_x41 is given by B_x41[B_t4(Fm\U₀)]:

Am 5(1;x²₄₁t41;x41y41t41;x41z41t41)W₄(1;x²₄₁t41;x41y41t41;x41z41t41)x41t41‡ c_m(1;x²₄₁t₄₁;x₄₁y₄₁t₄₁;x₄₁z₄₁t₄₁;x₄₁t₄₁)ˆ0:

With the above notations, this total transform is given by B_x41[B_t4(Fm\U0)]:

X

i‡j‡k‡hˆm 5

a_ijkhx^2j‡k‡h₄₁ y^k₄₁z^h₄₁

!

W₄(1;x²₄₁t₄₁;x₄₁y₄₁t₄₁;x₄₁z₄₁t₄₁)x₄₁t₄₁

‡ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

b_ijkhx^2j₄₁^0‡k0‡h0y^k₄₁⁰z^h₄₁⁰ ˆ0:

The following claims hold true; they are corollaries to Lemma 1 and 2 and consequences of the desingularization ofV.

CLAIM1. The composition of the two local blows-upB_x41 B_t4coincides, up to isomorphisms, with the desingularizations_jX on the affine open set V_x41, becauseV_x41 is nonsingular (see the tree of blow-ups at the end of section 3). In fact,Vx41is isomorphicto an open set onXand the two above morphisms can be identified onVx41.

CLAIM2. SinceF_mis anm-canonical adjoint toV, by definition we have D_mjX 0; so, from Lemma 2, we can say that:D_m‡E₉0.

CLAIM3. From Claims 1 and 2, we deduce (up to isomorphisms) that B_x41[B_t4(Fm\U0)] mE2‡E90:

This last inequality is equivalent to the following equality of polynomials

X

i‡j‡k‡hˆm 5

aijkhx^2j‡k‡h₄₁ y^k₄₁z^h₄₁

W₄(1;x²₄₁t41;x41y41t41;x41z41t41)x41t41

‡ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

b_ijkhx^2j₄₁^0‡k0‡h0y^k₄₁⁰z^h₄₁⁰ ˆx^m₄₁(:::)

CLAIM 4. Since W₄(1;x²₄₁t41;x41y41t41;x41z41t41)ˆx³₄₁(:::), the latter equality of polynomials is equivalent to the inequalities

2j‡k‡h‡3‡1m 2j⁰‡k⁰‡h⁰m

; i:e: ji‡1

j⁰i⁰

:

Next, let us consider the two blows-upp3andp4. As in section 3, we can assume that the first blow-up that we perform isp3atA1, so we can use the local blows-upB_x1;B_y2;B_z3;B_t4 in section 1.

As in the above case ofp₁andp₂, here too forp₃andp₄, we find that the

total transform ofFm\U1 with respect toBt4 is given by B_t4(Fm\U1):Am 5(x4t4;1;y4t4;z4t4)W₄(x4t4;1;y4t4;z4t4)t4

‡c_m(x₄t₄;1;y₄t₄;z₄t₄;t₄)ˆ0:

The triple curveC1infinitely nearA1in affine coordinates (x4;y4;z4;t4) is given by (section 3) x4ˆ0

z₄ˆ0 t₄ ˆ0 8<

: :

The blow-upp4 alongC1is locally given by the formulas:

Bx41 :

x₄ˆx₄₁ y₄ˆy₄₁ z4ˆx41z41

t4ˆx41t41

8>

><

>>

: ; Bz42 :

x₄ˆx₄₂z₄₂ y₄ˆy₄₂ z4ˆz42

t4ˆz42t42

; Bt43 :

x₄ ˆx₄₃t₄₃ y₄ˆy₄₃ z4ˆz43t43

t4ˆt43

8>

><

>>

: :

8>

><

>>

:

The total transform ofB_t4(F_m\U₁) with respect toB_x41 is given by B_x41[B_t4(F_m\U₁)]:

X

i‡j‡k‡hˆm 5

a_ijkhx^2i‡k‡2h₄₁ y^k₄₁z^h₄₁

W₄(x²₄₁t₄₁;1;x₄₁y₄₁t₄₁;x²₄₁z₄₁t₄₁)x₄₁t₄₁

‡ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

bijkhx²ⁱ₄₁^0‡k0‡2h0y^k₄₁⁰z^h₄₁⁰ ˆ0:

From the analogous four claims written above and from the equality W₄(x²₄₁t₄₁;1;x₄₁y₄₁t₄₁;x²₄₁z₄₁t₄₁)ˆx⁴₄₁(:::);

we obtain the inequalities

2i‡k‡2h‡4‡1m 2i⁰‡k⁰‡2h⁰m

; i:e: i‡hj

i⁰‡h⁰j⁰

:

Let us move on now to consider the two blows-upp5 andp6. Once again, we can assume that the blow-upp3atA2is performed first, so we can again use the local blows-upBx1;By2;Bz3;Bt4 in section 1.

As in the above cases, here for p₅ and p₆ we obtain that the total transform ofF_m\U₂, with respect toB_y2, is given by

B_y2(Fm\U2):Am 5(x2y2;y2;1;y2z2)W₄(x2y2;y2;1;y2z2)y2t2

‡c_m(x2y2;y2;1;y2z2;y2t2)ˆ0:

The triple curveC2infinitely nearA2in affine coordinates (x2;y2;z2;t2) is given by (section 3) x2ˆ0

y₂ˆ0 t₂ˆ0 8<

: :

The blow-upp6alongC2is locally given by the formulas:

Bx21 :

x₂ˆx₂₁ y₂ˆx₂₁y₂₁ z2ˆz21

t2ˆx21t21

8>

><

>>

: ; By22 :

x₂ˆx₂₂y₂₂ y₂ˆy₂₂ z2ˆz22

t2ˆy22t22

8>

><

>>

: ; Bt23 :

x₂ˆx₂₃t₂₃ y₂ˆy₂₃t₂₃ z2ˆz23

t2ˆt23

8>

><

>>

: :

The total transform ofB_y2(F_m\U₂) with respect toB_x21 is given by B_x21[B_y2(F_m\U₂)]:

X

i‡j‡k‡hˆm 5

a_ijkhx^2i‡j‡h₂₁ y^j₂₁z^h₂₁

W₄(x²₂₁y21;x21y21;1;x21z21)x²₂₁y21t21

‡ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

b_ijkhx²ⁱ₂₁^0‡j0‡h0y₂₁^j0z^h₂₁⁰ ˆ0:

From the same four claims written above, and from the equality W₄(x²₂₁y21;x21y21;1;x21z21)ˆx²₂₁(:::);

we obtain the inequalities

2i‡j‡h‡2‡2m 2i⁰‡j⁰‡h⁰m

; i:e: ik‡1

i⁰k⁰

:

Finally, considering the two blows-upp7andp8, as in the case ofp5

andp₆, we obtain the inequalities i‡2j‡k‡2‡2m i⁰‡2j⁰‡k⁰m

; i:e: jh‡1

j⁰h⁰

: Joining the above inequalities, we obtain

i‡hji‡1k‡2; jh‡1 i⁰‡h⁰j⁰i⁰k⁰; j⁰h⁰

: …†

From the inequalities in the first line of (**), we deduce j2,i1, h1. Bearing in mind thati‡j‡k‡hˆm 5,

i) there are no values ofi;j;k;hsatisfying (**) and corresponding tom, form8;

ii) the values [iˆ1,jˆ2,kˆ0,hˆ1] correspond tomˆ9;

iii) there are no values ofi;j;k;hsatisfying (**) and corresponding tomˆ10;

iv) the two sets of values [iˆ2, jˆ3, kˆ0, hˆ1] and [iˆ1, jˆ3,kˆ0,hˆ2] satisfy (**) and correspond tomˆ11, and so on; there are values ofi,j,k,hsatisfying (**) that correspond to any value ofm12.

As for the inequalities in the second line of (**), and given that i⁰‡j⁰‡k⁰‡h⁰ˆm,

1) there are no values ofi⁰,j⁰,k⁰,h⁰satisfying (**) and corresponding to mˆ1;

2) the two sets of values [i⁰ˆj⁰ˆ1, k⁰ˆh⁰ˆ0] and [j⁰ˆh⁰ˆ1, i⁰ˆk⁰ˆ0] satisfy (**) and correspond tomˆ2;

3) the two sets of values [i⁰ˆj⁰ˆk⁰ˆ1, h⁰ˆ0] and [i⁰ˆj⁰ˆh⁰ˆ1, k⁰ˆ0] satisfy (**) and correspond tomˆ3;

4) there are 4 sets of values satisfying (**) and corresponding tomˆ4, there are also 4 sets of values satisfying (**) and corresponding to mˆ5, 8 sets satisfying (**) and corresponding tomˆ6 and 8 sets sa- tisfying (**) and corresponding tomˆ7.

5) The following sets [i⁰ˆj⁰ˆ3, k⁰ˆh⁰ˆ0], [i⁰ˆj⁰ˆh⁰ˆ2, k⁰ˆ0], [i⁰ˆj⁰ˆ2,k⁰ˆh⁰ˆ1], [i⁰ˆ2,j⁰ˆ3,k⁰ˆ0,h⁰ˆ1] are 4 of the 8 sets of values satisfying (**) that correspond tomˆ6.

The following sets [i⁰ˆj⁰ˆ3, k⁰ˆ1, h⁰ˆ0], [i⁰ˆh⁰ˆ2, j⁰ˆ3, k⁰ˆ0], [i⁰ˆ1,j⁰ˆ3,k⁰ˆ1,h⁰ˆ2], [i⁰ˆ1,j⁰ˆh⁰ˆ3,k⁰ˆ0] are 4 of the 8 sets of values satisfying (**) that correspond tomˆ7.

CONSEQUENCES. Let us just recall that we have written the equation of anm-canonical adjointF_m as follows:

F_m:A_{m 5}(X₀;X₁;X₂;X₃)W₄(X₀;X₁;X₂;X₃)X₄‡c_m(X₀;X₁;X₃;X₄)ˆ0;

where

A_{m 5}(X₀;X₁;X₂;X₃)X₄ˆ

X

i‡j‡k‡hˆm 5

a_ijkhX₀ⁱX₁^jX₂^kX₃^h

X₄

and

c_m(X0;X1;X2;X3)ˆ X

i⁰‡j⁰‡h⁰‡l⁰ˆm

bijkhX₀ⁱ⁰X₁^j0X₂^k0X₃^h0:

From i),...,vi), we deduce that the formAm 5is zero if and only ifm8 and mˆ10.

Since them-genusPmof a desingularizationXofVis the number of the linearly independent forms defining m-canonical adjoints toV (cf. [S1]), from 1), ..., 4), we deduce the following results regarding the plurigenera of a desingularizationXofV.

From 1), we can establish that there are no 1-canonical adjoints (also called canonical adjoints) toV; this implies that the geometricgenus ofXis pgˆ0.

From 2), we find that F₂:c₂(X₀;X₁;X₃;X₄)ˆX₁(l₁X₀‡l₂X₃)ˆ0, wherel_i2k; this implies that the bigenus ofXisP₂ˆ2.

From 3), we learn thatF3:c₃(X0;X1;X3;X4)ˆX0X1(m₁X2‡m₂X3)ˆ0, m_i2k; this implies that the trigenus ofXisP₃ˆ2.

From 4), we obtain thatP4ˆP5ˆ4,P6ˆ8 andP7ˆ8.

In addition,Xhas the plurigeneraP8ˆ13,P9ˆ15,P10ˆ19,P11ˆ22.

6. The m-canonical transformationW_jmKXj; m2.

Let us useam:V !P^{Pm 1}to indicate the rational transformation associated with the linear system of m-canonical adjoints Fm to V. The following triangle

is commutative.

Let us consider the linear system ofm-canonical adjointsF_m. From i) and 1), ..., 4) and the Consequences, we can see that if 2m5, thenF_m is given by c_m(X₀;X₁;X₃;X₄)ˆ0; moreover, the rational transformation amhas the genericfiber of dimension1. From the commutativity of the above triangle,W_jmKXjalso has the genericfiber of dimension1.

From i) and 5) and the Consequences, we know thatFm, formˆ6;7, is again given byc_m(X₀;X₁;X₃;X₄)ˆ0, and that the rational transformation a_m, as well asW_jmKXj, is generically 2:1. As a consequence of this and of the