THE CHERN NUMBERS OF THE NORMALIZATION OF AN ALGEBRAIC THREEFOLD

THE CHERN NUMBERS OF THE NORMALIZATION OF AN ALGEBRAIC THREEFOLD

WITH ORDINARY SINGULARITIES by

Shoji Tsuboi

Abstract. — By a classical formula due to Enriques, the Chern numbers of the non- singular normalization X of an algebraic surface S with ordinary singularities in P³(C) are given byR

Xc²₁ =n(n−4)²−(3n−16)m+ 3t−γ,R

Xc2=n(n²−4n+ 6)−(3n−8)m+ 3t−2γ, wheren= the degree ofS,m= the degree of the double curve (singular locus)DSofS,t= the cardinal number of the triple points ofS, and γ=the cardinal number of the cuspidal points ofS. In this article we shall give similar formulas for an algebraic threefoldX with ordinary singularities inP⁴(C) (Theorem 1.15, Theorem 2.1, Theorem 3.2). As a by-product, we obtain a numerical formula for the Euler-Poincar´e characteristic χ(X,TX) with coefficient in the sheaf TX of holomorphic vector fields on the non-singular normalizationX ofX(Theorem 4.1).

Résumé (Les nombres de Chern de la normalisée d’une variété algébrique de dimension3à points singuliers ordinaires)

Par une formule classique due `a Enriques, les nombres de Chern de la normalisation non singuli`ereX de la surface alg´ebriqueSavec singularit´es ordinaires dansP³(C) sont donn´es par R

Xc²₁ =n(n−4)²−(3n−16)m+ 3t−γ,R

Xc2 =n(n²−4n+ 6)−(3n−8)m+ 3t−2γ, o`unest le degr´e deS,mest le degr´e de la courbe double (lieu singulier)DS deS,test le nombre de points triples deS, etγest le nombre de points cuspidaux de S. Dans cet article nous donnons des formules similaires pour une “threefold” alg´ebriqueXavec singularit´es ordinaires dansP<sup>4</sup>(C) (Th´eor`eme 1.15, Th´eor`eme 2.1, Th´eor`eme 3.2). Comme application, nous obtenons une formule num´erique pour la caract´eristique d’Euler-Poincar´eχ(X,TX) `a coefficients dans le faisceauTX de champs de vecteurs holomorphes de la normalisation non singuli`ere XdeX (Th´eor`eme 4.1).

2000 Mathematics Subject Classification. — Primary 14G17; Secondary 14G30, 32C20, 32G05.

Key words and phrases. — Chern number, threefold, hypersurface, ordinary singularity, normalization.

This work is supported by the Grant-in-Aid for Scientific Research (No. 13640083), The Ministry of Education, Science and Culture, Japan.

Introduction

An irreducible hypersurface X in the complex projective 4-spaceP⁴(C) is called an algebraic threefold with ordinary singularities if it is locally isomorphic to one of the following germs of hypersurface at the origin of the complex 4-spaceC⁴ at every point ofX:

(0.1)



















(i) w= 0 (simple point)

(ii) zw= 0 (ordinary double point) (iii) yzw= 0 (ordinary triple point) (iv) xyzw= 0 (ordinary quadruple point) (v) xy²−z²= 0 (cuspidal point)

(vi) w(xy²−z²) = 0 (stationary point)

where (x, y, z, w) is the coordinate onC⁴. These singularities arise if we project a non- singular threefold embedded in a sufficiently higher dimensional complex projective space to its four dimensional linear subspace by a generic linear projection ([R]), though the singularities (iv) and (vi) above do not occur in the surface case. This fact can also be proved by use of the classification theory of multi-germs oflocally stable holomorphic maps ([M-3], [T-1]). Indeed, in the threefold case, the pair of dimensions of the source and target manifolds belongs to the so-callednice range([M-2]). Hence the multi-germ of ageneric linear projection at the inverse image of any point ofX isstable,i.e., stable under small deformations ([M-4]).

In [T-2] we have proved, for an algebraic threefoldX with ordinary singularities inP⁴(C) which is free from quadruple points, a formula expressing the Euler number χ(X) of the non-singular normalizationX ofX in terms of numerical characteristics ofX and its singular loci. Note that, by the Gauss-Bonnet formula, the Euler number χ(X) is equal to the Chern numberR

Xc3, wherec3denotes the top Chern class ofX. In §1 we shall extend this formula to the general case where X admits quadruple points. In this general case, we need to blow upX twice. First, along the quadruple point locus, and secondly, along the triple point locus. It turns out that the existence of quadruple points adds only the term 4#Σqto the formula, where #Σqdenotes the cardinal number of the quadruple point locus Σq. Using Fulton-MacPherson’s inter- section theory, especially, theexcess intersection formula ([F], Theorem 6.3, p. 102), the blow-up formula (ibid., Theorem 6.7, p. 116), the double point formula (ibid., Theorem 9.3, p. 166) and theramification formula (ibid., Example 3.2.20, p. 62), we compute thepush-forwards f∗[D]² andf∗[D]³forDthe inverse image of the singular locus of X by the normalization map in order to know the Segre classes s(J , X)i

(06i62) of the singular subschemeJ defined by the Jacobian ideal ofX. In§2 we shall give a formula for the Chern numberR

Xc³₁=−[KX]³, where [KX] is the canonical class ofX. The expressions forf∗[D]²andf∗[D]³obtained in§1 enable us to compute it, because [KX] =f^∗[X+KY]−[D] by thedouble point formula, where

KY is the canonical divisor of P⁴(C). In §3 we shall give a formula for the Chern numberR

Xc1c2. In fact, we shall calculate the Euler-Poincar´e characteristicχ(X, KX) with coefficient in the canonical line bundle of X, which is equal to −(1/24)R

Xc1c2

by the Riemann-Roch theorem. In §4, as a by-product, we shall give a numerical formula for the Euler-Poincar´e characteristic χ(X,TX) with coefficient in the sheaf TX of holomorphic tangent vector fields onX.

Notation and Terminology

Throughout this article we fix the notation as follows:

Y :=P⁴(C): the complex projective 4-space,

X: an algebraic threefold with ordinary singularities inY,

J: the singular subscheme ofX defined by the Jacobian ideal ofX, D: the singular locus ofX,

T: the triple point locus ofX, which is equal to the singular locus ofD,

C: the cuspidal point locus ofX, precisely, its closure, since we always considerC contains the stationary points,

Σq: the quadruple point locus ofX, Σs: the stationary point locus ofX, n_X:X→X: the normalization ofX,

f : X → Y: the composite of the normalization map n_X and the inclusion ι : X ,→ Y,

J: the scheme-theoretic inverse ofJ byf,

D,T,C and Σq: the inverse images ofD,T,C and Σqbyf, respectively, Σs=T∩C: the intersection ofT andC.

We put

n:= degX (the degree ofX in P⁴(C)), m:= degD, t:= degT , γ:= degC.

Note thatT andCare non-singular curves, intersecting transversely at Σs, and that the normalizationXofXis also non-singular. Calculating by use of local coordinates, we can easily see the following:

(i) J containsD, and theresidual schemetoD inJ is the reduced schemeC,i.e., IJ =ID⊗IXIC, whereIJ,ID,IC are the ideal sheaves ofJ,D andC, respectively (cf.[F], Definition 9.2.1, p. 160);

(ii) D is a surface with ordinary singularities, whose singular locus isT,

(iii) D is the double point locus of the map f : X → Y, i.e., the closure of {q∈X |#f⁻¹(f(q))>2};

(iv) the mapf|D:D→D is generically two to one, simply ramified atC;

(v) the mapf|T :T →T is generically three to one, simply ramified at Σs.

Furthermore, we need the following diagram consisting of two fiber squares:

(0.2)

Xy⁰⁰−−−−→^f00 Y⁰⁰

τ_T0

 y^σT0 Xy⁰ −−−−→^f0 Y⁰

τΣq

 y^σΣq X −−−−→

f Y, which is defined as follows:

σΣq :Y⁰→Y: the blowing-up ofY along the quadruple point locus ΣqofX, X⁰: the proper inverse image ofX byσΣq,

X⁰:=X×_XX⁰: the fiber product ofX andX⁰ overX,

n_X⁰ :X⁰→X⁰: the projection to the second factor ofX×_XX⁰, which is nothing but the normalization ofX⁰,

f⁰:X⁰ →Y⁰: the composite of the normalization mapn_X⁰ and the inclusion ι⁰ :X⁰,→Y⁰,

Σq: the inverse image of the quadruple point locus ΣqofX byf,

τΣq :X⁰→X: the projection to the first factor ofX×_XX⁰, which is nothing but the blowing-up ofX along Σq,

D⁰,T⁰,C⁰and Σs⁰: the proper inverse images ofD,T,Cand ΣsbyσΣq, respectively.

D⁰,T⁰ andC⁰: the proper inverse images ofD,T andCbyτΣq, which are equal to the inverse images ofD⁰,T⁰ andC⁰ byf⁰, respectively,

Σs⁰: the inverse image of ΣsbyτΣq, which is equal to T⁰∩C⁰, σ_T⁰ :Y⁰⁰→Y⁰: the blowing-up ofY⁰ alongT⁰,

X⁰⁰: the proper inverse image ofX⁰ byσ_T⁰,

X⁰⁰:=X⁰×_X⁰X⁰⁰: the fiber product ofX⁰ andX⁰⁰ overX⁰,

n_X⁰⁰:X⁰⁰→X⁰⁰: the projection to the second factor ofX⁰×_X⁰X⁰⁰, which is nothing but the normalization ofX⁰⁰

f⁰⁰:X⁰⁰→Y⁰⁰: the composite of the normalization mapn_X⁰⁰ and the inclusion ι⁰⁰:X⁰⁰,→Y⁰⁰,

τT⁰ :X⁰⁰→X⁰: the projection to the first factor ofX⁰×_X⁰X⁰⁰, which is nothing but the blowing-up ofX⁰ alongT⁰,

D⁰⁰, T⁰⁰,C⁰⁰and Σs⁰⁰: the proper inverse images ofD⁰,T⁰,C⁰ and Σs⁰ byσ_T, respectively,

D⁰⁰, T⁰⁰andC⁰⁰: the proper inverse images ofD⁰,T⁰ andC⁰ byτT⁰, which are equal to the inverse images of D⁰⁰,T⁰⁰ andC⁰⁰ byf⁰⁰, respectively,

Σs⁰⁰: the inverse image of Σs⁰ byτT⁰, which is equal to T⁰⁰∩C⁰⁰.

We also use the following notation throughout this article:

[α]: the rational equivalence class of an algebraic cycleα,

α·β: the intersection class of two algebraic cycle classes [α] and [β].

Finally, we give the definitions ofregular embeddingsandlocal complete intersection morphisms of schemes.

Definition 0.1. — We say a closed embedding ι : X → Y of schemes is a regular embedding of codimension d if every point in X has an affine neighborhood U in Y, such that if A is the coordinate ring of U, I the ideal of A defining X, then I is generated by a regular sequence of lengthd.

If this is the case, the conormal sheafI/I², whereI is the ideal sheaf ofX in Y, is a locally free sheaf of rankd. Thenormal bundle to X inY, denoted byNXY, is the vector bundle onXwhose sheaf of sections is dual toI/I². Note that the normal bundleNXY is canonically isomorphic to the normal coneCXY for a (closed) regular embeddingι:X→Y since the canonical map from Sym(I/I²) toS^·:= Σ^∞_k=0I^k/I^k+1 is an isomorphism (cf.[F], Appendix B, B.7).

Definition 0.2. — A morphism f : X → Y is called a local complete intersection morphism of codimension d iff factors into a (closed) regular embeddingι:X →P of some constant codimension e, followed by a smooth morphism p : P → Y of constant relative dimensiond+e.

1. The computation of R

Xc3

In [T-2] we have proved, for an algebraic threefoldX with ordinary singularities in P⁴(C) which is free from quadruple points, a formula expressing the the Euler number χ(X) of the non-singular normalizationX ofX in terms of numerical characteristics of X and its singular loci. We recall its proof briefly. We have first proved the following:

Theorem 1.1 ([T-2], Theorem 2.1). — We have a linear pencil L :=S

λ∈P¹Xλ on X, consisting of hyperplane sectionsXλ of X in P⁴(C), whose pull-backL:=S

λ∈P¹Xλ

toX by the normalization mapf :X →X has the following properties: There exists a finite set {λ1, . . . , λc} of points ofP¹ such that

(i) Xλ is non-singular forλwith λ6=λi (16i6c), and

(ii) Xλi is a surface with only one isolated ordinary double point which is contained inXrf⁻¹(C∞)for any iwith 16i6c,

where c is the class of X, i.e., the degree of the top polar class [M3] of X in P⁴(C) (cf.[P]), and C∞ the base point locus of the linear pencil L, which is an irreducible curve with m(= degD) ordinary double points in P²(C) whose degree is equal to n(= degX).

Letσ:Xb →X be the blowing-up alongC∞:=f⁻¹(C∞), andLb:=S

λ∈P¹Xcλ the proper inverse ofL:=S

λ∈PXλ. ThenLbgives a fibering ofXb overP¹(C). Hence the Euler numberχ(X) ofb Xb is given by

χ(X) =b χ(P¹(C))χ(Xcλ) + Σ^c_j=1(χ(Xdλj)−χ(Xcλ))

= 2χ(Xcλ)−c

where Xcλ denotes a generic fiber of the fiber space Xb → P¹. The second equality above follows from the fact that a topological 2-cycle vanishes when λ → λj for j= 1, . . . , c. We putEb:=σ⁻¹(C∞). Then, sinceXbrEb'XrC∞,

χ(X)b −χ(X) =χ(E)b −χ(C∞)

=χ(P¹(C))χ(C∞)−χ(C∞)

=χ(C∞) Hence,

χ(X) =χ(Xb)−χ(C∞) = 2χ(Xcλ)−χ(C∞)−c (1.1)

= 2χ(Xλ)−χ(C∞)−c.

Since C∞ is a curve whose degree is equal to n with m ordinary double points in P²(C), the genusg(C∞) is given by

g(C∞) =1

2(n−1)(n−2)−m.

Hence,

(1.2) χ(C∞) = 2−2g(C∞) = 2−(n−1)(n−2) + 2m.

Note thatXλ is a surface with ordinary singularities in a hyperplaneHλ'P³(C) of degreen, whose numerical characteristics related to its singularities are as follows:

– the degree of its double curveDλ=m – #{triple points ofXλ}=t,

– #{cuspidal points ofXλ}=γ.

Therefore, by the classical formula,

(1.3) χ(Xλ) =n(n²−4n+ 6)−(3n−8)m+ 3t−2γ By (1.1), (1.2) and (1.3), we have the following:

Proposition 1.2 ([T-2], Proposition 2.2)

(1.4)

χ(X) = 2n(n²−4n+ 6)−2(3n−8)m+ 6t−4γ

−2 + (n−1)(n−2)−2m−c

=n(2n²−7n+ 9)−2(3n−7)m+ 6t−4γ−c

Even ifX admits quadruple points, Theorem 1.1 and Proposition 1.2 above can be proved without change of their proofs in [T-2]. Hence what we have to do is compute theclass c of X, i.e., the degree of the top polar class [M3] of X in P⁴(C). By the result due to R. Piene ([P], Theorem (2.3)), the top polar class [M3] ofX is given by (1.5) [M3] = (n−1)³h³−3(n−1)²h²∩s2−3(n−1)h∩s1−s0,

where h denotes the hyperplane section class and si i-th Segre class s(J, X)i (0 6 i 6 2). Since f∗s(J, X)i = s(J, X)i (0 6 i 6 2), it suffices to compute the Segre classess(J, X)i and their push-forwards byf. To compute the Segre classs(J, X)i, the following proposition is useful.

Proposition 1.3 ([F], Proposition 9.2, p. 161). — Let D⊂W ⊂V be closed embeddings of schemes, withV ak-dimensional variety, andD a Cartier divisor onV. LetR be the residual scheme toD in W. Then, for allm,

s(W, V)m=s(D, V)m+ Σ^k−m_j=0

k−m j

[−D]^j·s(R, V)m+j

inAm(W), the m-th rational equivalence class group of algebraic cycles onW. In our case, sinceD=f⁻¹(D) is a Cartier divisor, its normal coneCDX toDinX is isomorphic toOX(D)|D, the restriction toD of the line bundleOX(D) associated toD. Therefore, the total Segre classs(D, X) ofDin X is given as follows:

s(D, X) =c(OX(D)_|D)⁻¹∩[D]

= [D]−c1(OX(D)|D)∩[D] +c1(OX(D)|D)²∩[D]

= [D]−[D]²+ [D]³. SinceC is non-singular,

c(NC/X)⁻¹∩[C] = [C]−c1(NC/X)∩[C].

Hence, applying Proposition 1.3 above to W =J,D=f⁻¹(D) andR=C, we have (1.6)





s(J, X)2= [D]

s(J, X)1=−[D]²+ [C]

s(J, X)0= [D]³−c1(NC/X)∩[C]−3D·C

whereNC/X is the normal bundle ofC inX. Sincef∗[D] = 2[D], it follows from the first identity in (1.6) that

s(J, X)2= 2[D].

In what follows we use the notation in the diagram (0.2) freely without mention.

Lemma 1.4

(1.7) σ_Σq^∗ [D] = [D⁰] + 6j⁰_∗X

q

[H_q⁰],

whereH_q⁰ is a hyperplane ofEq :=σ⁻¹_Σq(q)'P³(C)for each quadruple pointq, and j⁰ the inclusion mapΣqEq,→Y⁰.

Proof. — Since the multiplicity ofDat each quadruple pointqofX is 6, (1.7) follows from theblow-up formula ([F], Theorem 6.7, p. 116 and Corollary 6.7.1, p. 117).

We consider the following fiber square:

(1.8)

E_T^{0 j}

00

−−−−→ Y⁰⁰

 y

p⁰⁰

 y^σT0 T⁰ −−−−→

ι⁰ Y⁰,

where E_T⁰ = P(N_T⁰Y⁰) is the exceptional divisor of the blowing-up σ_T⁰, which is a P²(C)-bundle over T⁰, and p⁰⁰ : E_T⁰ → T⁰ is the projection to the base space of this bundle. We denote by ON_T0Y⁰(1) the canonical line bundle on E_T⁰, and by ON_T0Y⁰(−1) its dual, or thetautological line bundle onE_T⁰.

Lemma 1.5. — σ^∗

T⁰[D⁰]is expressed as

(1.9) σ_T^∗0[D⁰] = [D⁰⁰] + 3j⁰⁰_∗[ξ_T⁰] +j⁰⁰_∗p^00∗[α0]

where[ξ_T⁰] =c1(ON_T0Y⁰(1))∩[E_T⁰]and[α0] an algebraic0-cycle class onT⁰. Proof. — By theblow-up formula,

(1.10) σ_T^∗0[D⁰] = [D⁰⁰] +j⁰⁰_∗{c(E⁰⁰)∩p^00∗s(T⁰, D⁰)}2

whereE⁰⁰=p^00∗N_T⁰Y⁰/NE_T0, Y⁰⁰=p^00∗N_T⁰Y⁰/ON_T0Y⁰(−1) ands(T⁰, D⁰) is the total Segre class ofT⁰ inD⁰. Since

c1(E⁰⁰) =p^00∗c1(N_T⁰Y⁰)−c1(ON_T0Y⁰(−1)) =p^00∗c1(N_T⁰Y⁰) +c1(ON_T0Y⁰(1)), we have

(1.11)

{c(E⁰⁰)∩s(T⁰, D⁰)}2=p^00∗s0(T⁰, D⁰) +c1(E⁰⁰)∩p^00∗s1(T⁰, D⁰)

=p^00∗{s0(T⁰, D⁰) +c1(N_T⁰Y⁰)∩s1(T⁰, D⁰)}

+c1(ON_T0Y⁰(1))∩p^00∗s1(T⁰, D⁰)

To compute s(T⁰, D⁰), we consider the normalization map n_D⁰ : D^0∗ → D⁰. D^0∗ is non-singular. Hence, if we putT^0∗:=n⁻¹

D⁰(T⁰), we have s(T^0∗, D^0∗) =c(N_T^0∗D^0∗)⁻¹∩[T^0∗]

= (1−c1(N_T^0∗D^0∗))∩[T^0∗]

= [T^0∗]−T^0∗·T^0∗. Therefore,

s(T⁰, D⁰) =n_D⁰_∗s(T^0∗, D^0∗) = 3[T⁰]−n_D⁰_∗(T^0∗·T^0∗),

and so, (1.12)

(s0(T⁰, D⁰) =−n_D⁰

∗(T^0∗·T^0∗) s1(T⁰, D⁰) = 3[T⁰]

By (1.11) and (1.12), if we put [α0] :=−n_D⁰_∗(T^0∗·T^0∗) + 3c1(N_T⁰Y⁰)∩[T⁰], {c(E⁰⁰)∩s(T⁰, D⁰)}2=p^00∗[α0] + 3[ξ_T⁰].

Consequently, by (1.10), we obtain (1.9).

By Lemma 1.4 and Lemma 1.5 we have the following:

Lemma 1.6

(1.13) σ^∗_T0σ^∗_Σq[D] = [D⁰⁰] + 3j⁰⁰_∗[ξ_T⁰] +j⁰⁰_∗p^00∗[α0] + 6`⁰⁰_∗X

q

[H_q⁰⁰],

where[ξ_T⁰] =c1(ON_T0Y⁰(1))∩[E_T⁰]and[α0] an algebraic0-cycle class onT⁰,H_q⁰⁰the proper inverse image ofH_q⁰ byσ_T⁰, and `⁰⁰the inclusion map ΣqE_q⁰ ,−→Y⁰⁰whereE_q⁰ is the proper inverse image ofEq by σ_T⁰.

Proposition 1.7

(1.14) f^00∗[D⁰⁰] =f^00∗[X⁰⁰]·D⁰⁰−[D⁰⁰]²−[C⁰⁰]

Proof. — SinceD⁰⁰is regularly embedded inY⁰⁰,i.e., C_D⁰⁰Y⁰⁰'N_D⁰⁰Y⁰⁰, whileD⁰ is not, we can apply the excess intersection formula ([F], Theorem 6.3, p. 102) to D⁰⁰. Then, denoting the tangent bundle of a non-singular algebraic variety, sayZ, byTZ, we have

(1.15)

f^00∗[D⁰⁰] =c1(f^00∗N_D⁰⁰Y⁰⁰/ND⁰⁰X⁰⁰)∩[D⁰⁰]

={c1(f^00∗TY⁰⁰)−c1(f^00∗T_D⁰⁰)−c1(TX⁰⁰) +c1(TD⁰⁰)} ∩[D⁰⁰]

={c1(f^00∗TY⁰⁰)−c1(TX⁰⁰)} ∩[D⁰⁰]−C⁰⁰,

where the last equality follows from the ramification formula ([F], Example 3.2.20, p. 62). On the other hand, by the double point formula ([F], Theorem 9.3, p. 166, Example 9.3.4, p. 167),

(1.16) [D⁰⁰] =f^00∗[X⁰⁰]− {c1(f^00∗TY⁰⁰)−c1(TX⁰⁰)} ∩[X⁰⁰].

By (1.15) and (1.16), we obtain (1.14).

Proposition 1.8

(1.17) f^0∗σ^∗_Σq[D] =f^0∗[X⁰]·D⁰−[D⁰]²−[C⁰] + [T⁰] + 6k⁰_∗X

q

[H_q⁰],

whereH_q⁰ is a hyperplane of τ_Σq⁻¹(q) :=Eq 'P²(C)for each pointq ofΣq, andk⁰ the inclusion map ΣqEq ,→X⁰.

Proof. — We first note that

(1.18) f^0∗σ_Σq^∗ [D] =τT⁰∗τ_T^∗0f^0∗σ^∗_Σq[D]

=τT⁰∗f^00∗σ^∗

T⁰σ_Σq^∗ [D]

The first equality above follows from the fact that τT⁰ is the blowing-up ofX⁰ along T⁰, and the second one from the commutativity of the upper fiber square in (0.2).

Therefore, it suffices to compute the image of each term on the right hand side in (1.13) by τT⁰∗f^00∗. First, we will compute the image by f^00∗. f^00∗[D⁰⁰] is given by (1.14). To computef^00∗(3j⁰⁰_∗[ξ_T⁰] +j⁰⁰_∗p^00∗[α0]), we consider the following fiber square:

(1.19)

ET⁰ j⁰⁰

−−−−→X⁰⁰

 y

p⁰⁰

 y^τT0 T⁰ −−−−→

ι⁰ X⁰,

where ET⁰ =P(NT⁰X⁰) is the exceptional divisor of the blowing-upτT⁰, which is a P¹(C)-bundle overT⁰, and p⁰⁰ :ET⁰ →T⁰ is the projection to the base space of this bundle. There is a set of morphisms from the diagram in (1.19) to the one in (1.8) induced by those in the upper fiber square in (0.2). We denote by g⁰ and g⁰⁰ the restriction of f⁰ : X⁰ → Y⁰ to T⁰ and that of f⁰⁰ : X⁰⁰ → Y⁰⁰ to ET⁰, respectively.

Note that the morphismg⁰⁰ :ET⁰ → E_T⁰ maps each fiber ofp⁰⁰ :ET⁰ → T⁰ to that ofp⁰⁰:E_T⁰ →T⁰, and sog^00∗[ξ_T⁰] = [ξT⁰], where [ξT⁰] =c1(ON_T0X⁰(1))∩[ET⁰]. Since f⁰⁰:X⁰⁰→Y⁰⁰ andg⁰⁰:ET⁰ →E_T⁰ arelocal complete intersection morphisms of the same codimension, we can apply Proposition 6.6, (c) in [F] (p. 113) to the fiber square

(1.20)

ET⁰ g⁰⁰

−−−−→E_T⁰

 y

j⁰⁰

 y^j00 X⁰⁰ −−−−→

f⁰⁰ Y⁰⁰. Then,

(1.21) f^00∗j⁰⁰_∗[ξ_T⁰] =j_∗⁰⁰g^00∗[ξ_T⁰] =j_∗⁰⁰[ξT⁰], and (1.22) f^00∗j⁰⁰_∗p^00∗[α0] =j⁰⁰_∗g^00∗p^00∗[α0] =j_∗⁰⁰p^00∗g^0∗[α0].

To computef^00∗(6`⁰⁰_∗Σq[H_q⁰⁰]), we consider the following fiber squares:

(1.23)

ΣqE_q⁰ −−−−→^{`00</sup> X<sup>00</sup> ΣqE<sup>0</sup><sub>q</sub> <sup>`}

00

−−−−→Y⁰⁰

 y

q⁰⁰



y^{τT0 q00}y y^σT0 ΣqEq −−−−→

k⁰ X⁰, ΣqEq −−−−→

k⁰

Y⁰.

As before there is a set of morphisms from the diagram on the left to the one on the right in (1.23) by those in the upper fiber square in (0.2). We denote byh⁰ and h⁰⁰ the restriction off⁰ to ΣqEq and that off⁰⁰:X⁰⁰→Y⁰⁰ to ΣqE_q⁰, respectively. Since

f⁰⁰ :X⁰⁰ →Y⁰⁰ and h⁰⁰: ΣqE_q⁰ →ΣqE_q⁰ are local complete intersection morphisms of the same codimension, we have

(1.24) f^00∗`<sup>00</sup><sub>∗</sub>[H<sub>q</sub><sup>00</sup>] =`⁰⁰_∗h^00∗[H_q⁰⁰].

Similarly, applying the same arguments for f⁰:X⁰→Y⁰ andh⁰ : ΣqEq →ΣqEq, we have

(1.25) f^0∗k⁰_∗[H_q⁰] =k_∗⁰h^0∗[H_q⁰] =k_∗⁰[H_q⁰].

Sinceh^00∗q^00∗=q^00∗h^0∗and [H_q⁰⁰] =q^00∗[H_q⁰],

(1.26)

`<sup>00</sup><sub>∗</sub>h<sup>00∗</sup>[H<sub>q</sub><sup>00</sup>] =`⁰⁰_∗h^00∗q^00∗[H_q⁰]

=`⁰⁰_∗q^00∗h^0∗[H_q⁰]

=`⁰⁰_∗q^00∗[H_q⁰].

Further, sinceτT⁰ :X⁰⁰ →X⁰ and g⁰⁰: ΣqE_q⁰ →ΣEq are local complete intersection morphisms of the same codimension,

(1.27) `⁰⁰_∗q^00∗[H_q⁰] =τ_T^∗0k_∗⁰[H_q⁰].

Therefore, by (1.24), (1.26) and (1.27),

(1.28) f^00∗`⁰⁰_∗[H_q⁰⁰] =τ_T^∗0k_∗⁰[H_q⁰].

Consequently, by (1.13), (1.14), (1.21), (1.22) and (1.28),

f^00∗σ_T^∗0σ_Σq^∗ [D] =f^00∗[X⁰⁰]·D⁰⁰−[D⁰⁰]²−[C⁰⁰] + 3j_∗⁰⁰[ξT⁰] +j_∗⁰⁰p^00∗g^0∗[α0] + 6τ_T^∗0k⁰_∗Σq[H_q⁰].

Since τT⁰∗[C⁰⁰] = [C⁰], τT⁰∗j_∗⁰⁰[ξT⁰] = [T⁰], τT⁰∗j_∗⁰⁰p^00∗g^0∗[α0] = 0 and τT⁰∗τ_T^∗0k_∗⁰[H_q⁰] = k⁰_∗[H_q⁰], by (1.18) and the equality above, we have

(1.29) f^0∗σ_q^∗[D] =τT⁰∗(f^00∗[X⁰⁰]·D⁰⁰])−τT⁰∗[D⁰⁰]²−[C⁰] + 3[T⁰] + 6k_∗⁰Σq[H_q⁰].

Sinceτ_T^∗0[D⁰] = [D⁰⁰] + 2[ET⁰],

(1.30) τT⁰∗(f^00∗[X⁰⁰]·D⁰⁰) =τT⁰∗(f^00∗[X⁰⁰]·τ_T^∗0[D⁰]−2f^00∗[X⁰⁰]·ET⁰).

On the other hand, sinceσ^∗

T⁰[X⁰] = [X⁰⁰] + 3[E_T⁰], f^00∗[X⁰⁰] =f^00∗σ_T^∗0[X⁰]−3[ET⁰].

Hence,

(1.31)

τT⁰∗(f^00∗[X⁰⁰]·τ_T^∗0D⁰) =τT⁰∗(f^00∗[X⁰⁰])·D⁰

=τT⁰∗(f^00∗σ_T^∗0[X⁰]−3[ET⁰])·D⁰

=τT⁰∗(f^00∗σ^∗

T⁰[X⁰])·D⁰

=τT⁰∗(τ_T^∗0f^0∗[X⁰])·D⁰

=f^0∗[X⁰]·D⁰,

and (1.32)

τT⁰∗(f^00∗[X⁰⁰]·ET⁰) =τT⁰∗((f^00∗σ^∗

T⁰[X⁰])·ET⁰−3[ET⁰]²)

=τT⁰∗(τ_T^∗0f^0∗[X⁰]·ET⁰) + 3τT⁰∗j_∗⁰⁰[ξT⁰]

=f^0∗[X⁰]·τT⁰∗[ET⁰] + 3ι⁰_∗[T⁰] = 3[T⁰].

Therefore, by (1.30), (1.31) and (1.32),

(1.33) τT⁰∗(f^00∗[X⁰⁰]·D⁰⁰) =f^0∗[X⁰]·D⁰−6[T⁰].

Furthermore, we have

(1.34)

τT⁰∗[D⁰⁰]²=τT⁰∗((τ_T^∗0[D⁰]−2[ET⁰])²)

=τT⁰∗((τ_T^∗0[D⁰])²−4τ_T^∗0[D⁰]·[ET⁰] + 4[ET⁰]²)

= (τT⁰∗τ_T^∗0[D⁰])·[D⁰]−4[D⁰]·τT⁰∗[ET⁰]−4τT⁰∗[ξT⁰]

= [D⁰]²−4[T⁰].

Consequently, by (1.29), (1.33) and (1.34), we obtain (1.17).

Proposition 1.9

(1.35) f^∗[D] =f^∗[X]·D−[D]²−[C] + [T]

Proof. — Since τΣq∗f^0∗σ_Σq^∗ [D] = τΣq∗τ_Σq^∗ f^∗[D] =f^∗[D], τΣq∗[C⁰] = [C], τΣq∗[T⁰] = [T] andτΣq∗[H_q⁰] = 0, by Proposition 1.8, we have

(1.36) f^∗[D] =τΣq∗(f^0∗[X]·D⁰)−τΣq∗[D⁰]²−[C] + [T].

Sinceτ_Σq^∗ [D] = [D⁰] + 3[ΣqEq], we have

(1.37)

τΣq∗[D⁰]²=τΣq∗((τ_Σq^∗ [D]−3[ΣqEq])²)

=τΣq∗(τ_Σq^∗ [D])²−6τΣq∗(τ_Σq^∗ D·ΣqEq) + 9τΣq∗[ΣqEq]²

= [D]²−6D·τΣq∗[EΣq]−9k_∗⁰τΣq∗[ΣqH_q⁰]

= [D]²,

whereH_q⁰ is a hyperplane ofEq 'P²(C), and

(1.38) τΣq∗(f^0∗[X⁰]·D⁰) =τΣq∗(f^0∗[X⁰]·τ_Σq^∗ [D])−3τΣq∗(f^0∗[X⁰]·ΣqEq).

On the other hand, sinceσΣq[X] = [X⁰] + 4[ΣqEq], f^0∗[X⁰] =f^0∗σ_Σq^∗ [X]−4 ΣqEq

Hence,

τΣq∗(f^0∗[X⁰]·τ_Σq^∗ [D]) =τΣq∗(f^0∗σ_Σq^∗ [X]·τ_Σq^∗ [D])−4τΣq∗(ΣqEq·τ_Σq^∗ [D])

=τΣq∗(f^0∗σ_Σq^∗ [X])·[D]−4τΣq∗(ΣqEq)·[D]

(1.39)

=τΣq∗τ_Σq^∗ f^∗[X]·[D] =f^∗[X]·[D],

and (1.40)

τΣq∗(f^0∗[X⁰]·ΣqEq) =τΣq∗(f^0∗σ^∗_Σq[X]·ΣqEq)−4τΣq∗[ΣqEq]²

=τΣq∗(τ_Σq^∗ f^∗[X]·ΣqEq) + 4τΣq∗(k⁰_∗ΣqH_q⁰)

=f^∗[X]·τΣq∗[ΣqEq] = 0.

Therefore, by (1.38), (1.39) and (1.40),

(1.41) τΣq∗(f^0∗[X⁰]·D⁰) =f^∗[X]·[D].

Consequently, by (1.36), (1.37) and (1.41), we obtain (1.35).

Since f∗[X] = [X], f∗[D] = 2[D], f∗[T] = 3[T] and f∗[C] = [C], by Proposition 1.9, we have the following:

Corollary 1.10

f∗[D]²= [X]·[D] + 3[T]−[C]

By Proposition 1.9,

(1.42) [D]²=f^∗[X]·D−f^∗[D]−[C] + [T].

Hence, by the second equality in (1.6),

s(J, X)1=−f^∗[X]·D+f^∗[D] + 2[C]−[T], and so, by theprojection formula,

s(J, X)1=−X·D−3[T] + 2[C].

We are now going to compute s(J , X)0. Since s(J , X)0 = f∗s(J, X)0, it suffices to know the push-forward of each term of the right hand side of the last identity in (1.6). By (1.42),

(1.43) [D]³=f^∗[X]·[D]²−f^∗[D]·D−D·C+D·T.

To realizef∗[D]³, we compute the push-forward of each term on the right hand side of (1.43). By theprojection formula and Corollary 1.10,

(1.44) f∗(f^∗[X]·[D]²) = [X]·f∗[D]²

= [X]²·[D] + 3[X]·T −[X]·C.

Sincef∗[D] = 2[D], by theprojection formula,

(1.45) f∗(f^∗[D]·D) = [D]·f∗[D]

= 2[D]².

To realizef∗[D·C], we computef^∗[C]. Since C isregularly embedded inY, we can apply theexcess intersection formula to it. Then,

(1.46)

f^∗[C] =c1(f^∗N_CY /NCX)∩[C]

={c1(f^∗TY)−c1(f^∗T_C)−c1(TX) +c1(TC)} ∩[C]

={c1(f^∗TY)−c1(TX)} ∩[C]

=f^∗[X]·C−D·C,

where the last equality but one follows from the fact C ' C and the last equality from thedouble point formulaforf :X →Y. Therefore, by (1.46) and theprojection formula, we have

(1.47) f∗(D·C) =X·f∗[C]−C·f∗[X]

=X·C−C·X = 0

To realizef∗(D·T), we compute f^∗[T]. Since T is notregularly embedded in Y, we cannot apply theexcess intersection formulatoT. But, sinceT⁰ isregularly embedded in Y⁰, we can apply it toT⁰. Then, by the same way as in the case ofC,

(1.48) f^0∗[T⁰] =f^0∗[X⁰]·T⁰−D⁰·T⁰−[Σs⁰]

Here the term [Σs⁰] comes from {c1(f^0∗T_T⁰)−c1(TT⁰)} ∩[T⁰] = [Σs⁰], which is the ramification formula forf_|T⁰ 0.

Lemma 1.11

(i) σ^∗_Σq[T] = [T⁰] + 4Σq[H_q⁰]², (ii) τ_Σq^∗ [T] = [T⁰] + 3Σq[H_q⁰],

where H_q⁰ is a hyperplane of Eq := σ_Σq⁻¹(q)' P³(C) for each quadruple point q and H_q⁰ that ofEq :=τ_Σq⁻¹(q)'P²(C)for each pointq off⁻¹(Σq).

Proof. — Since the multiplicity ofT (resp.T) at each quadruple pointq(resp. at each pointqoff⁻¹(Σq)) is 4 (resp. 3), (i) (resp. (ii)) follows from theblow-up formula([F], Theorem 6.7, p. 116, and Corollary 6.7.1, p. 117).

Proposition 1.12

(1.49) f^∗[T] =f^∗[X]·T−D·T −[Σs] + [Σq]

Proof. — Sincef^0∗(4 Σq[H_q⁰]²) = 4Σq[H_q⁰]², by Lemma 1.11, (i) and (1.48), (1.50) f^0∗σ^∗_Σq[T] =f^0∗[T⁰] + 4 Σq[H_q⁰]²

=f^0∗[X⁰]·T⁰−D⁰·T⁰−[Σs⁰] + 4 Σq[H_q⁰]². Since

(1.51) f^∗[T] =τΣq∗τ_Σq^∗ f^∗[T] =τΣq∗f^0∗σ^∗_Σq[T],

it suffices to compute the push-forward of each term on the right hand side in (1.50) by τΣq∗ in order to knowf^∗[T]. Sinceσ_Σq^∗ [X] = [X⁰] + 4[ΣqEq] andf^0∗[ΣqEq] = [ΣqEq], by Lemma 1.11, (ii),

(1.52)

τΣq∗(f^0∗[X⁰]·T⁰) =τΣq∗((f^0∗σ_Σq^∗ [X]−4[ΣqEq])·(τ_Σq^∗ [T]−3[ΣqH_q⁰]))

=τΣq∗((τ_Σq^∗ f^∗[X])·τ_Σq^∗ [T]−4[ΣqEq]·τ_Σq^∗ [T]

−3τ_Σq^∗ f^∗[X]·[ΣqH_q⁰] + 12[ΣqEq]·[ΣqH_q⁰])

=f^∗[X]·T−12[Σq].

Here the second equality follows from the commutativity of the lower fiber square in (0.2) and the third one from theprojection formula and the following facts:

(1.53)









τΣq∗[ΣqEq] = 0, τΣq∗[ΣqH_q⁰] = 0,

[ΣqEq]·[ΣqH_q⁰] =−Σq[H_q⁰]², τΣq∗(Σq[H_q⁰]²) = [Σq].

Sinceτ_Σq^∗ [D] = [D⁰] + 3[ΣqEq], by Lemma 1.11, (ii), D⁰·T⁰= (τ_Σq^∗ [D]−3[ΣqEq])·(τ_Σq^∗ [T]−3Σq[H_q⁰])

=τ_Σq^∗ [D]·τ_Σq^∗ [T]−3(τ_Σq^∗ [D]·Σq[H_q⁰])−3([EΣq]·τ_Σq^∗ [T]) + 9[EΣq]·Σq[H_q⁰] Hence, by theprojection formula and (1.53),

(1.54) τΣq∗(D⁰·T⁰) =D·T−9[Σq]

Consequently, by (1.51), (1.50), (1.52), (1.54) and the fourth equality in (1.53), f^∗[T] =f^∗[X]·T −12[Σq]−D·T + 9[Σq]−[Σs] + 4[Σq]

=f^∗[X]·T −D·T−[Σs] + [Σq].

Corollary 1.13

(1.55) f∗[D]³= [X]²·[D]−2[D]²+ 5X·T−X·C−[Σs] + 4[Σq].

Proof. — By Proposition 1.12,

D·T =f^∗[X]·T−f^∗[T]−[Σs] + [Σq].

Hence,

(1.56) f∗(D·T) = 3X·T−X·T−[Σs] + 4[Σq]

= 2X·T−[Σs] + 4[Σq]

By (1.43), (1.47), (1.56) and Corollary 1.10,

f∗[D]³= [X]·f∗[D]²−2[D]²+ 2X·T−[Σs] + 4[Σq]

= [X]²·D+ 3X·T−X·C−2[D]²+ 2X·T−[Σs] + 4[Σq]

= [X]²·D+ 5X·T−X·C−2[D]²−[Σs] + 4[Σq]

Since (1.57)

s(J, X)0=f∗s(J, X)0

=f∗[D]³−f∗c1(NC/X)∩[C]−3f∗(D·C)

=f∗[D]³−f∗c1(NC/X)∩[C] (cf. (1.47)),

what remains is to compute f∗c1(NC/X)∩[C] in order to know s(J , X)0. By the adjunction formula, thedouble point formula forf :X→Y and (1.46),

c1(NCX)∩[C] =−KX·C+ [kC]

= (−f^∗[X+KY] +D)·C+ [kC]

=−f^∗[KY]·C−f^∗[C] + [kC],

whereKY,KX andkCare the canonical divisors ofY,XandC, respectively. There- fore, by theprojection formula and the factC'C,

(1.58) f∗(c1(NCX)∩[C]) =−KY ·C−X·C+ [k_C] Substituting (1.55) and (1.58) into (1.57), we have

s(J, X)0= [X]²·D−2[D]²+ 5X·T+KY ·C−[k_C]−[Σs] + 4[Σq].

We collect the results concerning the Segre classes ofX obtained up to this point in the following proposition:

Proposition 1.14. — The Segre classes of the singular subscheme J, defined by the Jacobian ideal, of an algebraic threefold X with ordinary singularities in the four dimensional projective spaceY =P⁴(C) are given as follows:





s(J, X)2= 2[D]

s(J, X)1=−X·D−3T+ 2C

s(J, X)0= [X]²·D−2[D]²+ 5X·T+KY ·C−[k_C]−[Σs] + 4[Σq].

Here D, T, C,Σs and Σq are the singular locus, triple point locus, cuspidal point locus,stationary point locusand quadruple point locusofX, respectively. KY is the canonical divisor of the projective 4-space Y, andk_C that ofC.

Note that the effect of the existence of quadruple points ofX is only the term 4[Σq]

in the expression ofs(J, X)0. Then, by Proposition 1.14,





degs2= 2m

degs1=−nm+ 2γ−3t

degs0=n²m−2m²+ 5nt−5γ−#Σs−degk_C+ 4#Σq,

where n = degX (the degree of X in Y), m = degD, t = degT, γ = degC, and #Σs = the cardinal number of Σs, and #Σq = the cardinal number of Σq.

Consequently, by (1.5), theclass c ofX is given by

c= deg[M3] = (n−1)³degX−3(n−1)²degs2−3(n−1) degs1−degs0

= (n−1)³n−(4n²−9n−2m+ 6)m+ (4n−9)t−(6n−11)γ

+ #Σs+ degk_C−4#Σq.

By this formula together with Proposition 1.2, we have the following: