164
Applications of Arithmetic Algebraic Geometry to Diophantine Approximations
Paul Vojta∗ Department of Mathematics University of California Berkeley, CA 94720 USA
Contents 1 History; integral and rational points 2 Siegel’s lemma 3 The index 4 Sketch of the proof of Roth’s theorem 5 Notation 6 Derivatives 7 Proof of Mordell, with some simplifications by Bombieri 8 Proof using Gillet-Soul´eRiemann-Roch 9 The Faltings complex 10 Overall plan 11 Lower bound on the space of sections 12 More geometry of numbers 13 Arithmetic of the Faltings complex 14 Construction of a global section 15 Some analysis 16 More derivatives 17 Lower bound for the index 18 The product theorem
Let us start by recalling the statement of Mordell’s conjecture, first proved by Faltings in 1983. Theorem 0.1. Let C be a curve of genus > 1 defined over a number field k . Then C(k) is finite. In this series of lectures I will describe an application of arithmetic algebraic geom- etry to obtain a proof of this result using the methods of diophantine approximations (instead of moduli spaces of abelian varieties). I obtained this proof in 1989 [V 4]; it was followed in that same year by an adaptation due to Faltings, giving the following more general theorem, originally conjectured by Lang [L 1]:
∗Partially supported by NSF grant DMS-9001372 165
Theorem 0.2 ([F 1]). Let X be a subvariety of an abelian variety A , and let k be a number field over which both of them are defined. Suppose that there is no ¯ ¯ nontrivial translated abelian subvariety of A ×k k contained in X ×k k . Then the set X(k) of k-rational points on X is finite. In 1990, Bombieri [Bo] also found a simplification of the proof [V 4]. While it does not prove any more general finiteness statements, it does provide for a very elementary exposition, and can be more readily used to obtain explicit bounds on the number of rational points. Early in 1991, Faltings succeeded in dropping the assumption in Theorem 0.2 that ¯ X ×k k not contain any translated abelian subvarieties of A , obtaining another conjec- ture of Lang ([L 2], p. 29). Theorem 0.3 ([F 2]). Let X be a subvariety of an abelian variety A , both assumed to be defined over a number field k . Then the set X(k) is contained in a finite union S i Bi(k) , where each Bi is a translated abelian subvariety of A contained in X . The problem of extending this to the case of integral points on subvarieties of semiabelian varieties is still open. One may also rephrase this problem as showing finiteness for the intersection of X with a finitely generated subgroup Γ of A(Q) . The same sort of finiteness question can then be posed for the division group
{g ∈ A(Q) | mg ∈ Γ for some m ∈ N }; this has recently been solved by M. McQuillan (unpublished); see also [Ra].
Despite the fact that arithmetic algebraic geometry is a very new set of techniques, the history of this subject goes back to a 1909 paper of A. Thue. Recall that a Thue equation is an equation f(x, y) = c, x, y ∈ Z where c ∈ Z and f ∈ Z[X,Y ] is irreducible and homogeneous, of degree at least three. Thue proved that such equations have only finitely many solutions. The lectures start, therefore, by recalling some very classical results. These include a lemma of Siegel which constructs small solutions of systems of linear equations and, later, Minkowski’s theorem on successive minima. Next follows a brief sketch of the proof of Roth’s theorem. It is this proof (or, more precisely, a slightly earlier proof due to Dyson) which motivated the new proof of Mordell’s conjecture. After that, we will consider how to apply the language of arithmetic intersection theory to this proof, and prove Mordell’s conjecture using some of the methods of Bombieri. This will be followed by the original (1989) proof using the Gillet-Soul´e Riemann-Roch theorem. These proofs will only be sketched, as they are written in detail elsewhere, and newer methods are available. Finally, we give in detail Faltings’ proof of Theorem 0.3, with a few minor simpli- fications. 166
In this paper, places v of a number field k will be taken in the classical sense, so that places corresponding to complex conjugate embeddings into C will be identified. Also, absolute values k·kv will be normalized so that kxkv = |σ(x)| if v corresponds to 2 a real embedding σ : k ,→ R ; kxkv = |σ(x)| if v corresponds to a complex embedding, −ef and kpkv = p if v is p-adic, where p is ramified to order e over a rational prime p and f is the degree of the residue field extension. With these normalizations, the product formula reads Y (0.4) kxkv = 1, x ∈ k, x 6= 0. v
A line sheaf on a scheme X means a sheaf which is locally isomorphic to OX ; i.e., an invertible sheaf. Similarly a vector sheaf is a locally free sheaf. More notations appear in Definition 2.3 and in Section 5.
§1. History; integral and rational points In its earliest form, the study of diophantine approximations concerns trying to prove that, given an algebraic number α , there are only finitely many p/q ∈ Q (written in lowest terms) satisfying an inequality of the form
p c − α < q |q|κ for some value of κ and some constant c > 0 . It took many decades to obtain the best value of κ : letting d = [Q(α): Q] , the progress is as follows: κ = d, c computable Liouville, 1844 d+1 κ = 2 + Thue, 1909 κ = min{ d + s − 1 | s = 2, . . . , d} + Siegel, 1921 √ s κ = 2d + Dyson, Gel’fond (independently), 1947 κ = 2 + Roth, 1955 Of course, stronger approximations may be conjectured; e.g.,
p −2 −1− − α < c|q| (log q) . q See ([L 3], p. 71). Beginning with Thue’s work, these approximation results can be used to prove finiteness results for certain diophantine equations, as the following example illustrates. Example 1.1. The (Thue) equation
3 3 (1.2) x − 2y = 1, x, y ∈ Z has only finitely many solutions. Indeed, this equation may be rewritten x √ 1 − 3 2 = √ √ . y y(x2 + 3 2xy + 3 4y2) 167
But for |y| large the absolute value of the right-hand side is dominated by some multiple of 1/|y|3 ; if (1.2) had infinitely many solutions, then the inequalities of Thue, et al. would be contradicted.
For a second example, consider a particular case of Mordell’s conjecture (Theorem 0.1). Example 1.3. The equation
4 4 4 (1.4) x + y = z , x, y, z ∈ Q in projective coordinates (or x4 + y4 = 1 in affine coordinates) has only finitely many solutions. The intent of these lectures is to show that Theorem 0.1 can be proved by the methods of diophantine approximations. At first glance this does not seem likely, since it is no longer true that solutions must go off toward infinity. But let us start by considering how, in the language of schemes, these two problems are very similar. In the first example, let W = Spec Z[X,Y ]/(X3 − 2Y 3 − 1) and B = Spec Z be schemes, and let π : W → B be the morphism corresponding to the injection
3 3 Z ,→ Z[X,Y ]/(X − 2Y − 1). Then solutions (x, y) to the equation (1.2) correspond bijectively to sections s: B → W of π since they correspond to homomorphisms
3 3 Z[X,Y ]/(X − 2Y − 1) → Z,X 7→ x, Y 7→ y and the composition of these two ring maps gives the identity map on Z . In the second example, let W = Proj Z[X,Y,Z]/(X4 + Y 4 − Z4) and B = Spec Z . Then sections s: B → W of π correspond bijectively to closed points on the generic fiber of π with residue field Q . In one direction this is the valuative criterion of properness, and in the other direction the bijection is given by taking the closure in W . These closed points correspond bijectively to rational solutions of (1.4). Thus, in both cases, solutions correspond bijectively to sections of π : W → B . The difference between integral and rational points is accounted for by the fact that in the first case π is an affine map, and in the second it is projective. Note that, in the second example, any ring with fraction field Q can be used in place of Z as the affine ring of B (by the valuative criterion of properness). But, in the 1 case of integral points, localizations of Z make a difference: using B = Spec Z[ 2 ] , for example, allows solutions in which x and y may have powers of 2 in the denominator.
§2. Siegel’s lemma Siegel’s lemma is a corollary of the “pigeonhole principle.” Actually, the idea dates back to Thue, but he did not state it explicitly as a separate lemma. 168
Lemma 2.1 (Siegel’s lemma). Let A be an M × N matrix with M < N and having entries in Z of absolute value at most Q . Then there exists a nonzero vector N x = (x1, . . . , xn) ∈ Z with Ax = 0 , such that
h M/(N−M)i |xi| ≤ (NQ) =: Z, i = 1,...,N.
Proof. The number of integer points in the box
(2.2) 0 ≤ xi ≤ Z, i = 1,...,N is (Z + 1)N . On the other hand, for all j = 1,...,N and for each such x , the jth coordinate yj of the vector y := Ax lies in the interval [−njQZ, (N − nj)QZ] , where th nj is the number of negative entries in the j row of A . Therefore there are at most (NQZ + 1)M < (Z + 1)N possible values of Ax . Hence there must exist vectors x1 6= x2 satisfying (2.2) and such that Ax1 = Ax2 . Then x = x1 − x2 satisfies the conditions of the lemma. In order to further emphasize the arithmetic-geometric nature of the subject, all results will be done in the context of a ring RS , obtained by localizing the ring R of integers of a number field k away from primes in a finite set S of places of k . We also will always assume that S contains the set of archimedean places of k . In the case of Siegel’s lemma, the generalization to number fields is sufficient; this was proved at least as early as LeVeque ([LeV], proof of Thm. 4.14). The form that we will use is due to Bombieri and Vaaler. First, however, we need to define some heights (cf. also Section 5). Recall that, classically, the height of an element x ∈ k \{0} is defined as 1 X h(x) = log max(kxk , 1). [k : ] v Q v Definition 2.3. (a). For vectors x ∈ kN , x 6= 0 , 1 X h(x) = log max kxikv. [k : ] 1≤i≤N Q v n (b). For P ∈ P (k) with homogeneous coordinates [x0 : ··· : xn], h(P ) is defined n+1 as the height of the vector (x0, . . . , xn) ∈ k . By the product formula (0.4), it is independent of the choice of homogeneous coordinates. (c). For x ∈ k , h(x) = h([1 : x]) , the height of the corresponding point in P1 . (d). For an M × N matrix A of rank M , h(A) is defined as the height of the vector consisting of all M × M minors of A .
For a number field k , let Dk/Q denote the discriminant and s the number of complex places. Then the generalization of Siegel’s lemma to number fields is the following. 169
Theorem 2.4 (Siegel-Bombieri-Vaaler, ([B-V], Theorem 9)). Let A be an M×N matrix of rank M with entries in k . Then there exists a basis {x1,..., xN−M } of the kernel of A (regarded as a linear transformation from kN to kM ) such that
N−M s X N − M 2 h(x ) ≤ h(A) + log p|D | . i [k : ] π k i=1 Q
The exact value of the constant will not be needed here; it has been included only for reference. Note that, in addition to allowing arbitrary number fields, this result gives infor- mation on all generators of the kernel of A ; this will be used briefly when discussing the proof of Theorem 0.2 (Section 18).
§3. The index Let Q(X,Y ) be a nonzero polynomial in two variables. Then recall that the multiplicity i j of Q at 0 is the smallest integer t such that aijX Y is a nonzero monomial in Q with i + j = t . This definition treats the two variables symmetrically, whereas here it will be necessary to treat them with weights which may vary. Therefore we will define a multiplicity using weighted variables, which is called the index. Definition 3.1. Let
X `1 `n X (`) Q(X1,...,Xn) = a`1,...,`n X1 ··· Xn =: a(`)X `1,...,`n≥0 (`)≥0
be a nonzero polynomial in n variables, and let d1, . . . , dn be positive real numbers. Then the index of Q at 0 with weights d1, . . . , dn is ( n ) X `i t(Q, (0,..., 0), d , . . . , d ) = min a 6= 0 1 n d (`) i=1 i
Often the notation will be shortened to t(Q, (0,..., 0)) when d1, . . . , dn are clear from the context. Note that, although stated for polynomials, the above definition applies equally well 0 02 to power series. Moreover, replacing some Xi with a power series b1Xi + b2Xi + ... ( b1 6= 0 ) does not change the value of the index. Likewise, the index is preserved if Q is multiplied by some power series with nonzero constant term (i.e., a unit). And finally, there is no reason why one cannot allow several variables in place of each Xi . Thus the index can be defined more generally for sections of line sheaves on products of varieties.
Definition 3.2. Let γ be a rational section of a line sheaf L , on a product X1 ×· · ·×Xn Q of varieties. Let P = (P1,...,Pn) be a regular point on Xi , and suppose that γ is regular at P . Let γ0 be a section which generates L in a neighborhood of P , and for each i = 1, . . . , n let zij , j = 1,..., dim Xi , be a system of local 170
parameters at Pi , with zij(Pi) = 0 for all j . Then γ/γ0 is a regular function in a neighborhood of P , so it can be written as a power series
X (`) γ/γ0 = a(`)X . (`)≥0
Here (`) = (`ij), i = 1, . . . , n , j = 1,..., dim Xi is a (dim X1 + ··· + dim Xn)- tuple. Also let d1, . . . , dn be positive real numbers. Then the index of γ at P with weights d1, . . . , dn , denoted t(γ, P, d1, . . . , dn) or just t(γ, P ) , is
( n dim Xi ) X X `ij min a 6= 0 . d (`) i=1 j=1 i
As noted already, this definition does not depend on the choices of γ0 or of local parameters zij . 1 As a special case of this definition, if n = 2 , if all Xi are taken to be P , and if no Pi is ∞ , then γ is just a polynomial in two variables and this definition specializes to the preceding definition after P is translated to the origin.
§4. Sketch of the proof of Roth’s theorem In a nutshell, the proof of Roth’s theorem amounts to a complicated system of inequal- ities involving the index. First we state the theorem, in general.
Theorem 4.1 (Roth [Ro]). Fix > 0 , a finite set of places S of k , and αv ∈ Q for each v ∈ S . Then for almost all x ∈ k , 1 X (4.2) − log min(kx − α k , 1) ≤ (2 + )h(x). [k : ] v v Q v∈S
If k = Q and S = {∞} , then this statement reduces to that of Section 1. We will only sketch the proof here; complete expositions can be found in [L 5] and [Sch], as well as [Ro]. 0 First, we may first assume that all αv lie in k . Otherwise, let k be some finite 0 0 extension field of k containing all αv , let S be the set of places w of k lying over v ∈ S , and for each w | v let αw be a certain conjugate of αv . (In order to write kx − αvkv when αv ∈/ k , some extension of k · kv to k(αv) must be chosen; then the αw should be chosen correspondingly.) With proper choices of αw , the left-hand side will remain unchanged when k is replaced by k0 , as will the right-hand side. The basic idea of the proof is to assume that there are infinitely many counterex- amples to (4.2), and derive a contradiction. In particular, we choose n good approxi- mations which satisfy certain additional constraints outlined below. The proof will be split into five steps, although the often the first two steps are merged, or the last two steps. The first two steps construct an auxiliary polynomial Q with certain properties. Let α1, . . . , αm be the distinct values taken on by all αv , v ∈ S . Also let d1, . . . , dn be positive integers. These will be taken large; independently of everything else in the proof, they may be taken arbitrarily large if their ratios are fixed. 171
The polynomial should be a nonzero polynomial in n variables X1,...,Xn , and its degree in each Xi should be at most di , for each i . The first requirement is that 1 the polynomial should have index ≥ n( 2 −1) at each point (αi, . . . , αi), i = 1, . . . , m . Such polynomials can be constructed by solving a linear algebra problem in which the variables are the coefficients of Q and the linear equations are given by the vanishing of various derivatives of Q at the chosen points (αi, . . . , αi) . A nonzero solution exists if the number of linear equations is less than the number of variables (coefficients of Q ). Step 1 consists of showing that this is the case; the exact inequalities on 1 and m will be omitted, however, since they will not be needed for this exposition. Thus, step 1 is geometrical in nature. Step 2 involves applying Siegel’s lemma to show that such a polynomial can be constructed with coefficients in RS and with bounded height. (Here we let the height of a polynomial be the height of its vector of coefficients.) From step 1, we know the values of M and N for Siegel’s lemma; the height will then be bounded by n X (4.3) h(Q) ≤ c1 di. i=1
Here the constant c1 (not a Chern class!) depends on k , S , n , and α1, . . . , αm . This bound holds because M/(N − M) will be bounded from above, and coefficients of constraints will be powers of α , multiplied by certain binomial coefficients. Step 3 is independent of the first two steps; for this step, we choose elements x1, . . . , xn ∈ k not satisfying (4.2). Further, the vectors #S − log min(kxi − αvkv, 1) v∈S ∈ R all need to point in approximately the same direction, for i = 1, . . . , n . This is easy to accomplish by a pigeonhole argument, since the vectors lie in a finite dimensional space. To be precise, there must exist real numbers κv , v ∈ S such that
− log min(kxi − αvkv, 1) ≥ κvh(xi), v ∈ S, i = 1, . . . , n and such that 1 X (4.4) κ = 2 + . [k : ] v 2 Q v∈S The points must also satisfy the conditions
h(x1) ≥ c2 and h(xi+1)/h(xi) ≥ r, i = 1, . . . , n − 1. These conditions are easily satisfied, since by assumption there are infinitely many x ∈ k not satisfying (4.2), and the heights of these x go to infinity. Having chosen x1, . . . , xn , let d be a large integer and let di be integers close to d/h(xi), i = 1, . . . , n . For step 4, we want to obtain a lower bound for the index of 172
Q at the point (x1, . . . , xn) . This is done by a Taylor series argument: if v ∈ S , then write X (`) Q(X1,...,Xn) = bv,(`)(X − αv) . (`)≥0
Bounds on the sizes kbv,(`)kv can be obtained from (4.3) in Step 2; moreover bv,(`) = 0 if `1 `n 1 + ··· + < n − 1 , d1 dn 2 by the index condition in step 1. Then the only terms with nonzero coefficients are those with high powers of some factors Xi − αv , so the falsehood of (4.2) implies a quite good bound on kQ(x1, . . . , xn)kv for v ∈ S . Indeed, the nonzero terms in this Taylor series are bounded by n Y exp(−κv`ih(xi)) · other factors i=1 n ! X `i ≤ exp −κ d · other factors v d i=1 i 1 ≤ exp(−κvdn( 2 − 1)) · other factors
At v∈ / S , we also have bounds on kQ(x1, . . . , xn)kv , depending on the denominators in x1, . . . , xn . If these bounds are good enough, then the product formula is contradicted, implying that Q(x1, . . . , xn) = 0 . Indeed, taking the product over all v , the other factors come out to roughly exp([k : Q]dn) ; by (4.4) this gives Y 1 kQ(x1, . . . , xn)kv ≤ exp −[k : Q](2 + 2 )dn( 2 − 1) · exp([k : Q]dn) < 1. v Applying the same argument to certain partial derivatives of Q similarly gives vanish- ing, so we obtain a lower bound for the index of Q at (x1, . . . , xn). Note that the choice of the di counterbalances the varying heights of the xi , so in fact each xi has roughly equal effect on the estimates in this step. Finally, in step 5 we show that this lower bound contradicts certain other properties of Q . One possibility is to use the height h(Q) from step 2.
Lemma 4.5 (Roth, [Ro]; see also [Bo]). Let Q(X1,...,Xn) 6≡ 0 be a polynomial in n variables, of degree at most di in Xi , with algebraic coefficients. Let x1, . . . , xn be algebraic numbers, and let t = t(Q, (x1, . . . , xn), d1, . . . , dn) be the index of Q at (x1, . . . , xn) . Suppose that 2 > 0 is such that
di+1 2n−1 ≤ 2 , i = 1, . . . , n − 1 di and
−2n−1 (4.6) dih(xi) ≥ 2 (h(Q) + 2nd1), i = 1, . . . , n. Then t ≤ 2 . n 2 173
Another approach is to use the index information from step 1. The following is a version which has been simplified, to cut down on extra notation. It is true more generally either in n variables, or on products of two curves. Lemma 4.7 (Dyson, [D]). Let Vol(t) be the area of the set
2 {(x1, x2) ∈ [0, 1] | x1 + x2 ≤ t}, 2 2 so that Vol(t) = t /2 if t ≤ 1 . Let ξ1, . . . , ξm be m points in C with distinct first coordinates and distinct second coordinates. Let Q be a polynomial in C[X1,X2] of degree at most d1 in X1 and d2 in X2 . Then m X d2 Vol(t(Q, ξ , d , d )) ≤ 1 + max(m − 2, 0). i 1 2 2d i=1 1
Historically, Dyson’s approach was the earlier of the two, but it has been revived in recent years by Bombieri. I prefer it for aesthetic reasons, although the method of using Roth’s lemma is much quicker. In particular, it was the form of Dyson’s lemma which suggested the particular line sheaf to use in Step 1 of the Mordell proof. Roth’s innovation in this area was the use of n good approximations instead of two; but for now we will just use two good approximations—it sufficed for Thue’s work on integral points.
§5. Notation The rest of this paper will make heavy use of the language and results of Gillet and Soul´eextending Arakelov theory to higher dimensions. For general references on this topic, see [So 2], [G-S 1], and [G-S 2]. In Arakelov theory it is traditional to regard distinct but complex conjugate em- beddings of k as giving rise to distinct local archimedean fibers. Here, however, we will follow the much older convention of general algebraic number theory, that com- plex conjugate embeddings be identified, and therefore give rise to a single archimedean fiber. This is possible to do in the Gillet-Soul´etheory, because all objects at complex conjugate places are assumed to be taken into each other by complex conjugation. Then, if X is an arithmetic variety and v is an archimedean place, let Xv denote ¯ ¯ the set X (kv) , identified with a complex manifold via some fixed embedding of kv into C . The notation also differs from Gillet’s and Soul´e’sin another respect. Namely, instead of using a pair D = (D, gD) to denote, say, an arithmetic divisor (i.e., an ˆ1 element of Z (X ) ), we will use the single letter D to refer to the tuple, and Dfin to refer to its first component: D = (Dfin, gD) . Likewise, L will usually refer to a metrized line sheaf whose corresponding non-metrized line sheaf is denoted Lfin , etc. This notation is probably closer to that in Arakelov’s original work than the more recent work of Gillet and Soul´eand others. Also, I feel that the objects with the additional structure at infinity are the more natural objects to be considering, and the notation should reflect this fact. 174
In the theory of the Gillet-Soul´eRiemann theorem (cf. [G-S 3] and [G-S 4]), it is natural to use the L2 norm to assign a metric to a global section γ of a metrized line sheaf; however, in this theory it is more convenient to use the supremum norm instead:
kγksup,v := sup kγ(P )k. P ∈X(k¯v ) ¯ Here the norm on the right is the norm of L on Xv , since P ∈ X (kv) . If instead we had P ∈ X(k) , we would write kγ(P )kv to specify the norm at the point (denoted Pv ) on Xv corresponding to P ∈ X(k) . Also, if E ⊆ X is the image of the section corresponding to P , then also let Ev equal Pv , as a special case of the notation Xv . For future reference, we note here that often we will be considering Q-divisors or Q-divisor classes; these are divisors with rational coefficients (note that we do not tensor with Q : this kills torsion, which leads to technical difficulties when converting to a line sheaf). Unless otherwise specified, divisors will always be assumed to be Cartier divisors. If L is such a Q-divisor class, then writing O(dL) will implicitly imply an assumption that d is sufficiently divisible so as to cancel all denominators in L . Also, the notations Γ(X,L) and hi(X,L) will mean Γ(X, O(L)) and hi(X, O(L)) , respectively. And finally, given any sort of product, let pri denote the projection morphism to the ith factor.
§6. Derivatives In adapting the proof of Roth’s theorem to prove Mordell’s conjecture, we replace P1 with an arbitrary curve C . Therefore instead of dealing with polynomials, we need to consider something more intrinsic on Cn , namely, sections of certain line sheaves. Step 4 of Roth’s proof therefore needs some notion of partial derivatives of sections of line sheaves at a point. To begin, let π : X → B be an arithmetic surface corresponding to the curve C . By a theorem of Abhyankar [Ab] or [Ar], we may assume that X is a regular surface. Let W be some arithmetic variety which for now we will assume to be X ×B · · · ×B X (but actually it will be a slight modification of that variety); it is then a model for Cn . Let γ be a section of a metrized line sheaf L on W , let (P1,...,Pn) be a rational point on Cn , and let E ⊆ W be the corresponding arithmetic curve, so that E =∼ B via the restriction of q : W → B . Now the morphism π : X → B is not necessarily smooth, but that is not a problem here, since we are dealing with rational points. Indeed, for i = 1, . . . , n let Ei denote the arithmetic curve in X corresponding to Pi ; then the intersection number of Ei with any fiber is 1 (or rather log qv ), which means that Ei can only meet one local branch of the fiber and can meet that branch only at a smooth point. Thus the completed local ring ObX,Ei,v is generated over ObB,v by the local generator of the divisor Ei ; hence π is smooth in a neighborhood of Ei ; likewise q is smooth in a neighborhood of E . Let x be a closed point in E , and let ObB,q(x) be the completed local ring of B at q(x) . For i = 1, . . . , n let zi be local equations representing the divisors Ei . Then, as noted above, the completed local ring ObW,x of W at x can be written ∼ ObW,x = ObB,q(x)[[z1, . . . , zn]]. 175
If γ0 is a local generator for L at x , then γ/γ0 is an element of this local ring, which can then be written γ X (`) = b(`)z . γ0 (`)≥0 (`) Then each term γ0b(`)z lies in the subsheaf
−1 −1 L ⊗ O(−`1 pr1 E1 − · · · − `n prn En) ⊆ L .
In general this element depends on the choices of γ0 and z1, . . . , zn . However, if 0 0 0 b(`0) = 0 for all tuples (` ) 6= (`) satisfying `1 ≤ i1 , ... , `n ≤ in (i.e., if (`) is a leading term), then the restriction to E of this term is independent of the above choices. This is the definition of the partial derivative