CORE Metadata, citation and similar papers at core.ac.uk Provided by Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania) LE MATEMATICHE Vol. LV (2000) – Fasc. II, pp. 407–418 ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS CATERINA CUMINO Dedicated to Silvio Greco in occasion of his 60-th birthday. We study the behaviour of analytic branches of a projective variety with respect to hypersurface sections and we give conditions under which their number and their orders are preserved. Introduction. Let X ⊆ Pnk be an algebraic variety over an algebraically closed �eld k of any characteristic. Starting point of this note was a question, posed in [10], if there is a Seidenberg type theorem for analytic branches; more precisely when is it true that the number of branches of X at a singular point P equals the number of branches at P of a “general” hypersurface section of X ? We approach this problem from different points of view and by different methods. In Section 1 we study the case of a closed point P belonging to a singular subvariety Y ⊆ X , with dim Y ≥ 1; using standard local algebra, we give some suf�cient conditions under which the number of branches at P and their orders are preserved by hypersurface sections; then we show by examples that this fact does not hold in general, but that such conditions are always veri�ed when Y is 1−codimensional and char k = 0; in these particular hypotheses and if Work partially supported by GNSAGA-INDAM and by MURST. 1991 Mathematics Subject Classi�cation: 14H20, 14B05. 408 CATERINA CUMINO moreover X is a hypersurface we prove that our results follow also by Zariski’s equisingularity theory and they are classically well known in P3 for multiple curves on surfaces (see [7]). In Section 2 we discuss the case of isolated points. When char k = 0 we use local Bertini type theorems, especially a result of Flenner (see [9]) about normality, and we get in particular suf�cient conditions under which the branches of X at P are well behaved with respect to the general element of a linear system on X . We observe also that, in any characteristic, when dim X = 2, something quite different may happen. Section 3 deals with analytic branches of an algebraic variety X and linear systems on X from a global point of view and in any characteristic. Main tool of our work is a Bertini theorem for geometrically unibranch schemes, due to Zhang (see [17]). We obtain in particular a global statement on the good behaviour of branches of a projective variety under hyperplane sections. 0. We recall some generalities about the branches of a scheme at its points. For more details see [10], [2]. De�nition 0.1. Let X be a k-scheme, X be the normalization of X red and ν : X → X be the canonical morphism; let x ∈ X . A (geometric) branch of X at x is a (geometric) point of the k(x )-scheme ν −1 (x ). The number of branches of X at x is denoted by b(x ) (or b X (x ) when necessary). If A is a local ring, the (geometric) branches of A are the (geometric) branches of spec(A) at its closed point. If A is a ring, we denote by min(A) the set of minimal primes of A and by c(A) their number; max(A) shall denote the set of maximal ideals of A. Lemma 0.2. Let hA be the henselization of the local ring A; there is a canonical bijection between the set of branches of spec(A) and min(hA). De�nition 0.3. Let A be a local reduced ring and p ∈ min(hA), we call order of the branch corresponding to p the multiplicity e(hA/ p). The branch is said to be linear if hA/ p is a regular ring. De�nition 0.4. A local ring A is said to be unibranch (UB) if b(A) = 1. A scheme X is said to be UB at x if O X,x is such. ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS 409 1. In this section we examine the behaviour of branches at closed points of an algebraic variety X ⊆ Pnk , over an algebraically closed �eld k, with respect to sections with an hypersurface F . We show at �rst some suf�cient condition under which the equality b X (P) = b X ∩F (P) holds, for a point P belonging to a reduced irreducible subvariety Y ⊆ X of dimension ≥ 1, when k is a �eld of any characteristic. Secondly we show that, when Y is a 1-codimensional subvariety of an hypersurface X and char k = 0, our results are a consequence of the classical theory of equisingularity. Theorem 1.1. Let (A, m, k) be a local reduced ring. Suppose that: 1) A is regular; 2) the conductor of A has only one associated prime p; 3) A/ p is regular and (A/ p A)red is étale over A/ p. Let f ∈ A be an element such that f := f mod p in A/ p belongs to a regular system of parameters. Then there is a canonical map {branches o f A} ←→ {branches of A/ f A} which is one to one and preserves the orders of branches. Proof. (a) Since f is not in p, it follows by (2) that the conductor is not contained in any minimal associated prime q to f A; hence (A/ f A)q = (A/ f A)q , for any such q ; therefore the integral closures of (A/ f A)red and (A/ f A)red coincide. (b) Put B = A/ p and C = (A/ p A)red = A/I ; let m 1 , · · · , m n be the maximal ideals of C. Consider, for each i = 1, · · · , n, the following diagram: B h � B/ f B h� � Cm i � � Cm ⊗ B B/ f B i Observe that Cm i ⊗ B B/ f B = Cm i / f Cm i and that, by hypothesis (3), B/ f B is regular and h is étale; h � too is étale, therefore Cm i / f Cm i is regular; ([8] I V2 6.5.2, I V4 17.6.1). (c) For each m ∈ max(A), put R = Am and R � = CmC . By hypothesis (1) and (3), R and R � are regular. Now R � / f R � = R/( f, t1 , · · · , ts ) is by (b) 410 CATERINA CUMINO a regular local ring, hence f belongs to a regular system of parameters of R, therefore R/ f R is a regular ring. (d) By hypothesis f ∈ m ⊆ rad(A) and by (c) A/ f A is a regular ring, hence we obtain the bijective map max(A) ←→ max(A/ f A) ←→ max(A/ f A) and, by (a), the correspondence of the thesis. Let now A and f be as above; let hA be the henselization of A and let q ∈ min(hA) be the minimal prime corresponding to a �xed branch of A. Put D = (hA)/q and D � = D/ f D; since f is in particular a super�cial element with respect to the maximal ideal of D, by [16] p. 287, we have that e(D) = e(D � ) and the thesis follows. De�nition 1.2. Let X ⊆ Pnk be a reduced projective variety and let Y ⊆ X be a reduced and irreducible subvariety; let P be a closed point of Y ; we say that a hypersurface H is transversal to Y at P if H does not contain Y nor it is tangent to it at P . Remark 1.3. In 1.1 hypothesis (3) is necessary. In fact, consider for example, in the af�ne space k 3 , the surface V having equation X Z 2 − Y 2 = 0; let A = k[U 2 , U V , V ] be the coordinate ring of V and let p = (U V , V ) be the ideal which corresponds to the double line Y = Z = 0; at the origin O the homomorphism k[U 2 ] → k[U ] is rami�ed and there is only one branch of order 2; moreover not all the sections which are transversal to the double line have, at O , only one branch of order 2: for example the sections of V with planes having equation a X + bY = 0 (a �= 0, b �= 0), are cubic curves, which are reducible in a parabola and a line tangent to each other at O . Theorem 1.4. Let X be a reduced projective variety over an algebraically closed �eld k, let ν : X → X be the normalization of X and let Y ⊆ X be a reduced and irreducible singular subvariety such that: (a) Y �⊆ ν(Sing(X)); (b) ν induces a morphism ν −1 (Y )red → Y , which is étale over a non empty open U ⊆ Y . Then there exists a non empty open V ⊆ Y such that, for any closed point P ∈ V and for each hypersurface H transversal to Y at P , we have a map {branches o f X at P} ←→ {branches o f X ∩ H at P} which is one to one and order preserving. ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS 411 Proof. Since X is normal and ν is a closed morphism, the closed subset ν(Sing(X) has codimension ≥ 2; consider then the open non empty subset W = X − ν(Sing(X)): we may restrict W in such a way that, for each closed point P ∈ W , Y is the unique reduced irreducible singular subvariety which contains P ; moreover we may suppose that P is regular for Y . By hypothesis ν −1 (Y )red → Y is étale over a non empty open U ⊆ Y . Put now V = U ∩ W and A = O X, P for each closed point P ∈ V and let p be the prime ideal of A corresponding to Y ; put moreover B = A/ p and let f be an element of the maximal ideal of A such that f = f mod p belongs to a regular system of parameters. A and f verify the hypothesis of 1.1 and the thesis follows. Remark 1.5. (1) When codim(Y ) = 1, hypothesis (a) is always veri�ed, being codim(ν(Sing(X)) ≥ 2. (2) If char k = 0, there always exists an open non empty set U ⊆ Y such that ν −1 (U )red → U is étale. If char k > 0, such a U may be empty and there are cases of subvarieties not well behaved with respect to transversal sections, as the following example shows. Example 1.6. Assume that k is an algebraically closed �eld of characteristic p > 0. Put A = k[X, Y, Z ]/(Y p + X Z p ) = k[x , y, z] = k[U p , U V , V ] and W = spec(A). The non normal locus of W is the line L : y = z = 0 of multiplicity p on W , which corresponds to the ideal p = (U V , V ). Let Q = (a, 0, 0), with a �= 0, be a closed point of L. Observe that A = k[U, V ], A/ p = k[U p ], A/ p A = k[U ]; then the tangent cone at the point Q is spec(k[X, Y, Z ]/(Y + bZ ) p ), where b p = a and, at Q, there is only one branch of order p. Consider now the surface F : (a − X )(1 +Y + bZ + Y Z p ) + Y (Y + bZ ) p−1 = 0; F is L-transversal at Q and the corresponding section of W has at Q one linear branch and one branch of order p − 1, having the same tangent of multiplicity p. We have in fact: (V , F) = (Y p + X Z p , (Y + bZ ) p (1 + Y + bZ + Y Z p ) + Y Z p (Y + bZ ) p−1) = = (Y p + X Z p , (Y + bZ ) p−1((Y + bZ )(1 + Y + bZ + Y Z p ) + Y Z p )) Consider now the same example with char k �= p: in this case we obtain an L-transversal section of W , which has, at Q, p linear branches corresponding to p distinct tangents; hence the situation is similar to the case of char k = 0. Assume now that the ground �eld k is algebraically closed of characteristic and let Y ⊆ X be an irreducible singular 0; let X be a hypersurface of Pn+1 k subvariety of codimension 1. In these hypothesis Theorem 1.4 is a consequence 412 CATERINA CUMINO of the classical theory of equisingularity developed by Zariski in the early 1960’s (see [13]). Let P ∈ Y be a simple point of Y , put A = Ô X, P and let p be the prime ideal of Y in A. De�nition 1.7. ([15] 3.1) The elements y1 , · · · , yn−1 of the maximal ideal m of A are said Y -transversal parameters if their images in A/ p form a regular system of parameters. Set A(y) = A/(Ay1 + · · · + Ayn−1 ) and call spec(A(y) ) an Y -transversal section of X at P ; observe that dim(A(y) ) = 1 (see[15] 3.5), hence spec(A(y) ) is a plane algebroid curve if it has no multiple components. Let Q be the generic point of Y ; then spec( � A p ) is the only Y −transversal section of X at Q, since in this case dim( � A p ) = 1 and hence the empty set is the only set of Y -transversal parameters; moreover spec( � A p ) is a plane algebroid curve de�ned over the �eld k( p). De�nition 1.8. ([15] 4.1) X is said to be equisingular at P along Y if there exists a Y -transversal section spec(A(y) ) of X at P , such that spec(A(y) ) is a � � curve and such that spec( A (y) ) and spec( A p ) have equivalent singularities at P and at Q respectively. Here equivalence of singularity of plane curves is intended in the sense de�ned by Zariski (see [13]), by comparing the successive locally quadratic transformations which resolve the singularities of the two curves at P and at Q respectively. Proposition 1.9. If X is equisingular at P along Y , then all Y -transversal sections of X at P are curves with equivalent singularities at P . Proof. See [15] 5.3. It can be shown in numerous ways (see[14]) that given a hypersurface X , whose singular locus Y is non singular and of codimension 1, the points of Y where X is equisingular form a dense open subset of Y . Corollary 1.10. Let X and Y be as above, then there exists a non empty open V ⊆ Y such that: (a) for each closed point P ∈ V , there is a one to one correspondence {branches of X at P} ←→ {branches o f any Y−transversal section o f X at P} ←→ { geometric branches of X at the generic point of Y } (b) Theorem 1.4 holds for each closed point P ∈ V . ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS 413 Proof. (a) Let y1 , · · · , yn−1 be a set of Y-transversal parameters of A. By hypothesis A is a Cohen-Macaulay ring; moreover dim(A) = n and dim(A/(Ay1 + · · · + Ayn−1 )) = 1; hence y1 , · · · , yn−1 form a regular sequence in A. Since A is a local ring, y 2 , · · · , y n−1 is a regular sequence in A/Ay1 ; in the same way y 3 , · · · , y n−1 is a regular sequence in A/(Ay1 + Ay2 ) and so on. As X is equisingular along Y at x , the ring A/(Ay1 , · · · , Ayn−1 ) is reduced; by [10] 6.5 it follows that A/(Ay1 , · · · , Ayn−i ) is reduced, for each i = 2, · · · , n and that c(A) ≤ c(A/Ay1 ) ≤ · · · ≤ c(A/(Ay1 , · · · , Ayn−1 )) On the other hand, as a consequence of equisingularity, we have c(A/ (Ay1 , · · · , Ayn−1 )) = c( � A p ) and, shrinking if necessary the open set V , we have , by [2] 3.2, c(A) = c( � A p ), hence the conclusion follows by [2] 3.3(i). (b) By the proof of (a), we have in particular that c(A) = c(A/Ay1 ). Remark 1.11. (1) 1.10 generalizes a classical result about “general points” of a multiple curve on a surface (see e.g. [7] vol.II, pg. 640). (2) Statement 1.10 does not hold in general in positive characteristic, since it is based on a concept of equivalence of plane curve singularities which works only in the zero characteristic case, because of pathologies due to inseparability (see for example [1]). 2. Let P be a closed point of an algebraic variety X ⊆ Pnk : we know (see example 1.6) that the equality b X (P) = b X ∩F (P) does not hold in general. In this section we examine various situations from this point of view ; we discuss in particular the case of isolated points. Proposition 2.1. Let P ∈ X be any closed point, assume that k is an algebraically closed �eld of any characteristic and let F be any hypersurface through P and not tangent to any branch of X at P ; then if the branches of X at P are linear, X ∩ F has at P the same number of branches and they are all linear. 414 CATERINA CUMINO Proof. Put A =hO X, P , let q be a minimal prime of A. Put B = A/q , let m be the maximal ideal of B and let f ∈ m be an element corresponding to F . For such an f put B � = B/ f B ; by hypothesis f is super�cial with respect to m and e(B) = 1, hence e(B) = e(B � ) = 1; therefore B � is regular, in particular it is the ring of a linear branch. Put now A� = A/ f A; we have: e(A) = �q e(A/q) and e(A� ) = �q � e(A� /q � ), where q (resp. q � ) ranges in the set of minimal primes of A (resp. A� ) (see [2] 2.7); since f is super�cial, we have e(A) = e(A� ) and hence b(A) = b(A� ). Lemma 2.2. Let k be a �eld of characteristic 0, (A, m) a local excellent k-algebra, B a �nite normal A-algebra and let x 1 , · · · , x n be elements of m; for λ = (λ1 , · · · , λn ) ∈ k n , put x λ = �λi x i ; suppose moreover m = rad(x 1 , · · · , x n ). Then: (i) there is a non empty open set U ⊆ k n such that, if λ ∈ U , spec(B/x λ B) − V (m B/x λ B) is normal; (ii) if moreover depth(B M ) ≥ 3, for each M ∈ max(B) and λ ∈ U , then B ⊗ A A/x λ A is normal. Proof. See[9] 3.4 and [5] 1.4, 1.8. Theorem 2.3. Let X ⊆ Pnk be a reduced irreducible variety over an algebraically closed �eld k of characteristic zero, such that dim(X ) ≥ 3 and let ν : X → X be the normalization morphism. Let P ∈ X be an isolated singularity and assume that depth(O X ,Q ) ≥ 3 for each Q ∈ ν −1 (P). Let S be a linear system on X , which has P as base point and which separates the tangent vectors at P (see [12] 7.3, p. 152); then, for a general element F of S, there is a map: {branches o f X at P} ↔ {branches o f F at P} which is one to one and order preserving. Proof. If P is an isolated normal singularity the thesis follows immediately by 2.2(ii). If P is an isolated non-normal singularity put A := O X, P and let m be the maximal ideal of A; since P is an isolated singularity, the conductor of A in A is m-primary; moreover for a regular element f ∈ m, the conductor is not contained in any minimal associated prime to f A and we deduce (as in the proof of 1.1, part (a)), that the integral closures of (A/ f A)red and of (A/ f A)red coincide. Let now m = (x 1 , · · · , x n ); for λ ∈ k n , put x λ = �λi x i ; then by 2.2 there exists a non empty open set U such that for λ ∈ U, A/x λ A is normal, therefore we have the conclusion by the correspondence max(A) ↔ max(A/x λ A) ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS 415 Nothing we can say if dim(X ) = 2 using the above methods; in this hypothesis, however we can observe that something quite different might happen: we present in particular a situation at which the equality b X (P) = b X ∩F (P) does never hold. Proposition 2.4. Assume that k is an algebraically closed �eld of any characteristic. Let X ⊆ Pnk be a 2-dimensional integral cone and let P be its vertex. Then, for any general hypersurface F of a suf�ciently large degree containing P , X ∩ F has at P exactly e(O X, P ) linear branches. Proof. Put A = O X, P and let f ∈ O X, P be an element corresponding to an hypersurface F passing through P; let B = A/ f A. Then A/ f A is a 1dimensional ring and, since X is a cone, we have Gr(B) = A/ f˜ A, where f˜ is the initial form of f in A. By Bertini’s Theorem (see e. g. [11] 6.11 (2) with d = 2) and by an argument similar to [4] Lemma 5, it follows that for any general hypersurface F of a suf�ciently large degree passing through P , X ∩ F\P is a (geometrically) reduced scheme; this means in particular that Proj Gr(B) is reduced. As it is well known, the closed points of Proj Gr(B) are in 1-1 correspondence with the tangents to X ∩ F at P , hence, by [3] 2.3 and 2.4 (i), it follows that the branches of B are all linear and their number is e(B); moreover e(B) = e(A), since f is super�cial. Corollary 2.5. Assume that P ∈ X is an isolated singular point of the surface X and that the projectivized tangent cone to X at P is integral, then for any general hypersurface F of suf�ciently large degree passing through P , X ∩ F has at P e(O X, P ) linear branches. Proof. Put A = O X, P ; it is suf�cient to apply the proof of 2.4 to the “cone” spec(Gr(A)), since spec (Gr(A/ f A)) is the tangent cone to X ∩ F at P. 3. Using Bertini type theorems, we can study from a global point of view the behaviour of branches of an algebraic variety X with respect to the general element of a linear system on X in any characteristic, in particular we deduce a result on the branches of a projective variety under hyperplane sections. Theorem 3.1. Let X be a scheme of �nite type over an algebraically closed �eld k of any characteristic and let f : X −→ Pnk be a morphism. Then there exists a non empty open subset U ⊆ (Pnk )∨ such that, for each hyperplane H ∈ U and for each closed point x ∈ f −1 (H ), the branches of X at x are in one to one correspondence with the branches of f −1 (H ) at x . 416 CATERINA CUMINO Proof. Let P = Pnk and let Z be the reduced subscheme of P × P∨ whose set of closed points is {(x , H ) ∈ P × P∨ |x ∈ H }. Consider the commutative diagram X� X ×P Z v �X � � X ×P Z f �P � �Z � P∨ where ν is the normalization morphism. Clearly, for each hyperplane H ⊆ P corresponding to an element s of P∨ , we have, by the commutativity of the diagram above: f −1 (H ) ∼ = Xs ∼ = X ×P Z s . By the theorem = X ×P Z s and X s ∼ of generic �atness and by [6] Lemma 4, there exists a non empty open U ⊆ P such that, for each s ∈ U , the induced morphism ν|X s : X s → X s is birational. Moreover the �ber of ν|X s over a point x ∈ X s can be identi�ed with the �ber of ν over x ∈ X . Now, by [17] Theorem 1.2, we may assume that, for each s ∈ U , X s is geometrically UB, or equivalently (see[8]I 3.5) that the normalization morphism (X s )red → X s is a universal homeomorphism. This completes the proof. Corollary 3.2. Let V be a scheme of �nite type over an algebraically closed �eld k; let S be a �nite dimensional linear system on V and let V −→ Pnk be the rational map corresponding to S. Let D be a general element 1 of S, considered as a subscheme of V . Then at each closed point x ∈ D, but perhaps at the base points of S, the branches of X at x are in one to one correspondence with the branches of D at x . Proof. Let X be the complementary of the base locus of S. Then X is open and we may apply Theorem 3.1 to the morphism X → Pnk induced by S. Corollary 3.3. Let X ⊆ Pnk be a projective reduced variety over an algebraically closed �eld k of any characteristic. Then the number of branches of X at its closed points is preserved by the generic hyperplane section. Remark 3.4. (1) If char k = 0 the last step of the proof of Theorem 3.1 follows also by [6] Theorem 1 and the following Corollary 1 with P = N ormal: in this hypothesis, indeed, after shrinking the open set U if necessary, X s is normal, for each s ∈ U . 1 General element means as usual element of a suitable dense open subset of the projective space parameterizing S. ANALYTIC BRANCHES AND HYPERSURFACE SECTIONS 417 (2) If char k = 0 Corollary 3.3 is a consequence of the local statements of the �rst section. Let indeed H be a generic hyperplane; by [9] 5.2 we have that H ∩ N or(X ) ⊆ N or(H ∩ X ), moreover H does not contain any component of the non normal locus of X , nor it is tangent to any of them; let Y be a singular subvariety of X and let V ⊆ X be an open set as in 1.3, then V ∩ H is non empty ([5] 3.4) and the conclusion follows. Note added in proofs. recently G. Castaldo and G. 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Dipartimento di Matematica, Politecnico di Torino, 10129 Torino (ITALY)