A note on Bézout`s theorem by Sławomir Rams, Piotr Tworzewski

ANNALES
POLONICI MATHEMATICI
87 (2005)
A note on Bézout's theorem
by
Sławomir Rams, Piotr Tworzewski
Tadeusz Winiarski (Kraków)
and
Abstrat. We present a version of Bézout's theorem basing on the intersetion theory
in omplex analyti geometry. Some appliations for produts of surfaes and urves are
also given.
1. Introdution. In the intersetion theory in omplex analyti geome-
try onstruted in [T℄ and ompleted by [AR℄, [R1 ℄, [R2 ℄, [R3 ℄, [R4 ℄, [R5 ℄, for
a system
N , one
X 1 , . . . , Xp
of irreduible analyti subsets of a omplex manifold
an dene the analytially ostrutible (upper semiontinuous in the
Zariski topology) funtion
d : N ∋ x 7→ d(x) := d(X1 , . . . , Xp )(x) ∈ N,
whih assigns to eah point
X 1 , . . . , Xp
at the point
x
x in N
the multipliity of intersetion of the sets
(see also [A℄, [ATW℄ and [CKT℄). By denition,
the intersetion produt of these sets is the unique analyti yle
X1 •· · ·•Xp
dened by the equality
ν(X1 • · · · • Xp , x) = d(X1 , . . . , Xp )(x)
where
for
ν(C, x) denotes the degree of the analyti yle C
x ∈ N,
at
x. One an extend
this denition in the natural way to the ase of analyti yles. Some details
of this onstrution will be given in the next setion of this paper.
One an observe that for a system of analyti yles
eah point
a∈N
C1 , . . . , Cp
on
N
and
we have by denition the Bézout type equality
ν(C1 • · · · • Cp , a) = d(C1 , . . . , Cp )(a),
saying that the multipliity of the intersetion produt of our system oinides with the intersetion multipliity at the point. Usually it is possible to
2000
Mathematis Subjet Classiation :
13H15, 14B05, 14C17, 32B99.
intersetion theory, omplex analyti geometry, Bézout's theorem, analyti yle, singularity.
This researh was partially supported by KBN Grants: 2P03A01522, 2P03A01625.
The rst author was supported by a Fellowship of the Foundation for Polish Siene.
Key words and phrases :
[219℄
220
S. Rams
nd relations between
et al.
d(C1 , . . . , Cp )(a) and the produt ν(C1 , a) · · · ν(Cp , a)
of the multipliities of the yles, and so we obtain inequalities or equalities
extending the lassial Bézout's theorem to the ase of analyti sets.
The organization of this paper is as follows. Setion 2 is of preparatory
nature; we ollet together basi onstrutions and fats on intersetion theory in omplex analyti geometry. In Setion 3 our main results are proved,
and then used, in Setion 4, to present some appliations for produts of
surfaes and urves. Some omments on the presented version of Bézout's
theorem an be found in [N2 ℄ and [N1 ℄.
2. Intersetion theory. In this paper analyti means omplex analyti,
and manifold means a seond-ountable omplex manifold. An analyti yle
M
on a manifold
is a formal sum
C=
X
αj Yj ,
j∈J
where
αj 6= 0
for
j∈J
are integers and
{Yj }j∈J
is a loally nite family of
nonempty, pairwise distint, irreduible analyti subsets of the manifold
The zero yle
N
C=0
is dened by
J = ∅.
M.
The family of all analyti yles
Z-module.
C . The sets
Yj are alled the omponents of C with multipliities αj , j ∈ J . We say that
the yle C is positive if αj > 0 for all j ∈ J . If all the omponents of C
have the same dimension k , C will be alled a k -yle. For a yle C and an
open subset V of M we an dene in the natural way the restrition C ∩ V
′
of C to V (f. [R3 ℄). If ϕ : M → M is a biholomorphism of manifolds then
we dene the image ϕ(C) of C by
X
ϕ(C) =
αj ϕ(Yj ).
on
will be onsidered with the natural struture of
The analyti set
S
j∈J Yj
is alled the support of the yle
j∈J
m-dimensional manifold and let Y be a pure k -dimensional analyti subset of M . For x ∈ N we denote by ν(Y, x) the degree of
Y at the point x [D, p. 194℄. This degree is equal to the lassial algebrai
Samuel multipliity, and the so-alled Lelong number of Y at x. In this
Now, let
M
be an
paper we will onsider a natural extension of this denition to the ase of an
arbitrary analyti yle. Namely, if
M,
then the sum
ν(C, x) =
C =
X
P
j∈J
αj Yj
is an analyti yle on
αj ν(Yj , x)
j∈J
is well dened and we all it the degree of the yle
For the yle
C
C
at the point
there exists a unique deomposition
C = C(m) + C(m−1) + · · · + C(0) ,
x.
221
Bézout's theorem
where
at
x
C(j)
is a
j -yle
for
j = 0, . . . , m.
We dene the extended degree of
C
by the formula
Denote by
νe(C, x) = (ν(C(m) , x), . . . , ν(C(0) , x)) ∈ Zm+1 .
ν(C)
and
νe(C)
the funtions
ν(C) : M ∋ x 7→ ν(C, x) ∈ Z,
νe(C) : M ∋ x 7→ νe(C, x) ∈ Zm+1 .
ν(C, x) = |e
ν (C, x)|, where |µ| denotes the sum of the oordim+1
nates of µ ∈ Z
.
Let M be an m-dimensional manifold and let S be a losed s-dimensional
P
submanifold of M . For an arbitrary analyti yle C =
j∈J αj Yj in M the
part of C supported by S is dened as
X
αj Yj .
CS =
Observe that
j∈J, Yj ⊂S
Observe that every analyti yle has the deomposition
If
C
C = C S +(C −C S ).
is positive, then so are both parts of this deomposition.
Let
U
set of all
(1)
(2)
be an open subset of
H := (H1 , . . . , Hm−s )
M
suh that
U ∩ S 6= ∅. Denote
by
H(U ) the
satisfying the following onditions:
H
is a smooth hypersurfae in U ontaining U ∩ S for j = 1, . . . , m − s,
Tjm−s
j=1 Tx (Hj ) = Tx S for eah x ∈ U ∩ S .
Z of M of pure dimension k we denote by
H(U, Z) the set of all H ∈ H(U ) suh that ((U \ S) ∩ Z) ∩ H1 ∩ · · · ∩ Hj is an
analyti subset of U \S of pure dimension k−j (or empty) for j = 1, . . . , m−s.
For every H = (H1 , . . . , Hm−s ) ∈ H(U, Z) an analyti yle Z · H in
S ∩ U is dened by the following algorithm (f. [T℄, see also [F℄), where in
For a given analyti subset
eah step we have only proper intersetions, and so the intersetion produt
is given by the lassial theory (f. [D℄, [W1 ℄). For
H
dene
j
n
o
\
d := min j ∈ {0, 1, . . . , m − s} : ((U \ S) ∩ Z) ∩
Hi = ∅ ,
i=1
and onsider the following
Algorithm 1.
Step 0: Let
Step 1: Let
Step 2: Let
Z0 = Z ∩ U . Then Z0 = Z0S∩U + (Z0 − Z0S∩U ).
Z1 = (Z0 − Z0S∩U ) · H1 and Z1 = Z1S∩U + (Z1 − Z1S∩U ).
Z2 = (Z1 − Z1S∩U ) · H2 and Z2 = Z2S∩U + (Z2 − Z2S∩U ).
...................................................................
Step d : Let
S∩U ) · H .
Zd = (Zd−1 − Zd−1
d
We all a positive analyti yle
We have
Zd = ZdS∩U .
Z · H = Z0S∩U + Z1S∩U + · · · + ZdS∩U
the result of the above algorithm.
in
S ∩U
222
S. Rams
Definition 2. For
c∈S
et al.
we all
ge(c) = ge(Z, S)(c) := minlex {e
ν (Z · H, c) : H ∈ H(U, Z), c ∈ U } ∈ Ns+1
g(c) = |e
g (c)| the
c.
the extended index of intersetion and
of
Z
with the submanifold
S
index of intersetion
at the point
X1 , . . . , Xp on a omplex maniZ = X1 × · · · × Xp in M = N p and
For a system of irreduible analyti sets
N we an onsider the analyti
S = ∆N the diagonal of N p . By [T,
fold
set
Thm. 6.2℄ the funtion
d : N ∋ x 7→ d(x) := d(X1 , . . . , Xp )(x) = g(X1 ×· · ·×Xp , ∆N )(x, . . . , x) ∈ N
is analytially ostrutible and by [T, Prop. 2.1℄ we an state the following
X1 , . . . , Xp is the
unique analyti yle X1 •· · ·•Xp suh that ν(X1 •· · ·•Xp , x) = d(x) for x ∈ N .
Definition 3. The intersetion produt of the system
One an extend this denition in the natural way to the ase of analyti
yles (see [T, Def. 6.4℄). Namely, let
Ck =
X
(k)
(k)
k = 1, . . . , p,
αjk Xjk ,
jk ∈Jk
be analyti yles on a manifold
N.
Definition 4. The intersetion produt of the yles
ned by
C1 • · · · • Cp =
X
(1)
(p)
C1 , . . . , Cp
(1)
is de-
(p)
αj1 · · · αjp Xj1 • · · · • Xjp .
jk ∈Jk , k=1,...,p
We onlude this setion with a useful theorem whih follows diretly
from our denitions and [AR, Cor. 5℄.
C1 , . . . , Cp is a system of
′
subset of N and ϕ : V → N
Theorem 5. If
N, V
is an open
yles on a omplex manifold
is a biholomorphism , then
ϕ(C1 ∩ V ) • · · · • ϕ(Cp ∩ V ) = ϕ((C1 • · · · • Cp ) ∩ V ).
3. Bézout's theorem. In this setion an analyti yle
in
Cn
is alled homogeneous (resp. projetive) if all sets
Xj
A=
P
j∈J
αj Xj
Cn
are ones in
(resp. projetive varieties).
C1 , . . . , Cp be homogeneous yles. Then the yle
is homogeneous and
Theorem 6. Let
C1 • · · · • Cp
deg(C1 • · · · • Cp ) = deg C1 · · · deg Cp .
Proof. We maintain the notation of the previous setion. It sues to
prove the theorem for irreduible ones
X 1 , . . . , Xp .
Observe that for
X1 × · · · × Xp ,
g (Z, ∆Cn )(0)|.
d(X1 , . . . , Xp )(0) = g(Z, ∆Cn )(0, . . . , 0) = |e
Z =
223
Bézout's theorem
By [AR, Cor. 3℄ there exists a system
suh that
ge(Z, ∆Cn )(0) = νe(Z · H, 0).
H = (H1 , . . . , Hm−s )
of hyperplanes
The lassial Bézout's theorem used
in eah step of Algorithm 1 gives
|e
g (Z, ∆Cn )(0)| = ν(Z, 0) = deg X1 · · · deg Xp .
Then we have the equality
ν(X1 • · · · • Xp , 0) = d(X1 , . . . , Xp )(0) = deg X1 · · · deg Xp .
It is easy to hek that
X1 • · · · • Xp
is homogeneous, so
ν(X1 • · · · • Xp , 0) = deg X1 • · · · • Xp
and the theorem follows.
For a projetive yle
where
Ybj
C=
P
j∈J
αj Yj
we dene the yle
denotes the one over the variety
Yj .
b := P
b
C
j∈J αj Yj ,
In the ase of the analyti intersetion produt we have the following
version of Bézout's theorem.
C1 , . . . , Cp be projetive yles and let α0 be the multib1 • · · · • C
bp . Then
{0} in the intersetion produt C
Theorem 7. Let
pliity of the point
deg(C1 • · · · • Cp ) = deg C1 · · · deg Cp − α0 .
Proof. Sine the intersetion produt is linear (see [AR, Cor. 5℄), we an
C1 , . . . , Cp are varieties of dimensions d1 , . . . , dp respetively.
Q := (1 : q1 : . . . : qn ) ∈ C1 ∩ . . . ∩ Cp . To simplify
n p
p(n+1) and Q := (1, q , . . . , q ).
our notation we put V := ({1} × C ) ⊂ C
1
n
The Grassmannian of ane r -planes through the point Q is denoted by
r
p(n+1) ).
GQ (C
By [AR, Cor. 3℄ the set of ane hyperplanes H ontaining Q, meeting
the spae V properly and satisfying the ondition
p
p
Y
∆ Y
b
bi , ∆ (Q) ,
ν
Ci · H , Q = ge
C
assume that
We hoose a point
i=1
where
(v)1
1
i=1
stands for the rst oordinate of a vetor
v , is an
open and dense
p(n+1)−1
subset of GQ
(Cp(n+1) ). Sine the map
p(n+1)−2
H 7→ H ∩ V ∈ GQ
(V )
restrited to the set of hyperplanes that do not ontain
V
is ontinuous, we
an assume that
ν
p
Y
i=1
Ci · (H ∩ V )
∆
p
Y
, Q = ge
Ci , ∆ , Q .
i=1
1
224
S. Rams
et al.
H = (H1 , . . . , Hpn−P di )
Proeeding indutively we onstrut a system
ane hypersurfaes suh that
of
Y
Y
bi · H, Q = ge
bi , ∆ (Q),
νe
C
C
Y
Y
Ci , ∆ (Q),
Ci · H|V , Q = ge
νe
H|V := (H1 ∩ V, . . . , Hpn−P di ∩ V ). Here, by abuse of notation, both
n p
n p
the diagonal in (C ) and the one in (P ) are denoted by ∆. Moreover, we
identify Q, resp. Q, with the orresponding point in ∆.
where
Applying [TW, Thm. 2.2℄ in eah step of Algorithm 1 we get
Y
Y
bi · H, Q = νe
νe
C
Ci · H|V , Q
whih implies
Y
Y
bi , ∆ (Q) = ge
ge
C
Ci , ∆ (Q).
As an immediate onsequene we obtain the equality
b1 • · · · • C
bp =
C
where
C1 • · · · • Cp :=
P
X
j≥1
αj Ybj + α0 · {0} ,
j≥1 αj Yj . By Theorem 6, we have
deg(C1 • · · · • Cp ) = deg C1 · · · deg Cp − α0 .
4. Appliations. To show an appliation of Theorem 6 we analyze the
produt of an algebrai surfae and a urve. We start with the following
lemma:
Lemma 8. Let
S
be a projetive surfae and let
has no ommon omponent with
Sing(S).
b=C
b+
Sb • C
r
X
i=1
C ⊂S
be a urve that
Then
αi b
ai + α · {0},
where
(1)
(2)
(3)
ai ∈ Sing(S) ∪ Sing(C) for i = 1, . . . , r,
αi ≥ (ν(S, ai ) − 1) ν(C, ai ) for i = 1, . . . , r,
α ≥ 0.
Proof. (1) It is obvious that all omponents of the yle
If
a ∈ Reg(S) ∩ Reg(C),
then the germ
one smooth hypersurfae that meets
implies that
(1) holds.
b
C
Sba
ba
C
b
Sb • C
are ones.
an be ut out from
transversally along
ba .
C
Sba
by
The latter
appears in the intersetion produt with multipliity one and
225
Bézout's theorem
(2) Sine
inequality
b ∆)(1, ai , 1, ai ) = ν(C, ai ) + αi ,
g(Sb × C,
it sues to prove the
b ∆)(1, ai , 1, ai ) ≥ ν(S, ai ) · ν(C, ai ) .
g(Sb × C,
The latter is an immediate onsequene of [W2 , Property 3℄.
b ,
α ≥ 0 hoose H := (H1 , . . .) ∈ H(U, Sb × C)
of 0, suh that
b · H, 0) = ge(Sb × C,
b ∆)(0) .
νe((Sb × C)
(3) To prove that
is a neighbourhood
where
U
Hi to be hyperplanes. Let Zi , Zi∆ denote
b×C
b , ∆, where ∆
Algorithm 1 applied to H, S
By [AR, Corollary 3℄ we an assume
ith step of
n+1 )2 .
the diagonal in (C
∆
∆
One an see that Z1 = Z2 = 0.
the results of the
is
Thus
b = j(C)
b +
Z3 = (H1 ∩ H2 ∩ H3 ) · (Sb × C)
X
βk Ck ,
j : Cn+1 ∋ x 7→ (x, x) ∈ ∆ and Ck are two-dimensional ones.
S
ai ) ⊂ Ck for eah ai suh that αi > 0. Now
Observe that j(b
X
X
Z4 = H4 ·
βk Ck =
βk′ lk
where
where lk are lines. Consequently
b ∆)(0) =
ge(Sb × C,
deg(C),
X
lk ⊂∆
βk′ ,
βk′ .
X
lk 6⊂∆
b∈b
ai ∩ U . Observe that if b
ai 6= lk for all k , then αi = 0. Indeed, we
b
b
b b) and implies
have H ∈ H(U, S × C), whih yields the equality g(b) = ν(C,
b is the only omponent of the yle Sb • C
b passing through b.
that C
Choose
If
Sine
αi > 0
then
b
ai
is one of the lines lk , say
b
ai = lk0 ,
and
b ∆)(b) ≤lex (ν(C,
b b), β ′ , 0).
ge(Sb × C,
k0
b ∆)(b))1 ≥ ν(C,
b b), we get αi ≤ β ′ and
(e
g (Sb × C,
k0
X
X
b ∆)(0) − ν C
b+
α = g(Sb × C,
αi b
ai , 0 ≥
βk′ ≥ 0.
lk 6⊂∆
Now we an state
Corollary 9. Let
S
be a projetive surfae and let
that has no ommon omponent with
X
Sing(S).
C ⊂S
be a urve
Then
(ν(S, a) − 1) · ν(C, a) ≤ (deg(S) − 1) · deg(C) .
a∈Sing(S)∩C
Proof. We maintain the notation of Lemma 8. By Theorem 6 we have
b = deg(S) · deg(C),
deg(Sb • C)
226
S. Rams
et al.
so Lemma 8 yields
deg(S) · deg(C) = deg(C) +
X
αi + α.
By Lemma 8(2) we get
X
(ν(S, a) − 1) · ν(C, a) ≤ (deg(S) − 1) · deg(C).
a∈Sing(S)∩C
Remark 10. It is natural to ask to what extent the bounds from Lem-
ma 8 and Corollary 9 are sharp. This amounts to the question how to ontrol
the behaviour of the multipliity of the point
0
in the intersetion produt
of ones. Let us point out that the latter an be expressed with the help of
the Hilbert funtion (see [AR℄, [AM℄) and omputed using various omputer
algebra systems.
Referenes
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[A℄
[AR℄
[ATW℄
[CKT℄
[D℄
[F℄
[N1 ℄
[N2 ℄
[R1 ℄
[R2 ℄
[R3 ℄
[R4 ℄
[R5 ℄
[T℄
[TW℄
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[W2 ℄
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S. Rams and P. Tworzewski
Institute of Mathematis
Jagiellonian University
Reymonta 4
30-059 Kraków, Poland
E-mail: slawomir.ramsim.uj.edu.pl
piotr.tworzewskiim.uj.edu.pl
T. Winiarski
Institute of Mathematis
Pedagogial University of Craow
Podhor¡»yh 2
30-084 Kraków, Poland
E-mail: twiniarskiautoom.pl
Reçu par la Rédation le 30.11.2004
Révisé le 1.4.2005
(1626)

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