## 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
P

^{3}(C) are given byR

Xc^{2}_{1} =n(n−4)^{2}−(3n−16)m+ 3t−γ,R

Xc2=n(n^{2}−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 inP^{4}(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 dimension*3

**à**

**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 dansP^{3}(C)
sont donn´es par R

Xc^{2}_{1} =n(n−4)^{2}−(3n−16)m+ 3t−γ,R

Xc2 =n(n^{2}−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^{4}(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-spaceP^{4}(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-spaceC^{4} 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) xy^{2}−z^{2}= 0 (cuspidal point)

(vi) w(xy^{2}−z^{2}) = 0 (stationary point)

where (x, y, z, w) is the coordinate onC^{4}. 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
inP^{4}(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]^{3}forDthe 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

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

KY is the canonical divisor of P^{4}(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 :=P^{4}(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 P^{4}(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)

Xy^{00}−−−−→^{f}^{00} Y^{00}

τ_{T}0

y^{σ}^{T}^{0}
Xy^{0} −−−−→^{f}^{0} Y^{0}

τΣq

y^{σ}^{Σq}
X −−−−→

f Y, which is defined as follows:

σΣq :Y^{0}→Y: the blowing-up ofY along the quadruple point locus ΣqofX,
X^{0}: the proper inverse image ofX byσΣq,

X^{0}:=X×_{X}X^{0}: the fiber product ofX andX^{0} overX,

n_{X}^{0} :X^{0}→X^{0}: the projection to the second factor ofX×_{X}X^{0}, which is nothing
but the normalization ofX^{0},

f^{0}:X^{0} →Y^{0}: the composite of the normalization mapn_{X}^{0} and the inclusion
ι^{0} :X^{0},→Y^{0},

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

τΣq :X^{0}→X: the projection to the first factor ofX×_{X}X^{0}, which is nothing but
the blowing-up ofX along Σq,

D^{0},T^{0},C^{0}and Σs^{0}: the proper inverse images ofD,T,Cand ΣsbyσΣq, respectively.

D^{0},T^{0} andC^{0}: the proper inverse images ofD,T andCbyτΣq, which are equal to
the inverse images ofD^{0},T^{0} andC^{0} byf^{0}, respectively,

Σs^{0}: the inverse image of ΣsbyτΣq, which is equal to T^{0}∩C^{0},
σ_{T}^{0} :Y^{00}→Y^{0}: the blowing-up ofY^{0} alongT^{0},

X^{00}: the proper inverse image ofX^{0} byσ_{T}^{0},

X^{00}:=X^{0}×_{X}^{0}X^{00}: the fiber product ofX^{0} andX^{00} overX^{0},

n_{X}^{00}:X^{00}→X^{00}: the projection to the second factor ofX^{0}×_{X}^{0}X^{00}, which is nothing
but the normalization ofX^{00}

f^{00}:X^{00}→Y^{00}: the composite of the normalization mapn_{X}^{00} and the inclusion
ι^{00}:X^{00},→Y^{00},

τT^{0} :X^{00}→X^{0}: the projection to the first factor ofX^{0}×_{X}^{0}X^{00}, which is nothing but
the blowing-up ofX^{0} alongT^{0},

D^{00}, T^{00},C^{00}and Σs^{00}: the proper inverse images ofD^{0},T^{0},C^{0} and Σs^{0} byσ_{T},
respectively,

D^{00}, T^{00}andC^{00}: the proper inverse images ofD^{0},T^{0} andC^{0} byτT^{0}, which are equal
to the inverse images of D^{00},T^{00} andC^{00} byf^{00}, respectively,

Σs^{00}: the inverse image of Σs^{0} byτT^{0}, which is equal to T^{00}∩C^{00}.

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/I^{2}, 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/I^{2}. 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/I^{2}) toS^{·}:= Σ^{∞}_{k=0}I^{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
P^{4}(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∈P^{1}Xλ on X,
consisting of hyperplane sectionsXλ of X in P^{4}(C), whose pull-backL:=S

λ∈P^{1}Xλ

toX by the normalization mapf :X →X has the following properties: There exists
a finite set {λ1, . . . , λc} of points ofP^{1} 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 P^{4}(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 P^{2}(C) whose degree is equal to
n(= degX).

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

λ∈P^{1}Xcλ the
proper inverse ofL:=S

λ∈PXλ. ThenLbgives a fibering ofXb overP^{1}(C). Hence the
Euler numberχ(X) ofb Xb is given by

χ(X) =b χ(P^{1}(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 → P^{1}. 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(P^{1}(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
P^{2}(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'P^{3}(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(n^{2}−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(n^{2}−4n+ 6)−2(3n−8)m+ 6t−4γ

−2 + (n−1)(n−2)−2m−c

=n(2n^{2}−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 P^{4}(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)^{3}h^{3}−3(n−1)^{2}h^{2}∩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] = [D^{0}] + 6j^{0}_{∗}X

q

[H_{q}^{0}],

whereH_{q}^{0} is a hyperplane ofEq :=σ^{−1}_{Σq}(q)'P^{3}(C)for each quadruple pointq, and j^{0}
the inclusion mapΣqEq,→Y^{0}.

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

−−−−→ Y^{00}

y

p^{00}

y^{σ}^{T}^{0}
T^{0} −−−−→

ι^{0} Y^{0},

where E_{T}^{0} = P(N_{T}^{0}Y^{0}) is the exceptional divisor of the blowing-up σ_{T}^{0}, which is
a P^{2}(C)-bundle over T^{0}, and p^{00} : E_{T}^{0} → T^{0} is the projection to the base space of
this bundle. We denote by ON_{T}0Y^{0}(1) the canonical line bundle on E_{T}^{0}, and by
ON_{T}0Y^{0}(−1) its dual, or thetautological line bundle onE_{T}^{0}.

* Lemma 1.5. —* σ

^{∗}

T^{0}[D^{0}]is expressed as

(1.9) σ_{T}^{∗}0[D^{0}] = [D^{00}] + 3j^{00}_{∗}[ξ_{T}^{0}] +j^{00}_{∗}p^{00∗}[α0]

where[ξ_{T}^{0}] =c1(ON_{T}0Y^{0}(1))∩[E_{T}^{0}]and[α0] an algebraic0-cycle class onT^{0}.
Proof. — By theblow-up formula,

(1.10) σ_{T}^{∗}0[D^{0}] = [D^{00}] +j^{00}_{∗}{c(E^{00})∩p^{00∗}s(T^{0}, D^{0})}2

whereE^{00}=p^{00∗}N_{T}^{0}Y^{0}/NE_{T}0, Y^{00}=p^{00∗}N_{T}^{0}Y^{0}/ON_{T}0Y^{0}(−1) ands(T^{0}, D^{0}) is the total
Segre class ofT^{0} inD^{0}. Since

c1(E^{00}) =p^{00∗}c1(N_{T}^{0}Y^{0})−c1(ON_{T}_{0}Y^{0}(−1)) =p^{00∗}c1(N_{T}^{0}Y^{0}) +c1(ON_{T}_{0}Y^{0}(1)),
we have

(1.11)

{c(E^{00})∩s(T^{0}, D^{0})}2=p^{00∗}s0(T^{0}, D^{0}) +c1(E^{00})∩p^{00∗}s1(T^{0}, D^{0})

=p^{00∗}{s0(T^{0}, D^{0}) +c1(N_{T}^{0}Y^{0})∩s1(T^{0}, D^{0})}

+c1(ON_{T}0Y^{0}(1))∩p^{00∗}s1(T^{0}, D^{0})

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

D^{0}(T^{0}), 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(T^{0}, D^{0}) =n_{D}^{0}_{∗}s(T^{0∗}, D^{0∗}) = 3[T^{0}]−n_{D}^{0}_{∗}(T^{0∗}·T^{0∗}),

and so, (1.12)

(s0(T^{0}, D^{0}) =−n_{D}^{0}

∗(T^{0∗}·T^{0∗})
s1(T^{0}, D^{0}) = 3[T^{0}]

By (1.11) and (1.12), if we put [α0] :=−n_{D}^{0}_{∗}(T^{0∗}·T^{0∗}) + 3c1(N_{T}^{0}Y^{0})∩[T^{0}],
{c(E^{00})∩s(T^{0}, D^{0})}2=p^{00∗}[α0] + 3[ξ_{T}^{0}].

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^{∗}_{T}0σ^{∗}_{Σq}[D] = [D^{00}] + 3j^{00}_{∗}[ξ_{T}^{0}] +j^{00}_{∗}p^{00∗}[α0] + 6`^{00}_{∗}X

q

[H_{q}^{00}],

where[ξ_{T}^{0}] =c1(ON_{T}0Y^{0}(1))∩[E_{T}^{0}]and[α0] an algebraic0-cycle class onT^{0},H_{q}^{00}the
proper inverse image ofH_{q}^{0} byσ_{T}^{0}, and `^{00}the inclusion map ΣqE_{q}^{0} ,−→Y^{00}whereE_{q}^{0}
is the proper inverse image ofEq by σ_{T}^{0}.

**Proposition 1.7**

(1.14) f^{00∗}[D^{00}] =f^{00∗}[X^{00}]·D^{00}−[D^{00}]^{2}−[C^{00}]

Proof. — SinceD^{00}is regularly embedded inY^{00},i.e., C_{D}^{00}Y^{00}'N_{D}^{00}Y^{00}, whileD^{0} is
not, we can apply the excess intersection formula ([F], Theorem 6.3, p. 102) to D^{00}.
Then, denoting the tangent bundle of a non-singular algebraic variety, sayZ, byTZ,
we have

(1.15)

f^{00∗}[D^{00}] =c1(f^{00∗}N_{D}^{00}Y^{00}/ND^{00}X^{00})∩[D^{00}]

={c1(f^{00∗}TY^{00})−c1(f^{00∗}T_{D}^{00})−c1(TX^{00}) +c1(TD^{00})} ∩[D^{00}]

={c1(f^{00∗}TY^{00})−c1(TX^{00})} ∩[D^{00}]−C^{00},

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) [D^{00}] =f^{00∗}[X^{00}]− {c1(f^{00∗}TY^{00})−c1(TX^{00})} ∩[X^{00}].

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

**Proposition 1.8**

(1.17) f^{0∗}σ^{∗}_{Σq}[D] =f^{0∗}[X^{0}]·D^{0}−[D^{0}]^{2}−[C^{0}] + [T^{0}] + 6k^{0}_{∗}X

q

[H_{q}^{0}],

whereH_{q}^{0} is a hyperplane of τ_{Σq}^{−1}(q) :=Eq 'P^{2}(C)for each pointq ofΣq, andk^{0} the
inclusion map ΣqEq ,→X^{0}.

Proof. — We first note that

(1.18) f^{0∗}σ_{Σq}^{∗} [D] =τT^{0}∗τ_{T}^{∗}0f^{0∗}σ^{∗}_{Σq}[D]

=τT^{0}∗f^{00∗}σ^{∗}

T^{0}σ_{Σq}^{∗} [D]

The first equality above follows from the fact that τT^{0} is the blowing-up ofX^{0} along
T^{0}, 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 τT^{0}∗f^{00∗}. First, we will compute the image by f^{00∗}. f^{00∗}[D^{00}] is given by
(1.14). To computef^{00∗}(3j^{00}_{∗}[ξ_{T}^{0}] +j^{00}_{∗}p^{00∗}[α0]), we consider the following fiber square:

(1.19)

ET^{0}
j^{00}

−−−−→X^{00}

y

p^{00}

y^{τ}^{T0}
T^{0} −−−−→

ι^{0} X^{0},

where ET^{0} =P(NT^{0}X^{0}) is the exceptional divisor of the blowing-upτT^{0}, which is a
P^{1}(C)-bundle overT^{0}, and p^{00} :ET^{0} →T^{0} 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 g^{0} and g^{00} the
restriction of f^{0} : X^{0} → Y^{0} to T^{0} and that of f^{00} : X^{00} → Y^{00} to ET^{0}, respectively.

Note that the morphismg^{00} :ET^{0} → E_{T}^{0} maps each fiber ofp^{00} :ET^{0} → T^{0} to that
ofp^{00}:E_{T}^{0} →T^{0}, and sog^{00∗}[ξ_{T}^{0}] = [ξT^{0}], where [ξT^{0}] =c1(ON_{T}0X^{0}(1))∩[ET^{0}]. Since
f^{00}:X^{00}→Y^{00} andg^{00}:ET^{0} →E_{T}^{0} 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)

ET^{0}
g^{00}

−−−−→E_{T}^{0}

y

j^{00}

y^{j}^{00}
X^{00} −−−−→

f^{00} Y^{00}.
Then,

(1.21) f^{00∗}j^{00}_{∗}[ξ_{T}^{0}] =j_{∗}^{00}g^{00∗}[ξ_{T}^{0}] =j_{∗}^{00}[ξT^{0}], and
(1.22) f^{00∗}j^{00}_{∗}p^{00∗}[α0] =j^{00}_{∗}g^{00∗}p^{00∗}[α0] =j_{∗}^{00}p^{00∗}g^{0∗}[α0].

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

(1.23)

ΣqE_{q}^{0} −−−−→^{`}^{00} X^{00} ΣqE^{0}_{q} ^{`}

00

−−−−→Y^{00}

y

q^{00}

y^{τ}^{T0} ^{q}^{00}y y^{σ}^{T}^{0}
ΣqEq −−−−→

k^{0} X^{0}, ΣqEq −−−−→

k^{0}

Y^{0}.

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 byh^{0} and h^{00}
the restriction off^{0} to ΣqEq and that off^{00}:X^{00}→Y^{00} to ΣqE_{q}^{0}, respectively. Since

f^{00} :X^{00} →Y^{00} and h^{00}: ΣqE_{q}^{0} →ΣqE_{q}^{0} are local complete intersection morphisms of
the same codimension, we have

(1.24) f^{00∗}`^{00}_{∗}[H_{q}^{00}] =`^{00}_{∗}h^{00∗}[H_{q}^{00}].

Similarly, applying the same arguments for f^{0}:X^{0}→Y^{0} andh^{0} : ΣqEq →ΣqEq, we
have

(1.25) f^{0∗}k^{0}_{∗}[H_{q}^{0}] =k_{∗}^{0}h^{0∗}[H_{q}^{0}] =k_{∗}^{0}[H_{q}^{0}].

Sinceh^{00∗}q^{00∗}=q^{00∗}h^{0∗}and [H_{q}^{00}] =q^{00∗}[H_{q}^{0}],

(1.26)

`^{00}_{∗}h^{00∗}[H_{q}^{00}] =`^{00}_{∗}h^{00∗}q^{00∗}[H_{q}^{0}]

=`^{00}_{∗}q^{00∗}h^{0∗}[H_{q}^{0}]

=`^{00}_{∗}q^{00∗}[H_{q}^{0}].

Further, sinceτT^{0} :X^{00} →X^{0} and g^{00}: ΣqE_{q}^{0} →ΣEq are local complete intersection
morphisms of the same codimension,

(1.27) `^{00}_{∗}q^{00∗}[H_{q}^{0}] =τ_{T}^{∗}0k_{∗}^{0}[H_{q}^{0}].

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

(1.28) f^{00∗}`^{00}_{∗}[H_{q}^{00}] =τ_{T}^{∗}0k_{∗}^{0}[H_{q}^{0}].

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

f^{00∗}σ_{T}^{∗}0σ_{Σq}^{∗} [D] =f^{00∗}[X^{00}]·D^{00}−[D^{00}]^{2}−[C^{00}] + 3j_{∗}^{00}[ξT^{0}] +j_{∗}^{00}p^{00∗}g^{0∗}[α0] + 6τ_{T}^{∗}0k^{0}_{∗}Σq[H_{q}^{0}].

Since τT^{0}∗[C^{00}] = [C^{0}], τT^{0}∗j_{∗}^{00}[ξT^{0}] = [T^{0}], τT^{0}∗j_{∗}^{00}p^{00∗}g^{0∗}[α0] = 0 and τT^{0}∗τ_{T}^{∗}0k_{∗}^{0}[H_{q}^{0}] =
k^{0}_{∗}[H_{q}^{0}], by (1.18) and the equality above, we have

(1.29) f^{0∗}σ_{q}^{∗}[D] =τT^{0}∗(f^{00∗}[X^{00}]·D^{00}])−τT^{0}∗[D^{00}]^{2}−[C^{0}] + 3[T^{0}] + 6k_{∗}^{0}Σq[H_{q}^{0}].

Sinceτ_{T}^{∗}0[D^{0}] = [D^{00}] + 2[ET^{0}],

(1.30) τT^{0}∗(f^{00∗}[X^{00}]·D^{00}) =τT^{0}∗(f^{00∗}[X^{00}]·τ_{T}^{∗}0[D^{0}]−2f^{00∗}[X^{00}]·ET^{0}).

On the other hand, sinceσ^{∗}

T^{0}[X^{0}] = [X^{00}] + 3[E_{T}^{0}],
f^{00∗}[X^{00}] =f^{00∗}σ_{T}^{∗}0[X^{0}]−3[ET^{0}].

Hence,

(1.31)

τT^{0}∗(f^{00∗}[X^{00}]·τ_{T}^{∗}0D^{0}) =τT^{0}∗(f^{00∗}[X^{00}])·D^{0}

=τT^{0}∗(f^{00∗}σ_{T}^{∗}0[X^{0}]−3[ET^{0}])·D^{0}

=τT^{0}∗(f^{00∗}σ^{∗}

T^{0}[X^{0}])·D^{0}

=τT^{0}∗(τ_{T}^{∗}0f^{0∗}[X^{0}])·D^{0}

=f^{0∗}[X^{0}]·D^{0},

and (1.32)

τT^{0}∗(f^{00∗}[X^{00}]·ET^{0}) =τT^{0}∗((f^{00∗}σ^{∗}

T^{0}[X^{0}])·ET^{0}−3[ET^{0}]^{2})

=τT^{0}∗(τ_{T}^{∗}0f^{0∗}[X^{0}]·ET^{0}) + 3τT^{0}∗j_{∗}^{00}[ξT^{0}]

=f^{0∗}[X^{0}]·τT^{0}∗[ET^{0}] + 3ι^{0}_{∗}[T^{0}] = 3[T^{0}].

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

(1.33) τT^{0}∗(f^{00∗}[X^{00}]·D^{00}) =f^{0∗}[X^{0}]·D^{0}−6[T^{0}].

Furthermore, we have

(1.34)

τT^{0}∗[D^{00}]^{2}=τT^{0}∗((τ_{T}^{∗}0[D^{0}]−2[ET^{0}])^{2})

=τT^{0}∗((τ_{T}^{∗}0[D^{0}])^{2}−4τ_{T}^{∗}0[D^{0}]·[ET^{0}] + 4[ET^{0}]^{2})

= (τT^{0}∗τ_{T}^{∗}0[D^{0}])·[D^{0}]−4[D^{0}]·τT^{0}∗[ET^{0}]−4τT^{0}∗[ξT^{0}]

= [D^{0}]^{2}−4[T^{0}].

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∗[C^{0}] = [C], τΣq∗[T^{0}] =
[T] andτΣq∗[H_{q}^{0}] = 0, by Proposition 1.8, we have

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

Sinceτ_{Σq}^{∗} [D] = [D^{0}] + 3[ΣqEq], we have

(1.37)

τΣq∗[D^{0}]^{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_{∗}^{0}τΣq∗[ΣqH_{q}^{0}]

= [D]^{2},

whereH_{q}^{0} is a hyperplane ofEq 'P^{2}(C), and

(1.38) τΣq∗(f^{0∗}[X^{0}]·D^{0}) =τΣq∗(f^{0∗}[X^{0}]·τ_{Σq}^{∗} [D])−3τΣq∗(f^{0∗}[X^{0}]·ΣqEq).

On the other hand, sinceσΣq[X] = [X^{0}] + 4[ΣqEq],
f^{0∗}[X^{0}] =f^{0∗}σ_{Σq}^{∗} [X]−4 ΣqEq

Hence,

τΣq∗(f^{0∗}[X^{0}]·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∗}[X^{0}]·ΣqEq) =τΣq∗(f^{0∗}σ^{∗}_{Σq}[X]·ΣqEq)−4τΣq∗[ΣqEq]^{2}

=τΣq∗(τ_{Σq}^{∗} f^{∗}[X]·ΣqEq) + 4τΣq∗(k^{0}_{∗}ΣqH_{q}^{0})

=f^{∗}[X]·τΣq∗[ΣqEq] = 0.

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

(1.41) τΣq∗(f^{0∗}[X^{0}]·D^{0}) =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_{C}Y /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, sinceT^{0} isregularly embedded
in Y^{0}, we can apply it toT^{0}. Then, by the same way as in the case ofC,

(1.48) f^{0∗}[T^{0}] =f^{0∗}[X^{0}]·T^{0}−D^{0}·T^{0}−[Σs^{0}]

Here the term [Σs^{0}] comes from {c1(f^{0∗}T_{T}^{0})−c1(TT^{0})} ∩[T^{0}] = [Σs^{0}], which is the
ramification formula forf_{|T}^{0} 0.

**Lemma 1.11**

(i) σ^{∗}_{Σq}[T] = [T^{0}] + 4Σq[H_{q}^{0}]^{2},
(ii) τ_{Σq}^{∗} [T] = [T^{0}] + 3Σq[H_{q}^{0}],

where H_{q}^{0} is a hyperplane of Eq := σ_{Σq}^{−1}(q)' P^{3}(C) for each quadruple point q and
H_{q}^{0} that ofEq :=τ_{Σq}^{−1}(q)'P^{2}(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_{q}^{0}]^{2}) = 4Σq[H_{q}^{0}]^{2}, by Lemma 1.11, (i) and (1.48),
(1.50) f^{0∗}σ^{∗}_{Σq}[T] =f^{0∗}[T^{0}] + 4 Σq[H_{q}^{0}]^{2}

=f^{0∗}[X^{0}]·T^{0}−D^{0}·T^{0}−[Σs^{0}] + 4 Σq[H_{q}^{0}]^{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] = [X^{0}] + 4[ΣqEq] andf^{0∗}[ΣqEq] = [ΣqEq],
by Lemma 1.11, (ii),

(1.52)

τΣq∗(f^{0∗}[X^{0}]·T^{0}) =τΣq∗((f^{0∗}σ_{Σq}^{∗} [X]−4[ΣqEq])·(τ_{Σq}^{∗} [T]−3[ΣqH_{q}^{0}]))

=τΣq∗((τ_{Σq}^{∗} f^{∗}[X])·τ_{Σq}^{∗} [T]−4[ΣqEq]·τ_{Σq}^{∗} [T]

−3τ_{Σq}^{∗} f^{∗}[X]·[ΣqH_{q}^{0}] + 12[ΣqEq]·[ΣqH_{q}^{0}])

=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_{q}^{0}] = 0,

[ΣqEq]·[ΣqH_{q}^{0}] =−Σq[H_{q}^{0}]^{2},
τΣq∗(Σq[H_{q}^{0}]^{2}) = [Σq].

Sinceτ_{Σq}^{∗} [D] = [D^{0}] + 3[ΣqEq], by Lemma 1.11, (ii),
D^{0}·T^{0}= (τ_{Σq}^{∗} [D]−3[ΣqEq])·(τ_{Σq}^{∗} [T]−3Σq[H_{q}^{0}])

=τ_{Σq}^{∗} [D]·τ_{Σq}^{∗} [T]−3(τ_{Σq}^{∗} [D]·Σq[H_{q}^{0}])−3([EΣq]·τ_{Σq}^{∗} [T]) + 9[EΣq]·Σq[H_{q}^{0}]
Hence, by theprojection formula and (1.53),

(1.54) τΣq∗(D^{0}·T^{0}) =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 =P

^{4}(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=n^{2}m−2m^{2}+ 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)^{3}degX−3(n−1)^{2}degs2−3(n−1) degs1−degs0

= (n−1)^{3}n−(4n^{2}−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: