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 [AM℄ [A℄ [AR℄ [ATW℄ [CKT℄ [D℄ [F℄ [N1 ℄ [N2 ℄ [R1 ℄ [R2 ℄ [R3 ℄ [R4 ℄ [R5 ℄ [T℄ [TW℄ R. Ahilles and M. Manaresi, Multipliities of bigraded rings and intersetion Math. Ann. 309 (1997), 573591. M. Montserrat Alonso Ferrero, On the index of ontat, Ann. Polon. Math. 83 (2004), 7786. R. Ahilles and S. Rams, Intersetion numbers, Segre numbers and generalized Samuel multipliities, Arh. Math. (Basel) 77 (2001), 391398. R. Ahilles, P. Tworzewski and T. Winiarski, On improper isolated intersetion in omplex analyti geometry , Ann. Polon. Math. 51 (1990), 2136. E. Cygan, T. Krasi«ski and P. Tworzewski, Separation of algebrai sets and ojasiewiz exponent of polynomial mappings , Invent. Math. 136 (1999), 7587. R. 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Math. 37 (1989), 95101. theory, 227 Bézout's theorem [V℄ [W1 ℄ [W2 ℄ W. Vogel, Results on Bézout's Theorem , Tata Leture Notes 74, Springer, Berlin, 1984. T. Winiarski, Continuity of total number of intersetion , Ann. Polon. Math. 47 (1986), 155178. , Loal properties of intersetion , in: Deformations of Mathematial Strutures, Kluwer, 1989, 141150. 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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