ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS IN TWO-DIMENSIONAL REGULAR LOCAL RINGS. Joseph Lipman Purdue University. Contents

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1 Bull.Soc.Math. de Belgique 45 (1993), ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS IN TWO-DIMENSIONAL REGULAR LOCAL RINGS Joseph Lipman Purdue University 0. Introduction 1. Preliminaries 2. The adjoint ideal 3. Covariance of the adjoint 4. More properties of the adjoint 5. The polar ideal Contents Introduction. In the study of the singularity of a complex plane curve germ (C, 0), given locally, say, by f(x, y) = 0, important roles are played by the adjoint curves (those having a local equation g =0suchthatg restricts on C to an element in the conductor ideal of the one-dimensional local ring O C,0 of germs of holomorphic functions), and by the polar curves (those given locally by an equation of the form af x + bf y + cf = 0 with a, b, c C). In this paper, we introduce corresponding complete ideals C and P in the convergent power series ring C{x, y}, and indeed in any two-dimensional regular local ring. The classical theory of base points of linear systems of curves on smooth surfaces, exposed at length in [4, book 4] and summarized in [16, Chaps. 1 and 2], inspired Zariski to create the theory of complete (=integrally closed) ideals in twodimensional regular local rings [15, pp ]. Over the past half century, this theory has been further developed, by Zariski himself and others [17, Appendix 5], [6, Chaps. II and V], [7], [5], [12], [8]. A complete ideal p in C{x, y} encodes local base conditions to be satisfied by certain complete linear systems of curves C through the origin; the adjoint ideal C p (defined in (2.1)) encodes the conditions which are then satisfied by the adjoints of a generic such C, andthepolar ideal P p (defined in (5.1)) does the same for curves whose multiplicities at the infinitely near base points of the linear system are at least as big as those of the generic polar of C. These polar multiplicities were worked out in [4, pp ] (and for an extensive modern treatment of local polars of plane curves, see [1], [2], [3]). They are just the integers which make up the point basis of P, i.e., the s i prescribed in Theorem (5.2) below in terms of the point basis {r i } of p, which{r i } may also be thought of as the sequence Partially supported by the National Security Agency 223

2 224 JOSEPH LIPMAN of infinitely near multiplicities of the generic curve C (weighted by the degrees of certain field extensions when we work with fields which are not algebraically closed). The factorization of P into simple complete ideals is related to the number of irreducible branches of the generic polar, cf. (5.3). The basic results on the adjoint ideal are given in (2.2) and (3.1), and further interesting properties appear in 4. When our two-dimensional regular local ring, with maximal ideal m, has an algebraically closed residue field, then C = P : m. The general case is treated in (2.3). Though the preceding loosely-described background is geometric, all our considerations will be purely algebraic and precise (except for some remarks at the very end). We assume throughout that the complete ideal p is simple. The extension to more general complete ideals corresponding to the extension from irreducible plane curve germs to reducible ones is left to the sufficiently motivated reader. 1. Preliminaries. We first fix some notation and terminology, and recall a few basic facts. More details and references can be found in [7]. (1.1) Let K be a field. Greek letters α, β, γ,... will denote two-dimensional regular local rings with fraction field K; and such objects will be called points. A quadratic transform of a point α is a localization α[x 1 m α ] n where x is an element of the maximal ideal m α of α, x/ m 2 α,andn is a maximal ideal in the ring α[x 1 m α ]. Any such quadratic transform is itself a point, in which m α generates a principal prime ideal. Apointβ is said to be infinitely near toapointα if β α. There exists then a unique sequence of the form (1.1.1) α =: α 0 α 1 α n := β (n 0) where for each i<n, α i+1 is a quadratic transform of α i. The maximal ideal m β contains m α, and the residue field extension α/m α β/m β has finite degree, denoted [β : α]. (1.2) For an α-ideal I of finite colength (i.e., I contains some power of m α ), the transform I β of I in a point β α is the ideal I(Iβ) 1,whichism β -primary unless Iβ is a principal ideal, in which case I β = β. If β is a quadratic transform of α, then the ideal m α β is principal, and I β = I(m α β) r,wherer:= ord α (I) isthe largest among those integers s such that I m s α. Transform is transitive: if γ β then (I β ) γ = I γ. Transform preserves products: (IJ) β = I β J β. An α-ideal I is complete (i.e., integrally closed) if IR α = I for every valuation ring R such that K R α. Any product of complete ideals is complete. I is simple if it is not the product of two other ideals. If the finite-colength ideal I is complete (resp. complete and simple) then so is I β. The point basis of I is the family of integers {ord β (I β )} β α, where ord β is the unique discrete valuation of K satisfying (1.2.1) ord β (x) = max{ n x m n β } (0 x β). Two complete finite-colength α-ideals coincide iff they have the same point basis.

3 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 225 The Hoskin-Deligne formula for the colength of an m α -primary complete ideal I with point basis {r β } β α is (1.2.2) λ(α/i) = β α [β : α]r β (r β +1)/2 (λ denotes length ). For such an I, then, there are only finitely many points β with r β 0. (Infactr β 0 iff there is a point γ β such that ord γ is a Rees valuation of I.) Using (1.2.2) or otherwise one can show that (1.2.3) λ(i/m α I)=ord α (I)+1 and (1.2.4) λ ( (I : m α )/I ) =ord α (I), cf. [7, 3]. (1.3) A valuation v of K dominates α if v(x) 0 for all x α and v(x) > 0for all x in the maximal ideal m α of α; in other words, the valuation ring R v contains α, and its maximal ideal m v contains m α. For such a v, av-ideal in α is an ideal of the form J α where J is a non-zero ideal in R v. The v-successor of a v-ideal I in α is the ideal I v := { x α xr v IR v } = { x α v(x) >v(i) } where v(i):= min{ v(y) y I } (which exists since I is finitely generated). The principal ideal IR v strictly contains I v R v, and it follows that I v =(I v R v) α, so that I is a v-ideal, clearly the largest one properly contained in I. Associated to any v we have then the sequence of v-ideals (1.3.1) α =: I 0 I 1 (= m α ) I 2 I n... in which I j+1 is the v-successor of I j for every j 0. (We also say that I j is the v-predecessor of I j+1.) For j>0, the ideal I j is m α -primary: clearly I j m α I j 1 whence, by induction, I j m j α.ifvisrank-one discrete, then every v-ideal appears somewhere in (1.3.1). (1.4) We ll call a valuation v of K a prime divisor of α if v dominates α and the residue field extension α/m α R v /m v is transcendental. For example, if β α then ord β is a prime divisor of α, cf. (1.2.1); and in fact every prime divisor of α is of this form for a unique β. Moreover, there is a one-one correspondence p v p between the set of simple m α -primary complete ideals and the set of prime divisors of α [17, p. 391, (E)]. (Actually, v p is the unique Rees valuation of p, cf. [6, p. 245, (21.3)] or [5, p. 333, Thm. 4.2]).

4 226 JOSEPH LIPMAN Thus we have three sets in one-one correspondence: {prime divisors of α} {points infinitely near to α} {simple m α -primary complete ideals} The infinitely near point β p corresponding to a simple m α -primary complete ideal p (i.e., ord βp = v p ) is characterized by any one of the following properties: β p is the largest among those β α such that p β β [17, p. 389, (B)]. β p is the unique point infinitely near to α in which the transform of p is the maximal ideal [17, p. 389, (B)]. A valuation v dominating α dominates β p iff p is a v-ideal. (This follows from [17, p. 390, (D)].) If β p β p then p p. (The converse doesn t hold.) Furthermore, for any v p -ideal I in α, theidealiβ p is principal iff p does not divide I (i.e., I p(i : p)) [17, p. 392, (F), (2)]. Lemma (1.5). Let β α, letv be a valuation dominating β, andleti be a v-ideal such that the ideal Iβ is principal. Then the v-successor of I is the ideal I := (m β I) α. Moreover, the length of the α-module I/I is [β : α]. Proof. Let z I be such that Iβ = zβ, and map Iα-linearly to β/m β by sending x I to (x/z) +m β. It is immediate that the kernel of this map is I,andthat the kernel of its composition with the injective map β/m β R v /m v is I v. Thus I v = I ; and we have an (α/m α )-linear injection I/I β/m β, whence the last assertion. Corollary (1.5.1). 1 For any v dominating β p (i.e., such that p is a v-ideal ) the sequence (1.3.1) of v-ideals in α coincides up to and including p with the corresponding sequence for v p. (1.6) Various proofs of the following useful relation can be found in [6, p. 247, Prop. (21.4)], [13, 7], [5, p. 334, Thm. 4.3], and [8, Cor. (4.8)]: (1.6.1) ord α (p) =[β p : α]v p (m α ). Here is one application. Define the intersection number of two m α -primary ideals I and J, with respective point bases {r β } and {s β },tobe (I J):= [β : α]r β s β. β α 1 Even for polynomial ideals this strengthens [15, p. 171, Thm. 6.2.], where only 0-dimensional valuations occur [ibid., bottom of p. 155].

5 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 227 Lemma (1.6.2). For any m α -primary ideals p, J, with p complete and simple, we have (p J) =[β p : α]v p (J). Proof. Since for β α, p β = β unless β p β, therefore (p J) = [β : α]ord β (p β )ord β (J β ) β p β α (1.6.1) = β p β α =[β p : α]v p (J), [β : α][β p : β]v p (m β )ord β (J β ) the last equality by an easy induction on the number of points between β p and α, cf. [7, pp , Lemma (1.11)]. We mention in passing a more general form of (1.6.1): Corollary (1.6.3). (Reciprocity) For any two simple m α -primary ideals p and q, [β p : α]v p (q) =[β q : α]v q (p). Geometrically, (1.6.1) can be understood thus: the left-hand side is the intersection number of the proper transform of a general element of p with the closed fiber f 1 {m α } of any map f : X Spec(α) obtained by a succession of point blowups, while the right hand side is that intersection number when f is the map obtained by successively blowing up all the points between α and β p, inclusive. A more precise intersection-theoretic interpretation of (1.6.3) is given in the proof of [6, p. 247, Prop. (21.4)]. 2. The adjoint ideal. We fix a point α, and denote its maximal ideal by m. The length of an α-module M will be denoted by λ(m). For any simple m-primary complete ideal p in α, we set f p := [β p : α], the degree of the residue field extension associated with β p α. By (1.5) and the statement immediately preceding it, if I is a v p -ideal not divisible by p, andi is the v p -successor of I, thenλ(i/i ) f p. In particular, if the residue field α/m is algebraically closed then λ(i/i )=f p =1. Definition (2.1). The adjoint ideal C p of a simple m-primary complete ideal p is the largest v p -ideal in α containing p such that whenever I is a v p -ideal with v p -successor I and C p I I p, then (i) v p (I )=v p (I) + 1, and (ii) λ(i/i )=f p. Remark. Corollary (2.2.1) below entails that the conductor property (i) holds for any successive pair I I of v p -ideals such that C p I. Property (ii), however, doesn t consider the case p = m. (For more information, see [12, Remark 3.3].)

6 228 JOSEPH LIPMAN Theorem (2.2). Let p be a simple m-primary complete ideal, and let q be the largest among those v p -ideals q such that ord α (q )=ord α (p). Then C p = q : m and mc p = q. In particular, ord α (C p )=ord α (p) 1. Corollary (2.2.1). For every integer n n 0 := v p (C p ), there exists an x α such that v p (x) =n. Proof. Since v p (zm) = v p (z) +v p (m) for any z α, it suffices to show that (2.2.1) holds for n = n 0, n 0 +1,..., n 0 + v p (m) 1, which by (2.1)(i) follows easily from the relation mc p = q p. Proof of (2.2). Setting v := v p, we have, by (1.2.4) and (1.6.1), (2.2.2) λ ( (q : m)/q ) =ord α (q) =ord α (p) =f p v(m). In view of the paragraph preceding (2.1), there must then be at least v(m) v-ideals contained in q : m and strictly containing q. On the other hand, m(q : m) q implies that v(q) v(q : m) v(m), and so there are at most v(m) such ideals. Hence there are precisely v(m) such ideals; and furthermore if I is any one of them, with v-successor I, then (i) and (ii) in (2.1) must hold. Thus q : m C p,and v(q : m) =v(q) v(m). (Incidentally, the same argument applies to any q in (2.2)). Now let P be the v-predecessor of q, so that v(p) = v(q) 1 = v(q : m)+v(m) 1 and λ(p/q) =f p. Then P : m is a v-ideal containing q : m, and again by (1.2.4) and (1.6.1) we have λ ( (P : m)/p ) =ord α (P) < ord α (p) =f p v(m), whence λ ( (P : m)/(q : m) ) <f p (cf. (2.2.2)). So if q : m C p, then by (ii) in (2.1) we must have P : m = q : m, and further, by (i), the v-predecessor I of q : m must satisfy v(i) =v(q : m) 1. But this is impossible, since then v(mi) =v(m)+v(i) =v(m)+v(q : m) 1=v(P) (see above), whence mi P, i.e., I P : m = q : m. Thusq : m = C p. Now ord α (C p ) < ord α (p) (sincec p q); and mc p q = 1+ord α (C p ) ord α (q) =ord α (p). So ord α (C p )=ord α (p) 1. Finally, λ ( (q : m)/mc p ) = λ(cp /mc p )=ord α (C p )+1=ord α (p) =λ ( (q : m)/q ), the second equality by (1.2.3) and the fourth by (2.2.2), so that mc p = q.

7 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 229 Some peculiarities of the case where α/m α is not algebraically closed are illustrated by: Corollary (2.3). With notation as in the proof of (2.2), and C := C p, the following conditions are equivalent: (i) f p > 1. (ii) P : m C. (iii) v(p : m) =v(c) 1. (iv) There exists z α with v(z) =v(c) 1. Proof. (i) (ii). Since q P C = q : m, therefore f p = λ(p/q) =λ(p/mc) λ(c/mc) λ(c/p) λ ( (P : m)/c ) = λ(c/mc) λ ( (P : m)/p ) =ord α (C)+1 ord α (P) (cf. (1.2.3), (1.2.4)) =1, with equality throughout iff P : m = C. (ii) (iii) follows easily (since P : m C = q : m) from v(m)+v(p : m) =v ( m(p : m) ) v(p) =v(q) 1=v(m)+v(C) 1. (iii) (iv) (i). The first implication is obvious. As for the second, with P the v-ideal P := { x α v(x) v(z) } we have v(p )=v(c) 1, whence by (2.1) and the paragraph preceding it, f p >λ(p /C) Covariance of the adjoint. Notation remains as in 2. This section is devoted to the proof of the following key result, to the effect that adjoint commutes with transform. Immediate consequences appear in 4. Theorem (3.1). Assume p m, so that there is a unique quadratic transform β of α dominated by v p. Denote the transform I β of an m-primary ideal I by I. Then p is a simple m β -primary complete ideal, and C p =(C p ). Proof. That p is a simple m β -primary complete ideal is proved e.g., in [17, p. 381, Prop. 5 and p. 386, Lemma 6]. Let q = mc p be as in (2.2), so that, mβ being principal, q =(C p ). We show first that q C p.setv := v p = v p,andlet q = q 0 > q 1 > q 2 > > q n = p be the sequence of all the v-ideals between q and p, cf. (1.3.1). Since all these ideals have the same order, say r, their transforms obtained by extending to β and then dividing by the principal ideal (mβ) r form a descending sequence q = q 0 > q 1 > q 2 > > q n = p whose members are v-ideals in β [17, p. 390, (D) (1)]. A similar argument holds for any quadratic transform γ of α; andsoifγ β then q γ i pγ = γ for all i. From (1.2.2) and transitivity of transform we find then that λ α (α/q i )=[β : α] ( 1 2 r(r +1)+λ β(β/q i) ) λ α (α/q i+1 )=[β : α] ( 1 2 r(r +1)+λ β(β/q i+1 )).

8 230 JOSEPH LIPMAN Subtracting the first of these equations from the second, and by (2.1)(ii), we get [β : α]λ β (q i /q i+1 )=λ α(q i /q i+1 )=f p, so that λ β (q i /q i+1 )=[β : α] 1 f p = f p. Also, v(q i )=v(q i β)=v(q i)+rv(m) v(q i+1 )=v(q i+1 β)=v(q i+1 )+rv(m) and so v(q i+1 ) v(q i )=v(q i+1) v(q i )=1. It follows, by (2.1), that indeed q C p, i.e., (C p ) C p. If q = β we are done. Otherwise, let P β be the v-predecessor of q,andlet P α be the inverse transform of P, i.e., the largest ideal in α whose transform in β is P [17, p. 390]. Clearly, P is not divisible by m and P =(m s P ) α where s:= ord α (P). Now s<r:= ord α (q); for if not, then we would have m s r+1 C p = m s r q (m s r qβ) α =(m s q ) α (m s P ) α = P; and since ord α (m s r+1 C p )=s =ord α (P), the initial forms of elements in P with order s would form an α/m α -vector space of dimension > 1, contradicting [17, p. 368, Prop. 3] since P is not divisible by m. Sot:= r 1 s 0, and v ( (C p ) ) v ( P ) = ( v(c p ) (r 1)v(m) ) ( v(m t P) (r 1)v(m) ) = v ( C p ) v ( m t P ). If v((c p ) ) v(p ) > 1, then by (2.1)(i), (C p ) = C p, q.e.d. So suppose that v(c p ) v(m t P) = 1, whence, by (2.1) again, the v-predecessor I of C p satisfies λ(i/c p ) <f p.notethatm t P =(m t Pβ) α [17, p. 369, Cor. 2], and that C p β = m r 1 q m r 1 P = m t Pβ, so that C p m t P I. Thus λ(m t P/C p ) <f p. Moreover, m t P and C p have the same order r 1, and since as before (C p ) γ = q γ = γ for any quadratic transform γ of α other than β, therefore (m t P) γ = γ too. Applying (1.2.2) to C p and to m t P, 2 we conclude as above that λ β ( P /(C p ) ) = λ β ( (m t P) /(C p ) ) =[β : α] 1 λ α ( m t P/C p ) < [β : α] 1 f p = f p. So once again we have by (2.1) that (C p ) = C p. 2 Here we use completeness of m t P; but m t P =(m t Pβ) α suffices, cf. [7, p. 223, (3.1) etc.].

9 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS More properties of the adjoint. Denote the point basis of the simple m α -primary complete ideal p by {r β }. From (2.2) and (3.1) we get: Corollary (4.1). Thepointbasisof C p is {r β 1}. Hence, by (1.2.2): Corollary (4.2). λ(α/c p )= β α [β : α]r β (r β 1)/2. Recall that f p := [β p : α]. Corollary (4.3). (Gorenstein property) f p v p (C p )=2λ(α/C p ). Proof. By (1.6.2), the left-hand side is (p C p )= β α [β : α]r β(r β 1). Corollary (4.4). (Noh, [11, Thm. 1]). If f p =1(for example, if α/m α is algebraically closed) then the semigroup S p := { v p (x) 0 x α } is symmetric. Proof. If f p = 1 then by (2.1) and (2.2.1), c := v p (C p )istheconductor of S p, i.e., S p contains every integer c, but c 1 / S p. Symmetry means, by definition, that c 1 s S p for every s S p, or, equivalently, that S p contains exactly half of the integers in the interval [0, c 1]. But when f p = 1, we have by (1.5) and the statement preceding it that λ(i/i ) = 1 for any two successive v p -ideals containing C p. Hence the symmetry of S p results from (4.3). (4.5). Let β α, and set p:= p β, C:= C p. The quadratic sequence α =: α 0 α 1 α n := β (n 0) in (1.1) gives rise to the sequence of regular surfaces Spec(α) =:X 0 f 0 X 1 f 1 f n X n+1 =: X where f i : X i+1 X i (0 i n) is obtained by blowing up the geometric point x i X i whose local ring is α i. Set m i := m αi. Note that m i O X is an invertible O X -ideal. Also po X and CO X are invertible; indeed, an easy induction yields po X = m r i i O ( X ri := ord αi (p α i ) ) and similarly, in view of (4.1), 0 i n CO X = 0 i n m r i 1 i O X.

10 232 JOSEPH LIPMAN Let E i be the curve E 1 i := fi {x i },andlete i be the proper transform of E i on X, i.e., the unique curve on X mapped isomorphically onto E i by the composed map f i+1 f i+2 f n. Let ι: E i X be the inclusion map. For any invertible O X -module L, (L E i ) denotes the degree, over α/m α, of the invertible sheaf ι L on the curve E i. For any divisor D on X, the intersection number (D E i )is,by definition, (O X (D) E i ). Proposition (4.5.1). Set K := p(co X ) 1 = 0 i n m i O X. Then (K E j )=(E j E j )+2[α j : α] (0 j n). Proof. 3 Standard intersection theory (cf. e.g., [6, 13]) yields the following facts. If i<jthen the O X -module m i O X is free (of rank one) in a neighborhood of E j, and hence (m i O X E j )=0. If i = j then (m i O X E j )=(m j O Xj+1 E j)= (E j E j)=[α j : α]. If i>jthen (m i O X E j )= [α i : α] ifx i lies on the proper transform of E j on X i (in which case we say that α i is proximate to α j, and write α i α j ); and otherwise (m i O X E j )=0. Thus (K E j )= (m i E j )=[α j : α] ( 1 [α i : α j ] ). α i α j j i n Now let E i j be the proper transform of E j on X i (i >j). Then Since we conclude that (E i+1 j E i+1 j )=(E i j E i j) [α i : α] if α i α j (E j+1 (E j E j )= [α j : α] =(E i j Ei j ) otherwise. j E j+1 j )=(E j E j)= [α j : α], α i α j [α i : α] =[α j : α] ( 1 α i α j [α i : α j ] ), and (4.5.1) results. 3 Alternatively, [α j : α] is the Euler characteristic of O Ej ; and hence (4.5.1) amounts to saying that K = ω 1 where ω is the relative canonical sheaf for X Spec(α), an assertion proved, in essence, in [10, p. 111] or [9, p. 202 and p. 206, (2.3)]. In particular, C = H 0 (X, pω).

11 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 233 Corollary (4.5.2). With E γ := E i when γ = α i, the adjoint ideal C factors as C = α γ β p c γ γ where c γ = [γ : α] 1 (E γ E γ ) 2 is one less than the number of δ β proximate to γ, eachsuchδ being counted with multiplicity [δ : γ]. Proof. The treatment of factorization of complete ideals given in [6, 18 19] gives C = α γ β pc γ γ with c γ =[γ : α] 1 (CO X E γ ) (4.5.1) = (p E γ ) (K E γ )=0 [γ : α] 1 (E γ E γ ) 2. The rest is contained in the proof of (4.5.1) The polar ideal. Notation remains as in 2. Definition (5.1). The polar ideal P p of a simple m-primary complete ideal p is the smallest among those m-primary v p -ideals P satisfying ord α (P) =ord α (p) 1, i.e., P p is the v p -predecessor of the ideal q = mc p of Theorem (2.2). Corollary (5.1.1). v p (P p )=v p (C p )+fp 1 ord α(p) 1, λ(α/p p )=λ(α/c p )+ord α (p) f p. Proof. Since P p C p, therefore, by (2.1) and (2.2), and also, v p (P p )=v p (q) 1=v p (C p )+v p (m) 1 (1.6.1) = v p (C p )+f 1 p ord α(p) 1, λ(α/p p )=λ(α/c p )+λ(c p /mc p ) f p (1.2.3) = λ(α/c p )+ord α (p) f p. Henceforth, we will write P (resp. C, v) forp p (resp. C p, v p ) when there is no possibility of confusion. Next we describe how to derive the point basis {s γ } γ α of P from the point basis {r γ } γ α of p. As mentioned in the Introduction, this will explain the appellation polar ideal. 4 Another justification not using geometric methods of this description of c γ in terms of proximity appears in [8, Example (3.2)].

12 234 JOSEPH LIPMAN Let β α be the point corresponding to p (so that v =ord β, cf. (1.4)) and let α =: α 0 α 1 α n := β be the corresponding quadratic sequence (1.1.1). Then r γ and s γ both vanish unless γ = α i for some i. (Such vanishing holds for the point basis of any v-ideal, cf. [17, p. 392, (F)].) So setting r i := r αi, s i := s αi, we need only look at the sequences (r i ) 0 i n,(s i ) 0 i n. We say, as in the proof of (4.5.1), that α i is proximate to α j, and write α i α j, if i>jand if the valuation ring of ord αj contains α i. When i>0, α i is proximate to α i 1 and to at most one other α j (because, for example, the closed points on X i in (4.5) form a normal-crossing divisor; or, one can use (5.2.2) below). If there is such an α j,wesaythatα i is a satellite, and otherwise that α i is free, with respect to (w.r.t.) α. Set m i := m αi. The integers v(m i )=[β : α i ] 1 r i (cf. (1.6)) appearing in the following Theorem may be thought of, informally, as representing the infinitely near multiplicities of a general element of p. At least that s what they do when α/m α is algebraically closed. We also let P i be the transform Pα i (which is a v-ideal in α i [17, p. 390, (1)]); and we set p i := p α i, P i := polar ideal in α i of p i. Theorem (5.2). With preceding notation, we have the following inductive prescription for determining r i s i : (i) P i P i p i, so that r i s i =0or1 (0 i n). (ii) r 0 s 0 =1. (iii) If v(m i+1 )=v(m i ) then r i+1 s i+1 = r i s i unless either i +1 = n or α i+1 is a satellite and α i+2 is free, in which cases r i+1 s i+1 =1(whether or not v(m i+1 )=v(m i )). (iv) If v(m i+1 ) <v(m i ) then r i+1 s i+1 r i s i. Moreover, if r i s i =1then P i = P i.andr i s i =1whenever α i+1 is free. Proof. Condition (ii) holds by the definition of P. For the rest, we proceed by induction on the number of points between α and β, everything being obvious if α = β. Suppose α β and that we can find a k>0such that P k = P k. The inductive hypothesis establishes (5.2) k, the statement (5.2) with the underlying pair α β replaced by α k β. Such a replacement does not affect p i or P i for any i k (use transitivity of transform (1.2)), and hence does not affect r i, s i, P i, or v(m i )=ord β (m i ). For i k then, assertions (i) and (iv) in (5.2), as well as the assertions following (iv), are clearly implied by the corresponding assertions in (5.2) k. So is assertion (iii), as we see after making the following observation: if w.r.t. α k, α i+1 is a satellite and α i+2 is free, then the same is true w.r.t. α 0 for if α i+2 α j for some j<k, i.e., α i+2 is contained in the valuation ring R j of ord αj, then also α i+1 R j,sothatα i+1 is proximate to α j as well as to two other points containing α k, contradiction. Thus, to establish (5.2) it will suffice to prove its assertions for i<k, and hence, to prove the following Lemma.

13 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 235 Lemma (5.2.1). Assume α β. Let a 0 and b betheuniqueintegerssuch that ( v(m 0 )=av(m 1 )+b 0 <b v(m1 ) ). Then: (i) {α 1,α 2,...,α a+1 } is the set of all points proximate to α 0 and contained in β; and so for 2 i a +1,α i is a satellite (w.r.t. α). (ii) v(m 1 )=v(m 2 )= = v(m a ) v(m a+1 )=b. (iii) If b = v(m 1 ) and 1 a n 2, then α a+1 is a satellite and α a+2 is free. (iv) For 1 i a +1, P i P i p i, and v(p i) v(p i )= i<j a+1 v(m j ). Thus P a+1 = P a+1,r a+1 s a+1 = 1, and for 1 i a, P i P i, so that r i s i =0. Proof. If α a+2 is proximate to α i for some i a, then so is α a+1, and therefore i =0ori = a. The first possibility is excluded by (i). So is the second under the hypotheses of (iii), since then by (ii), v(m a )=v(m a+1 ), and we can replace m 0 by m α in (i) to conclude. Hence (iii) follows from (i) and (ii). Now (i) and (ii) follow readily from the next result (applied with γ := α, δ the largest point between α and β proximate to α, andw := v). Lemma (5.2.2). Let γ δ be two points, with corresponding quadratic sequence γ =: γ 0 γ 1 γ d := δ. If w is any valuation dominating δ, then, with m i := m γi, ( ) w(m γ ) w(m 1 )+w(m 2 )+ + w(m d 1 )+w(m δ ), with equality iff w does not dominate that quadratic transform of δ which is proximate to γ; and consequently 5 ( ) w(m γ ) w(m 1 )=w(m 2 )= = w(m d 1 ) w(m δ ). Moreover, [γ i : γ 1 ]=1for i =1, 2,...,d. Proof. Let m 0 γ 1 = tγ 1,andletu γ 1 be such that {t, u} is a regular system of parameters in γ 1 (u exists because γ 1 /m 0 γ 1 is regular). The localization of γ 1 at the prime ideal m 0 γ 1 is the valuation ring of ord γ,andsoord γ (t) = 1, ord γ (u) =0. We show next, by induction on i, that for 1 i d, {t/u i 1,u} is a regular system of parameters in γ i, and [γ i : γ 1 ] = 1; and that for 1 i < d, m i γ i+1 = uγ i+1. Indeed, if i<dand if {t/u i 1,u} is a regular system of parameters in γ i,thensinceord γ is non-negative on δ (by definition of the condition γ δ), 5 ( ) can be deduced from the preceding statement with γ i in place of γ 0 (1 i d 2), since γ i+2, being proximate to γ 0 and γ i+1, can t be proximate to γ i.

14 236 JOSEPH LIPMAN hence on γ i+1, and since ord γ (u i /t) < 0, therefore γ i+1 cannot be a localization of the ring γ i [(t/u i 1 ) 1 m i ], so γ i+1 must be the localization of γ i [u 1 m i ] at its maximal ideal (t/u i,u). Now for i i<d,wehavew(m i )=w(m i γ i+1 )=w(u), which is independent of i, proving ( ). Furthermore, w(m γ )=w(t) =(d 1)w(u)+w(t/u d 1 ) = w(m 1 )+w(m 2 )+ + w(m d 1 )+w(t/u d 1 ). As above, the unique quadratic transform δ of δ which is proximate to γ is the localization of the ring δ[t/u d,u] at its maximal ideal (t/u d,u); and w dominates this δ iff w(t/u d ) > 0, i.e., iff w(t/u d 1 ) >w(u). Since w(m δ )=w ( (t/u d 1,u)δ ) =min{w(t/u d 1 ),w(u)}, the rest of (5.2.2) follows. It remains to prove (iv) in Lemma (5.2.1). We proceed by induction, beginning with i =1. SinceP contains the inverse transform p 0 of p 1, and since P is contained in the inverse transform of P 1 [17, p. 390, (3)], therefore [ibid, (4)] gives P 1 p 1. Let C i be the adjoint ideal in α i of p i. From (2.2) and (3.1) we get m r C 1 = C 0 α 1. Also, m r P 1 = P 0 α 1.So v(m 1 C 1 )=v(m 1 )+v(c 0 ) (r 0 1)v(m 0 ), v(p 1 )=v(p 0) (r 0 1)v(m 0 )=v(m 0 )+v(c 0 ) 1 (r 0 1)v(m 0 ) (cf. proof of (5.1.1)). Hence v(p 1) v(p 1 )=v(p 1) v(m 1 C 1 )+1=v(m 0 ) v(m 1 )=(a 1)v(m 1 )+b. Having already proved (ii) in (5.2.1), we then deduce (iv) for i =1. If n = 1 we are done. Otherwise, v(m 0 ) >v(m 1 ), so P 1 P 1 p 1,som 1 C 1 P 1 and ord α1 (P 1 )=ord α 1 (p 1 ). Since as in the proof of (5.1.1), v(p i )=v(m i C i ) 1, the rest follows from the next Lemma. Lemma (5.2.3). Let L be a v-ideal in α i (1 i<n) such that m i C i L p i. Then L α i+1 p i+1,and v(l α i+1 ) v(m i+1 C i+1 )=v(l) v(m i C i ) v(m i+1 ). Proof. From (2.2) and (3.1), we see that where m i+1 C i+1 = m i+1 (m i C i ) α i+1 = m i+1 m i C i (m i α i+1 ) r i r i := ord αi (p i )=ord αi (m i C i )=ord αi (L), so that L α i+1 = L(m i α i+1 ) r i. The conclusion results. This completes the proof of Lemma (5.2.1) and of Theorem (5.2).

15 ADJOINTS AND POLARS OF SIMPLE COMPLETE IDEALS 237 Remarks. (A) Let m:= p 0 p 1 p n := p be the sequence of simple complete ideals in α corresponding to the quadratic sequence (1.1.1) (where β = β p, cf. (1.4)). The v-ideal P:= P p factors uniquely as P = n i=1 cf. [17, p. 392, (2)]. Here a n =0sinceP p n = p. The remaining exponents a i can be characterized in terms of the point basis {s i } 1 i n of P, by means of proximity: a i = s i [α j : α i ]s j, α j α i cf. [8, Cor. (3.1)]. Hence [2, 1] yields the following geometric interpretation when α is the local ring of a smooth point on an algebraic surface over an algebraically closed characteristic 0 field: Let f be a generic element of p, andletζ be a generic polar of the curve germ f =0. Thenζ has n i=1 a i irreducible branches, of which precisely a i pass through the infinitely near points 6 α 1,α 2,...,α i but not through α i+1. (B) Now observe, keeping in mind that transform preserves products (1.2) and consequently point basis takes products to sums, that for any finite-colength α-ideal L, (P L) = n i=1 a i (p i L) (1.6.2) = p a i i, n a i [α i : α]ord αi (L). Assume, for simplicity, that [β : α] =1(sothat[α i : α] = 1 for all i). Set m:= m α. Define: i=1 e i := ord αi (p) ord αi (m) m i := ord αi (m) (1 i n), (1 i n). The integers e i, m i are the same as the integers e q,m q associated by Teissier [14, p. 270] to any one of the a i branches ζ q of ζ which part company at α i with the curve f = 0. To prove this, one needs to know that except for the α i,all other infinitely near points on ζ are nonsingular on ζ and free w.r.t. α, cf.[2,p.4, Thm. 2.1]. With v =ord β as before, we have then n a i m i =(P m) =ord α (P) =ord α (p) 1, i=1 6 A curve germ passes through a point α j if it has a defining equation g =0(g α) with gα j m t α j (t:= ord α (g)), i.e., the proper transform of g in α j is a non-unit.

16 238 JOSEPH LIPMAN and n i=1 a i e i =(P p) (P m) (1.6.2) = v(p) ord α (P) (5.1) = ( v(m)+v(c) 1 ) ( ord α (p) 1 ) (1.6.2) = v(c) (4.3) = 2λ(α/C) (4.2) = n r i (r i 1). i=1 These relations are in accord with the equations near the top of p. 270 in [14]. References 1. E. Casas-Alvero, Infinitely near imposed singularities and singularities of polar curves, Math. Ann. 287 (1990), , Base points of polar curves, Ann. Inst. Fourier41 (1991), , Singularities of polar curves, preprint. 4. F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, vol. II, N. Zanichelli, Bologna, C. Huneke, Complete ideals in two-dimensional regular local rings, Commutative Algebra, Proceedings of a Microprogram held June 15 July 2, 1987, Springer-Verlag, New York, 1989, pp J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), , On complete ideals in regular local rings, Algebraic Geometry and Commutative Algebra, vol. I, in honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1988, pp , Proximity inequalities for complete ideals in two-dimensional regular local rings, Commutative Algebra week, Mount Holyoke College, July 1992, Proceedings, to appear in Contemporary Mathematics. 9. and A. Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), S. Noh, The value semigroups of prime divisors of the second kind on 2-dimensional regular local rings, Transactions Amer. Math. Soc. (to appear). 12. S. Noh, Sequence of valuation ideals of prime divisors of the second kind in 2-dimensional regular local rings, J. Algebra (to appear). 13. M. Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math112 (1990), B. Teissier, Variétés polaires I. Invariants polaires des singularités d hypersurfaces, Inventiones Math. 40 (1977), O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938), , Algebraic Surfaces (2nd supplemented edition), Springer-Verlag, New York, and P. Samuel, Commutative Algebra, vol. 2, D. van Nostrand, Princeton, Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA address: lipman@math.purdue.edu