UNIFORM APPROXIMATION OF ABHYANKAR VALUATION IDEALS IN SMOOTH FUNCTION FIELDS

LAWRENCE EIN, ROBERT LAZARSFELD, AND KAREN E. SMITH

Introduction

In this paper we use the theory of multiplier ideals to show that the valuation ideals of a rank one Abhyankar valuation centered at a smooth point of a complex algebraic variety are approximated, in a quite strong sense, by sequences of powers of fixed ideals.

Let R be an n-dimensional regular local domain essentially of finite type over a ground field k of characteristic zero, and let ν be a rank one valuation centered on R. Recall that this is equivalent to asking that ν be an R-valued valuation on the fraction field K of R, taking non-negative values on R and positive values on the maximal ideal 𝔪 ⊆ R. A theorem of Zariski and Abhyankar states that

(1)  trans.deg ν + rat.rank ν ≤ dim K/k,

where the rational rank of ν is the rank of its value group, while its transcendence degree is the maximal dimension of the center ν on some model of K/k. One says that ν is an Abhyankar valuation if equality holds in (1), i.e. if

trans.deg ν + rat.rank ν = dim K/k.

Among all the valuations centered on R, these are the ones from which one expects the best behavior. For example, a divisorial valuation — corresponding to the order of vanishing along a prime divisor E ⊆ Y contracting to the closed point of X = Spec(R) under a proper birational map Y ⟶ X — is an Abhyankar valuation centered on R having transcendence degree n−1 and rational rank 1. At the other extreme, if α_{1}, …, α_{n} ∈ R are any Q-linearly independent positive real numbers, then there exists a unique valuation ν centered on k[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})} with ν(x_{i}) = α_{i}: here trans.deg ν = 0 and rat.rank ν = n. Abhyankar valuations have been the focus of considerable attention. For example, they are known to admit local uniformization in any characteristic [14], and already when dim R = 2 they involve a great deal of beautiful and intricate geometry [24].

Given a valuation ν as above, denote by Φ = ν(R) ⊆ R the value semigroup of ν on R. For each real number m ∈ Φ, let

𝔞_{m} = {f ∈ R | ν(f) ≥ m}

denote the ideal of R consisting of all elements of R whose values are at least m. Clearly 𝔞_{m}^{ℓ} ⊆ 𝔞_{ℓm} for every natural number ℓ ∈ N, but typically the inclusion is strict. However our main theorem shows that for Abhyankar valuations these two ideals lie surprisingly close to each other:

Research of Ein partially supported by NSF Grant DMS 99-70295.

Research of Lazarsfeld partially supported by NSF Grant DMS 97-13149.

Research of Smith partially supported by NSF Grant DMS 00-70722, and by a US Fulbright Fellowship.

Theorem A. Let ν be an Abhyankar valuation centered on R. Then there exists a fixed value e ∈ Φ such that

𝔞_{m}^{ℓ} ⊆ 𝔞_{mℓ} ⊆ 𝔞_{m−e}^{ℓ}

for all m ∈ Φ and all ℓ ∈ N.^{1}

Roughly speaking, the Theorem asserts that the valuation ideals 𝔞_{mℓ} are closely and uniformly approximated by powers of 𝔞_{m}. It follows from the Theorem that there is a fixed non-zero element δ ∈ R such that

(2)  δ^{ℓ}𝔞_{mℓ} ⊆ 𝔞_{m}^{ℓ}

for all m ∈ Φ and ℓ ∈ N. This points to the heuristic idea that the associated “Rees ring” of the Abhyankar valuation ν, while usually not finitely generated when n = dim R > 1, is “almost” finitely generated. It would be interesting to know if one can make this precise. Both the Theorem and (2) can fail for non-Abhyankar valuations: see Remark 2.6.

Theorem A can be interpreted as a strengthened form of a celebrated theorem of Izumi. In the setting of Theorem A, Izumi’s result (in a form due to Hübl and Swanson) asserts that there exists an index p such that 𝔞_{pℓ} ⊆ 𝔪^{ℓ} for all natural numbers ℓ, where as above 𝔪 is the maximal ideal of R.^{2} Theorem A says much more: not only is 𝔞_{pℓ} ⊆ 𝔪^{ℓ} for all ℓ, but 𝔞_{pℓ} is contained in the ℓ-th power of an ideal 𝔞_{p−e} very close to 𝔞_{p}. In other words 𝔞_{pℓ} ⊆ 𝔞_{p−e}^{ℓ}, where not only are the 𝔞_{p−e}’s getting deeper in R as a function of p (whereas 𝔪 stays fixed), but they are getting deeper at the same rate as the 𝔞_{p} themselves. The theorem also implies—at least for regular local rings essentially of finite type over a field— the well-known statement of Izumi’s Theorem comparing values of two valuations centered at 𝔪; see Corollary 2.5.

In an algebro-geometric context, Theorem A is most interesting for divisorial valuations. More generally, Theorem A can be applied to a composite of valuations to yield the following:

Corollary B. Let D be an effective divisor on a normal variety X and suppose that X →^{π} Y is a proper birational map contracting D to a smooth (but not necessarily closed) point of Y. Then there exists a natural number e such that

π_{∗}O_{X}(−mℓD) ⊂ [π_{∗}O_{X}(−(m − e)D)]^{ℓ}

for all natural numbers ℓ and all m ≥ e.

Thus even though the algebra

(3)  ⨁_{ℓ∈N} π_{∗}O_{X}(−mℓD)

1We adopt the convention that 𝔞_{m−e} = R when m < e.

2Izumi’s statement actually deals with less general valuations but more general rings. It is proved in [12], [22], and [11].

is not finitely generated in general, Corollary B gives some measure of control over it. Of course, it is a central problem of birational geometry to understand when such algebras are finitely generated: when (3) is finitely generated, the corresponding projective scheme is the stable image of X under the birational map to projective space over Y given by the linear series | − mD| for m ≫ 0.

Another viewpoint involves the concept of the volume of a rank one valuation centered on a local domain. By definition, the volume of ν on R is

vol_{R}(ν) := lim sup_{m→∞} \frac{length (R/𝔞_{m})}{m^{n}/n!},

where n is the dimension of the local ring R. In case 𝔞_{m} is the m-th power of a fixed ideal 𝔞, then of course this is simply the multiplicity of 𝔞. But in general the volume is not actually a multiplicity: indeed, it can be an irrational number; see Example 3.5 (iii). However Theorem A implies that the volume of an Abhyankar valuation is approximated arbitrarily well by the multiplicities of the ideals 𝔞_{m}. Specifically:

Corollary C. If ν is an Abhyankar valuation as above, then

vol_{R}(ν) = lim_{m→∞} \frac{e(𝔞_{m})}{m^{n}}.

This corollary in turn leads to an interesting upper bound in the spirit of Teissier on the multiplicity of an 𝔪-primary ideal I ⊆ R in terms of the volumes of its Rees valuations ν_{1}, …, ν_{r}.^{3} Specifically, we prove an inequality of Minkowski-type on the volumes of Abhyankar valuations which implies that the multiplicity of I satisfies

e(I)^{1/n} ≤ e_{1}vol_{R}(ν_{1})^{1/n} + ⋯ + e_{r}vol_{R}(ν_{r})^{1/n},

where e_{i} = ν_{i}(I), and as above n = dim R. Since an earlier version of this manuscript was written, Mircea Mustata has shown that Corollary C and the related Minkowskitype statements hold for arbitrary valuations (not just Abhyankar) and even much more generally. See [20].

The proof of Theorem A uses the theory of multiplier ideals. One can associate to the valuation ideals {𝔞_{m}} a sequence of asymptotic multiplier ideals {𝔧_{m}}_{m∈Φ}, as defined (in a slightly different setting) in [8]. The general theory — which we review in §1 — shows that these satisfy

(4)  𝔞_{m}^{ℓ} ⊆ 𝔞_{mℓ} ⊆ 𝔧_{m}^{ℓ}

for all m ∈ Φ and ℓ ∈ N. The main work of the present paper — which we carry out in §2 — is to establish that 𝔞_{m} and 𝔧_{m} have “bounded difference” in the sense that there is a fixed 𝔪-primary ideal 𝔡 ⊆ R such that

𝔡 ⊆ (𝔞_{m} : 𝔧_{m})

3Recall that the Rees valuations of I are defined by taking the normalized blowup X of I, and writing IO_{X} = O_{X}(−e_{1}E_{1} − ⋯ − e_{r}E_{r}) for some prime Weil-divisors E_{i} of X and some positive integers e_{i}.

Then ν_{i} is the valuation on the fraction field of R given by order of vanishing along E_{i}.

for all sufficiently large m ∈ Φ. Theorem A then follows from (4) upon taking e = ν(𝔡). In §3, we discuss the volume of a valuation: we believe that this is an invariant of independent interest. Finally we give in §4 some further applications, including the proof of Corollary B.

The present paper continues the project started in [8] of using ideas from higher dimensional complex geometry to search for possibly unexpected uniform behavior in Noetherian rings. Our techniques rest on resolution of singularities and vanishing theorems, and so are essentially limited to local rings coming from smooth complex varieties. Although one could expect the statements themselves to remain valid in less restrictive settings, we haven’t seriously investigated the extent to which such a generalization is possible. We hope however that the results appearing here will pique the interest of experts and encourage them to put the picture in a broader perspective. We note that the main theorem here was inspired by an attempt to “localize” a result of Fujita concerning the volumes of big line bundles: a proof via multiplier ideals appears in [6] (see also [17] and [4, 14.5]). The reader may consult [4], [23], [21], [13], [5], [17], [16], [7] and the references therein for numerous other recent applications of multiplier ideals to the global study linear series on a projective variety.

The authors are grateful to Dale Cutkosky, Will Traves and especially Mark Spivakovsky for numerous helpful and encouraging discussions. In particular, Cutkosky first explained to us a special case of Proposition 2.8, and Example 3.15 was worked out in a series of discussions with Spivakovsky and Traves. Also, we thank Ray Heitmann for questions and remarks that improved the presentation of our results.

1. The Asymptotic Multiplier Ideal of a graded family

In this section, we recall the notion of a graded family of ideals and examine the elementary properties of graded families of valuation ideals. We also review the construction and basic properties of the asymptotic multiplier ideals introduced in [8]. Because collections of valuation ideals are naturally indexed by semigroups slightly more general than the natural numbers, it is convenient to allow graded families that are indexed by additive subsemigroups of the real numbers. Thus the exposition here is slightly more general than what is stated in [8], but all the proofs are the same.

1.1. Graded Families indexed by a semi-group. Let Φ be an additive subsemigroup of the non-negative real numbers. Of course, one of the main examples is the semigroup of natural numbers N.

Definition 1.1. Fix a ring R. A graded family or graded system of ideals indexed by Φ is a collection 𝔞_{•} = {𝔞_{m}}_{m∈Φ} of ideals of R satisfying

(5)  𝔞_{m} · 𝔞_{ℓ} ⊆ 𝔞_{m+ℓ} for all m, ℓ ∈ Φ.

To avoid trivialities, we assume also that 𝔞_{m} ≠ 0 for m ≫ 0.

It is convenient to assume also that 𝔞_{0} = R, in which case condition (5) is equivalent to the statement that the R-module

⨁_{m∈Φ} 𝔞_{m}

has the natural structure of a Φ-graded R-algebra. We refer to this ring as the Rees algebra of the graded system 𝔞_{•}. The theory of asymptotic multiplier ideals is particularly useful when the Rees algebra fails to be finitely generated (or at least is not known to be so). Of course, one can also define graded families, and develop the theory of asymptotic multiplier ideals, for families of coherent ideal sheaves in the structure sheaf of a scheme. But since all the definitions are local in nature this involves no essential differences from the affine setting.

Example 1.2. The following are familiar examples of graded families indexed by the natural numbers.

(i). The simplest example is the collection {𝔞^{m}} of powers of a fixed ideal 𝔞. This should

be considered a trivial example of a graded family. (ii). A slightly less trivial example is the collection \{\overline{𝔞^{m}}\} of integral closures of powers of

a fixed ideal 𝔞. From the point of view of multiplier ideals, however, this example is no less trivial than the first, since the multiplier ideals are the same. See also Example 3.5 (i).

(iii). The graded family {𝔞^{(m)}} of symbolic powers of a fixed ideal 𝔞 was the main example

treated in [8]. (iv). The collection {𝔟_{m}} of defining ideals for the base loci of the complete linear series

|mD|, where D is a fixed big line bundle on a projective variety X, forms a graded family of ideals in the sheaf of rings O_{X}. This graded family plays a central role in Kawamata’s work on deformation of canonical singularities [13].

1.2. Graded systems arising from a valuation. In this paper, we are primarily interested in graded families of valuation ideals for rank one valuations centered on a regular local domain. Let us recall some of the basic terminology and give a few examples. Good general references on valuation theory are [26] and [27].

Let Γ be an ordered Abelian group, written additively. A valuation on a field K with values in Γ is a map of Abelian groups

ν : K^{∗} = K \ 0 ⟶ Γ

satisfying the additional condition that

(6)  ν(f + g) ≥ min{ν(f), ν(g)}.

Since only the image of ν is of importance, we assume that ν is surjective. An elementary but useful observation is that in fact,

(7)  ν(f + g) = min{ν(f), ν(g)} whenever ν(f) ≠ ν(g).

The set of all elements of K which have non-negative values forms of subring R_{ν}, called the valuation ring of ν. The valuation ring of ν is a local ring with maximal ideal consisting of all the positively valued elements of K, but it is not Noetherian in general. One can also define a valuation ring abstractly as a subring of K containing either x or \frac{1}{x} for every x ∈ K. The data of a valuation ring inside a field K is equivalent to the data of a valuation on K (up order preserving isomorphism of the value group). The rank of a valuation is by definition the Krull dimension of the valuation ring R_{ν}.

In this paper, we consider only rank one valuations on function fields. In this case, the ordered group Γ can and will be identified with an ordered subgroup of the real numbers, the field K is assumed to be a finitely generated field extension of some fixed ground field k, and we consider only those valuations vanishing on k. In particular, the valuation ring R_{ν} is a k-algebra, though not usually finitely generated.

Let ν be a valuation on a function field K/k. The valuative criterion for properness (see [9, p. 101]) ensures that for any complete algebraic variety X over k with function field K (that is, for any complete model of X), there is a unique map SpecR_{ν} ⟶ X. Thus a valuation chooses, in a consistent way, a (not necessarily closed) point on every complete model of K, namely, the the image W of the closed point of SpecR_{ν} under this map. The variety^{4} W is called the center of ν on X, and the local ring O_{W,X} is called the local ring of ν on X. The defining ideal I_{W} of W consists of all local sections of O_{X} of positive value.

For a local domain R contained in K, we say that the valuation ν is centered on R if ν takes non-negative values on R and strictly positive values on the maximal ideal of R. Thus the valuation is centered on any of its local rings.

Let ν be a valuation on a function field K/k and let X be any irreducible k-scheme with function field K. For each non-negative m ∈ R, the subset

𝔞_{m} = {f ∈ O_{X} | ν(f) ≥ m}

forms an ideal sheaf in O_{X}, called a ν-valuation ideal (or just a valuation ideal when ν is understood). When it is necessary to emphasize the model, we will write 𝔞_{m}(X). Note that if π : X ⟶ Y is a proper birational map between complete models of K, then π_{∗}𝔞_{m}(X) = 𝔞_{m}(Y).

For any subsemigroup Ψ of R (eg, Ψ = N or Ψ = R_{≥0}), the collection {𝔞_{m}}_{m∈Ψ} forms a graded family of ideals in O_{X}, called the graded family of ν on X. It is natural to index this graded system by Γ, but allowing the indexing set to be N or R will sometimes by more convenient. The value semigroup Φ = ν(O_{X}) is the ’optimal’ indexing set: every ν-valuation ideal appears exactly once as a member this graded family. When X is Noetherian, the value semigroup Φ is well-ordered, meaning that every subset has a minimal element.

Proposition 1.3. If ν is a rank one valuation on a function field K/k and X is any complete model of K, then the valuation ideals 𝔞_{m}(X) are primary to the ideal defining the center of ν on X. If ν is a rank one valuation centered on local domain R, then each of the valuation ideals 𝔞_{m} is primary to the maximal ideal of R.

4Here and elsewhere, we abuse terminology by failing to distinguish between an irreducible variety and its generic point.

Proof. Suppose that f and g are local sections of O_{X} with f ∉ 𝔞_{m} but fg ∈ 𝔞_{m}. Then ν(fg) = ν(f) + ν(g) ≥ m, but ν(f) is strictly less than m. This means that ν(g) is positive, so some positive multiple, say kν(g), exceeds m. Thus g^{k} is in 𝔞_{m}, proving that 𝔞_{m} is primary. Furthermore, since any element of I_{W} has a power in 𝔞_{m}, the radical of 𝔞_{m} is I_{W}. The argument for the second statement in the proposition is the same.

Example 1.4. We give four simple examples of graded families of valuation ideals in the local ring R = k[x, y]_{(x,y)} of the origin in the affine plane. Each arises from a different rank one valuation on the function field K = k(x, y).

(i). Let ν be the valuation given by ’order of vanishing at the origin’. Explicitly, for a polynomial f, ν(f) is the degree of the smallest degree non-zero monomial appearing in the unique expression of f as a sum of monomials x^{a}y^{b}. The value of any rational function \frac{f}{g} where f and g are polynomials is uniquely determined by virtue of ν being a group homomorphism: ν(\frac{f}{g}) = ν(f) − ν(g).

This valuation has value group Z and value semigroup N on R. The valuation ideals of ν are powers of the defining ideal of the origin:

𝔞_{m} = (x, y)^{m}.

(ii). Let π : X ⟶ SpecR be any proper birational map from a normal scheme X and let D be any prime divisor of X collapsed to the origin. Let ν_{D} be the valuation given by ’order of vanishing along D.’ Explicitly, for any f ∈ R, consider the image of π^{∗}f in the local ring O_{D,X} of X at the generic point of D. Then ν_{D}(f) is the maximal integer n such that π^{∗}(f) is divisible by t^{n} in O_{D,X}, where t is a uniformizing parameter for the discrete valuation ring O_{D,X}.

This valuation has value group Z, and value semigroup on R a subsemigroup of N. The valuation ideals are

𝔞_{m} = π_{∗}O_{X}(−mD),

which in general, can be tricky to understand. In the simple case where π is the blowup of the origin and D is the resulting exceptional divisor, this example recovers Example (i) above.

(iii). Let ν be the valuation on k(x, y) defined by the assignment ν(x) = 1 and ν(y) = π ∈ R. This uniquely determines a valuation on k(x, y): the value of each monomial x^{a}y^{b} is a + bπ, and because 1 and π are Z-independent, distinct monomials have distinct values, so the value of an arbitrary polynomial is determined by (7).

This valuation has value group Z + Zπ ⊂ R and value semigroup N + Nπ on R. The valuation ideals for ν_{π} are all monomial ideals

𝔞_{m} = ({x^{a}y^{b} | a + πb ≥ m}).

(iv). Let ν be the valuation given by ’order of vanishing along the analytic arc y = e^{x} −1’. Explicitly, for a polynomial f, we define ν(f) to be the highest power of t dividing the image of f under the map

k[x, y] ↪ k[[t]]
f(x, y) ↦ f(t, e^{t} − 1).

Here e^{t}−1 denotes the power series t+ \frac{t^{2}}{2!} + \frac{t^{3}}{3!} +… (assuming that k has characteristic zero).

This valuation has value group Z and value semigroup N on R. Its valuation ideals are given by

𝔞_{m} = (x^{m}, y − x − \frac{x^{2}}{2!} − ⋯ − \frac{x^{m−1}}{(m − 1)!}).

The preceding examples are simple examples of rank one valuations centered on the origin of the plane. The first three are Abhyankar valuations (see 2.1); all four have finitely generated value groups. By contrast, there are valuations with value group Q centered on the origin of the plane (see Example 3.15); while more complicated to describe, these are actually ’typical’ in a certain sense. The classification of valuations centered on the origin of the plane is a beautiful story; see [24]. For further examples of valuations, including higher rank valuations, consult [26] §10, or [27] VI §15.

1.3. Multiplier ideals. We now recall the construction and basic properties of multiplier ideals as well as the asymptotic constructions from [8]. We will give only a few proofs; the rest can be found for instance in [8], [17] or [16].

Consider a scheme X, smooth and essentially of finite type over a field of characteristic zero— mainly we have in mind the case where X is a smooth complex variety or the spectrum of a regular local k-algebra. Let 𝔞 ⊆ O_{X} be a (coherent) ideal sheaf on X. Given a rational number c > 0 we define multiplier ideals

J (X; c · 𝔞) = J (X; 𝔞^{c}) ⊆ O_{X}.

Intuitively these are ideals with remarkable cohomological properties which reflect in a somewhat subtle manner the singularities of the divisors of functions f ∈ 𝔞. The construction starts by taking log resolution μ : X′ ⟶ X of 𝔞. Recall that this means that μ is a projective birational map from a regular scheme X′ to X such that 𝔞O_{X′} = O_{X′}(−F), where F is an effective Cartier divisor on X′ with the property that the sum of F and the exceptional divisor of μ has simple normal crossing support. Such resolutions can be constructed (as we are in characteristic zero) by resolving the singularities of the blow-up of 𝔞. We write K_{X′/X} = K_{X′} − μ^{∗}K_{X} for the relative canonical divisor of X′ over X. Given 𝔞 and c > 0 as above, we now define

(8)  J (X; c · 𝔞) = J (c · 𝔞) = μ_{∗}O_{X′}( K_{X′/X} − [ c F ] ).

Here cF is viewed as an effective Q-divisor on X′, and its integer part [c F] is defined by replacing the coefficient of each component by the greatest integer less than or equal to it. This definition is independent of the log resolution μ; see e.g. [17]. When X is the affine scheme SpecR, we write also J (R, c · 𝔞), and when c = 1 we write simply J (𝔞).

Remark 1.5. Following Lipman [18] it is also possible to define J (X; c · 𝔞) without referring to a log resolution. For example, when R is a regular local ring,

(9)  J (R; c · 𝔞) = ⋂_{ν}{r ∈ K | ν(r) ≥ cν(𝔞) − ν(Jac_{R_{ν}/R})},

where the intersection is taken over all valuations of K given by order of vanishing along some prime divisor on a model X′ of K, and where Jac_{R_{ν}/R} denotes the Jacobian ideal of the extension R ↪ R_{ν}. This is a slight modification of Lipman’s definition of an adjoint ideal, in which we have allowed for the possibility of a coefficient c; see [18]. This approach makes sense for arbitrary regular Noetherian schemes (not just for classes of schemes that admit a good theory of resolution of singularities as we have assumed here.) However multiplier ideals derive their power from the properties they satisfy, and the important facts — which rest on vanishing theorems — are so far only known over fields of characteristic zero.

Remark 1.6. Multiplier ideals were originally defined analytically, as ideals of germs of holomorphic functions that are L^{2}-integrable for a certain weighted L^{2} space. See e.g. [4]. In this approach they appear as sheaves of multipliers, whence the name.

Among the various properties these ideals satisfy, we will need three in particular. First:

(10)  𝔞 ⊆ J (𝔞)

for any ideal 𝔞. This is elementary: it boils down to the fact that the relative canonical bundle K_{X′/X} of a log resolution is effective. More substantially for any rational c > 0 and any ℓ ∈ N one has the subadditivity relation:

(11)  J ( ℓc · 𝔞) ⊆ J ( c · 𝔞)^{ℓ}.

This is established in [6] using vanishing theorems. The third useful property, which follows easily from the definitions, is the behavior of these ideals under birational maps. Specifically, if μ : X′ ⟶ X is a proper birational map, with X′ regular, and if 𝔞 ⊆ O_{X} is any ideal, then

(12)  J (X; 𝔞) = μ_{∗}(J (X′; 𝔞′) ⊗ O_{X′}(K_{X′/X})),

𝔞′ = 𝔞 · O_{X′} being the pullback of 𝔞 to X′. The reader may consult [17], or [16, Part III] for proofs of these and other properties of multiplier ideals.

Given now a graded family 𝔞_{•} as above and an index m ∈ Φ, we will construct an asymptotic multiplier ideal 𝔧_{m} = 𝔧_{m}(𝔞_{•}) which reflects the asymptotic properties of all the ideals 𝔞_{pm} for p ∈ N. From (5) and (8), it is easy to check that for each m ∈ Φ, we have

(13)  J (𝔞_{m}) ⊆ J (\frac{1}{p} · 𝔞_{pm})

for all p ∈ N; see [8, §1]. This, together with the Noetherian property for O_{X}, implies that the set of ideals

\{J (\frac{1}{p} · 𝔞_{pm})\}_{p∈N}

has a unique maximal element. We then define the m-th asymptotic multiplier ideal 𝔧_{m}(𝔞_{•}) to be this maximal element. In other words,

(15)  𝔧_{m}(𝔞_{•}) = J (\frac{1}{p} · 𝔞_{pm}) for sufficiently divisible p ∈ N.

In fact, it is not necessary to assume that p is sufficiently divisible: assuming that 𝔞_{m} ≠ (0) for m ≫ 0, any very large p will do; see [8], Remark following Definition 1.4.

The essential property of these ideals is summarized in the next result, which was established in [8].

Theorem 1.7. For any graded system 𝔞_{•}, any index m ∈ Φ, and any natural number ℓ ∈ N one has inclusions:

(16)  𝔞_{m}^{ℓ} ⊆ 𝔞_{mℓ} ⊆ 𝔧_{m}^{ℓ}.

Sketch of proof. The first inclusion is definitional. For the second, note first that 𝔞_{mℓ} ⊆ 𝔧_{mℓ}: this is easily checked using (10) and (13). So it is enough to show that 𝔧_{mℓ} ⊆ 𝔧_{m}^{ℓ}. For this, choose any p ≫ 0. Then using the subadditivity relation (11) one finds:

𝔧_{mℓ} = J (\frac{1}{p} · 𝔞_{mℓp})
= J (\frac{ℓ}{ℓp} · 𝔞_{mℓp})
⊆ J (\frac{1}{ℓp} · 𝔞_{mℓp})^{ℓ}
= 𝔧_{m}^{ℓ},

as required.

Remark 1.8. The second inclusion in (16) does not hold in general if one works with the “absolute” multiplier ideal J (𝔞_{m}) in place of 𝔧_{m}.

2. Abhyankar Valuations

Let ν be a rank one valuation on a function field K/k. There are two basic invariants of ν. The rational rank of ν is the dimension of the Q-vector space Q ⊗_{Z} Γ. The transcendence degree of ν is the transcendence degree of the residue field of the valuation ring R_{ν} over k. Equivalently, the transcendence degree is the maximal dimension of the center of ν over all models of K/k. The basic result relating these invariants is the Zariski-Abhyankar inequality.^{5}

Theorem 2.1 (The Zariski-Abhyankar Inequality). For any valuation on a function field K/k

(17)  trans.deg ν + rat.rank ν ≤ dim K/k.

Furthermore, if equality holds in (17), then the value group Γ is a finitely generated (free) Abelian group. Here, dim K/k refers to the transcendence degree of K over k, or equivalently, to the dimension of any complete model for K/k.

A valuation satisfying equality in (17) is called an Abhyankar valuation. Abhyankar valuations generalize the familiar example of divisorial valuations, that is, valuations given by order of vanishing along some divisor on a normal model of K/k. Note that divisorial valuations have rational rank one and transcendence degree n−1, where n is the dimension dim K/k.

5In the form stated, Theorem 2.1 is due to Zariski; see [27]. Abhyankar [1] later proved a more general version of 2.1 for valuations centered on any local Noetherian domain (the non-function field case).

Example 2.2. For the valuations in Example 1.4, the rational rank and transcendence degree are easily computed: for (i) and (ii), the rational rank is 1 and the transcendence degree is 1; for (iii), the rational rank is 2 and the transcendence degree is zero; and for (iv), the rational rank is 1 and the transcendence degree is zero.

We now state the main technical result of the present paper:

Theorem 2.3. Let ν be an Abhyankar valuation on a function field K/k of characteristic zero. Let R be the local ring of the center of ν on some smooth model of K, and let {𝔞_{m}}_{m∈Φ} be the graded family of ν-valuation ideals in R. Then there exists a non-zero element δ ∈ R such that

δ · 𝔧_{m} ⊆ 𝔞_{m}

for all m ∈ Φ, where {𝔧_{m}}_{m∈Φ} are the asymptotic multiplier ideals associated to 𝔞_{•}.

Remark 2.4. For any ν-valuation ideal 𝔞 in domain R and for an arbitrary ideal 𝔟, the colon ideal (𝔞 : 𝔟) is also a ν-valuation ideal of R; see [27, p. 342]. Thus (taking Φ to be the full value semigroup ν(R)), there is a function α : Φ ⟶ Φ such that

(𝔞_{m} : 𝔧_{m}) = 𝔞_{α(m)}.

Theorem 2.3 says that for Abhyankar valuations centered on a regular local domain essentially of finite type over a field of characteristic zero, this function is bounded above. In other words, the graded family of (𝔪-primary) valuation ideals (𝔞_{m} : 𝔧_{m}) has a minimal element.

In view of the preceding Remark, Theorem A from the Introduction follows immediately upon combining Theorems 1.7 and 2.3. Indeed, one simply takes e = ν(δ), where δ is the non-zero element of R whose existence is guaranteed by Theorem 2.3. Since δ𝔧_{m} ⊂ 𝔞_{m} for all m, clearly 𝔧_{m} ⊂ 𝔞_{m−e} for all m, and so 𝔞_{m}^{ℓ} ⊂ 𝔧_{m}^{ℓ} ⊂ 𝔞_{m−e}^{ℓ}.

Theorem 2.3 also implies that Izumi’s theorem holds for non-divisorial Abhyankar valuations in our setting:

Corollary 2.5 (Izumi’s Theorem for Abhyankar valuations). Let ν and w be rank one Abhyankar valuations on a function field K/k of characteristic zero and let (R, 𝔪) be any regular local ring essentially of finite type over k on which both ν and w are centered. Then there exists C > 0 such that

(18)  ν(x) ≥ Cw(x)

for all non-zero elements x ∈ R.

Proof. Without loss of generality, we may assume that the minimal value obtained by w on R is 1. Indeed, because R is Noetherian, w must achieve some minimal value on R \ 0; now we can simply scale the values of w and ν by this minimal value.

Now we claim that there exists a value p such that

(19)  𝔞_{mp} ⊂ 𝔟_{m}

for all m ∈ N, where {𝔞_{m}} (respectively {𝔟_{m}}) denote valuation ideals of ν (respectively, w). Indeed, note that 𝔟_{1} = 𝔪, and so 𝔪^{ℓ} ⊂ 𝔟_{ℓ} for all ℓ ∈ N. Thus to prove (19), it is enough to show that there exists p such that

(20)  𝔞_{pℓ} ⊂ 𝔪^{ℓ}

for all ℓ. (In other words, we are reduced to the case where w is the 𝔪-adic valuation on R.) By Theorem 1.7, we see that (20) follows immediately provided that some 𝔧_{p} ⊂ 𝔪. But if all 𝔧_{p} are trivial, we have δ ∈ ∩𝔞_{m} = (0), contradicting Theorem 2.3.

Finally, (18) follows as in [11, Lemma 1.4] by setting C = 2p − 1 (and enlarging p if necessary so that p ≥ 2). Indeed, suppose on the contrary that there is some x ∈ R such that ν(x) > Cw(x). Set m = w(x) + 1. Then x ∈ 𝔞_{mp}, but x ∉ 𝔟_{m}.

Remark 2.6. Theorems A and 2.3 — and also the Izumi-type statement of Corollary 2.5 — can fail for non-Abhyankar valuations. In fact, consider the valuation ν given by order of vanishing along the exponential curve y = e^{x} − 1 (Example 1.4.iv). Here 𝔞_{p} ⊆ R = C[x, y]_{(x,y)} has colength p. So there cannot exist a non-trivial ideal 𝔧 ⊆ R having the property that 𝔞_{mℓ} ⊆ 𝔧^{ℓ} for fixed m and ℓ ≫ 0, since the colength of 𝔧^{ℓ} would grow quadratically in ℓ. Therefore the inclusion (4) in Theorem A can only hold with 𝔧_{m} = R for all m. On the other hand, ∩_{m}𝔞_{m} = (0), so there cannot exist a fixed non-zero element δ ∈ R with δ · 𝔧_{m} ⊆ 𝔞_{m} for all m. However for an arbitrary valuation ν, it is possible that the colon ideals 𝔡_{m} = (𝔞_{m} : 𝔧_{m}) “grow slowly”: see Remark 3.14 for a precise statement.

We now prove Theorem 2.3. The outline is this: Lemma 2.7 below guarantees that it is enough to find δ after blowing up, but after a suitable blow up, any Abhyankar valuation is “essentially monomial” by Proposition 2.8, where a direct computation can be carried out.

Lemma 2.7. Let π : X ⟶ Y be a proper birational map of smooth varieties over k. Assume that there exists an ideal 𝔡′ ⊆ O_{X} such that 𝔡′ · 𝔧_{m}(X) ⊂ 𝔞_{m}(X). Then there exists an ideal 𝔡 ⊆ O_{Y} such that 𝔡 · 𝔧_{m}(Y) ⊆ 𝔞_{m}(Y).

Proof. Let 𝔡 = π_{∗}(𝔡′ω_{X/Y}^{−1} ). Note that both 𝔡′ and ω_{X/Y}^{−1} are ideals of O_{X}, so 𝔡 is an ideal of O_{Y}. Now

𝔡 · 𝔧_{m}(Y) = π_{∗}(𝔡′ω_{X/Y}^{−1} ) · π_{∗}(J ( X, \frac{1}{p} · 𝔞_{mp}(Y) · O_{X} ) ⊗ ω_{X/Y} )

where p is sufficiently large. Here we have used the definition of the asymptotic multiplier ideal together with the transformation rule (12) for multiplier ideals under proper birational morphisms. Therefore

𝔡 · 𝔧_{m}(Y) ⊆ π_{∗}(𝔡′ω_{X/Y}^{−1} · J (X, \frac{1}{p} · 𝔞_{mp}(Y)O_{X}) ⊗ ω_{X/Y} )

= π_{∗}(𝔡′ · J (X, \frac{1}{p} · 𝔞_{mp}(Y) · O_{X})).

Because 𝔞_{mp}(Y)O_{X} ⊂ 𝔞_{mp}(X) for all m and p, the corresponding inclusion holds for the multiplier ideals. Note that the multiplier ideal appearing on the right is that associated to the pull-back of a valuation ideal on Y rather than the corresponding valuation ideal on X. However 𝔞_{mp}(Y) · O_{X} ⊆ 𝔞_{mp}(X) and consequently for p ≫ 0:

𝔡′ · J (X, \frac{1}{p}𝔞_{mp}(Y) · O_{X}) ⊆ 𝔡′ · J (X, \frac{1}{p}𝔞_{mp}(X)) = 𝔡′ · 𝔧_{m}(X).

But 𝔡′ · 𝔧_{m}(X) ⊆ 𝔞_{m}(X), and putting these inclusions together we find that

𝔡 · 𝔧_{m}(Y) ⊆ π_{∗}(𝔡′ · 𝔧_{m}(X)) ⊆ π_{∗}(𝔞_{m}(X)) = 𝔞_{m}(Y)

for all m, as required.

The next Proposition is probably well-known, at least for valuations of transcendence degree zero. We learned that case from Dale Cutkosky.

Proposition 2.8. Let ν be a rank one Abhyankar valuation on a function field K/k of characteristic zero. Given any model Y of K/k, there exists a smooth model X dominating Y and a regular system of parameters x_{1}, …, x_{r} for the local ring of ν on X such that ν(x_{1}), …, ν(x_{r}) freely generate the value group Γ.

Proof. Let r denote the rational rank of ν, so by Theorem 2.1, Γ ≅ Z^{r}. Fix f_{1}, …, f_{r} in the field K, whose values generate Γ. By replacing f_{i} by \frac{1}{f_{i}} if necessary, we can assume all v(f_{i}) > 0.

We can write each f_{i} as a fraction \frac{a_{i}}{b_{i}}, where the a_{i} and b_{i} are regular on some neighborhood of the center of ν on Y. By blowing up the ideals (a_{i}, b_{i}), we can make the fractions \frac{a_{i}}{b_{i}} regular on some neighborhood of the center. By blowing up further if necessary, we can assume that the dimension of the center is the transcendence degree of ν— which means its codimension equals the rational rank in the presence of the Abhyankar hypothesis. So we have created a model Y′ dominating Y where the elements f_{i} are regular on a neighborhood of the center of ν, and where the codimension of the center is exactly r, the rational rank of Γ.

Now we use embedded resolution of singularities to resolve the hypersurface defined by the product f_{1}f_{2} … f_{r} in a neighborhood of the center on Y′. This produces for us a smooth model X dominating Y′ such that the pullback of the hypersurface to this model has simple normal crossing support. In particular, for any closed point x of X, we have

f_{1}f_{2} … f_{r} = u x_{1}^{a_{1}}x_{2}^{a_{2}} … x_{N}^{a_{N}} ,

where x_{1}, …, x_{N} is a regular system of parameters at x, the exponents a_{i} are natural numbers, and u is a regular function invertible in a neighborhood of x. Because the local rings of X are unique factorizations domains, for each f_{i} we have

f_{i} = u_{i} x_{1}^{a_{i1}}x_{2}^{a_{i2}} … x_{N}^{a_{iN}}

for some a_{ij} ∈ N and some unit u_{i}.

In particular, choosing the point x to be in the center W of ν on X, then the elements u_{i} are also units in the local ring O_{W,X}. Because units in O_{W,X} have value zero, we see that

ν(f_{i}) = ∑_{j=1}^{N} a_{ij}ν(x_{j}).

So clearly, the elements ν(x_{j}) generate Γ.

We claim that exactly r of the elements x_{j} have non-zero value. Indeed, if fewer have non-zero value, then the rank of Γ can not be r. But if more have non-zero value, then there are at least r + 1 of the parameters x_{1}, …, x_{r+1} contained in the defining ideal of the center W. This would force W to have codimension greater than r, a contradiction.

Relabeling so that the parameters x_{1}, …, x_{r} are those with positive value, note finally that the images of these elements generate the maximal ideal in the local ring of X along W. Indeed, this maximal ideal is generated by the image of the defining ideal I_{W} of W, and we have already remarked that (x_{1}, …, x_{r}) ⊂ I_{W}. But since x_{1}, …, x_{r} are part of regular sequence of parameters in a neighborhood of W, they must generate the maximal ideal after localizing at I_{W}. Thus the proposition is proved: the elements x_{1}, …, x_{r} of K are a regular system of parameters for the local ring O_{W,X} and the values ν(x_{1}), …, ν(x_{r}) generate Γ.

We can now finish the proof of the main theorem.

Proof of Theorem 2.3. By Lemma 2.7 and Proposition 2.8, we can assume that we are in the following situation. The variety X is smooth, and the center W of ν on X is of codimension r equal to the rational rank of ν; furthermore, the local ring R of X along W has regular system of parameters x_{1}, …, x_{r} whose values generate the value group Γ. We wish to prove that there exists δ ⊂ O_{X} such that δ𝔧_{m} ⊂ 𝔞_{m}. Because 𝔞_{m} is primary to I_{W} (see Proposition 1.3), it is enough to check this after localizing along the defining ideal of W, so we consider the graded family of valuation ideals {𝔞_{•}} in the local ring (R, 𝔪.

Because the values of the parameters x_{1}, …, x_{r} are all Z-independent, the ideals 𝔞_{m} are generated by ’monomials’ in x_{1}, …, x_{r}. Indeed, fix any m ∈ Φ. By (1.3), some power of the maximal ideal of R, say 𝔪^{t}, is contained in 𝔞_{m}. Now consider an arbitrary element f of R (not already in 𝔪^{t}). Modulo 𝔪^{t}, f can be written as a sum of monomials in the regular system of parameters x_{1}, …, x_{r} with unit coefficients. Because the values of the x_{i} are independent, each of these monomials has a distinct value, and so the value of f is equal to the value of the unique smallest value monomial in this sum (Cf (7)). So each of the monomial “terms” of f are in 𝔞_{m}, and 𝔞_{m} is generated by monomials in the regular system of parameters for R.

We now claim that this setup is sufficiently close to the standard monomial case so as to be able to apply the computation derived in [10] for the multiplier ideal of a monomial ideal in a polynomial ring. Roughly this reason is this: the ring R is étale over a polynomial ring and the computation of multiplier ideals commutes with étale extension. We justify this carefully in the next paragraph.

Think of the parameters x_{1}, …, x_{r} as local sections of O_{X} and extend them to a full set of regular parameters in some affine neighborhood U of W on X. The inclusion

(21)  k[x_{1}, …, x_{r}, x_{r+1}, …, x_{n}] ↪ O_{X}(U)

induces a natural map

(22)  U ⟶ A^{n}

consisting of an open immersion followed by a finite map. Localizing at the prime ideal of W and its corresponding contraction to the polynomial ring, we have an inclusion

(23)  A = k[x_{1}, …, x_{r}, x_{r+1}, …, x_{n}]_{(x_{1},...,x_{r})} ↪ O_{W,X} = R.

Our claim above that 𝔞_{m} is a monomial ideal is tantamount to saying that 𝔞_{m} is the expansion of a monomial ideal 𝔞′_{m} ⊂ k[x_{1}, …, x_{n}]. The monomial ideal 𝔞′_{m} is itself a valuation ideal for the valuation on k(x_{1}, …, x_{n}) obtained by restricting ν to this subfield. Because the maximal ideal of A expands to the maximal ideal of R, the map of rings (23) is étale, which is to say, the morphism (22) is étale is a neighborhood of W [19]. Thus replacing U by a possibly smaller open neighborhood, we can assume the morphism

U ⟶ A^{n}

consists of compositions of open immersions with a finite étale map. But the computation of multiplier ideals commutes with pullback under both open immersions (obvious) and finite étale maps (straightforward; see [17], §5.4). So

J (A^{n}, c · 𝔞′_{m}) · O_{U} = J (U, c · 𝔞′_{m}O_{U}),

and passing to the local ring at W (which after all, amounts to taking a limit of pullbacks to smaller and smaller affine neighborhoods of W), we see that

J (A^{n}, c · 𝔞′_{m})R = J (R, c · 𝔞′_{m}R) = J (R, c · 𝔞_{m}).

So to compute the multiplier ideal of 𝔞_{m} in the local ring R, it is sufficient to compute the multiplier ideal of the monomial ideal 𝔞′_{m} in the polynomial ring k[x_{1}, …, x_{n}] and expand to R.

We recall the formula for the multiplier ideal of a monomial ideal from [10]. Let 𝔞 ⊂ k[x_{1}, …, x_{n}] be an ideal generated by monomials and let L denote its lattice of exponents:

L = {(a_{1}, …, a_{n}) | x_{1}^{a_{1}} … x_{n}^{a_{n}} ∈ 𝔞}.

Then the multiplier ideal J (c · 𝔞) is the ideal of the polynomial ring generated by those monomials x_{1}^{b_{1}} … x_{n}^{b_{n}} satisfying

(b_{1}, …, b_{n}) + (1, …, 1) ∈ {hull(cL)}^{int},

where {hull(cL)}^{int} denotes the interior of the convex hull of the lattice L scaled by the real number c.

In our case, the monomial ideals 𝔞′_{m} are generated by

{x_{1}^{a_{1}} … x_{n}^{a_{n}} | ∑ a_{i}ν(x_{i}) ≥ m},

so the multiplier ideals are given by

J (c · 𝔞′_{m}) = ({x_{1}^{b_{1}} … x_{n}^{b_{n}} | ∑(b_{i} + 1)ν(x_{i}) > cm}) .

In particular, for any real positive p,

J (c · 𝔞′_{m}) = J (\frac{c}{p} · 𝔞′_{mp})

so that the asymptotic multiplier ideals 𝔧′_{m} of the graded family {𝔞′_{m}} in the polynomial ring satisfy

𝔧′_{m} = J (𝔞′_{m}).

Expanding to R, we see that the asymptotic multiplier ideals in R are given by

𝔧_{m}(𝔞_{•}) = ({x_{1}^{b_{1}} … x_{n}^{b_{n}} | ∑(b_{i} + 1)ν(x_{i}) > m}) .

Finally, using this description of the asymptotic multiplier ideals, we observe that the element δ = x_{1} … x_{r} in R satisfies the condition that

δ𝔧_{m} ⊆ 𝔞_{m}

and Theorem 2.3 is proved.

3. The Volume of a Graded Family of 𝔪-primary ideals

In this section we introduce the volume of a graded family of 𝔪-primary ideals, and compare it to the multiplicities of the individual ideals of the graded system. In particular, we deduce Corollary C from the Introduction as a special case of a general phenomenon.

Definition 3.1. Let 𝔞_{•} = {𝔞_{m}}_{m∈Φ} be a graded family of 𝔪-primary ideals in a local Noetherian ring (R, 𝔪) of dimension n. The volume of 𝔞_{•} is the real number

(24)  vol(𝔞_{•}) = lim sup_{m∈Φ} \frac{length(R/𝔞_{m})}{m^{n}/n!}.

Remark 3.2. The volume of a graded system is the local analogue of the volume of a big divisor D on a projective variety X of dimension n, which is defined to be

vol(D) = lim sup_{m} \frac{h^{0}(X, O_{X}(mD))}{m^{n}/n!}.

When X is smooth and D is ample, this coincides up to constants with the volume of X determined by any Kähler form representing c_{1}(O_{X}(D)), which explains the terminology.

While Definition 3.1 works well for graded systems indexed by the natural numbers N, it does not have very good behavior for arbitrary graded systems {𝔞_{m}} indexed by more general semigroups Φ ⊆ R^{≥}. For example, suppose that Φ = N + N\sqrt{2}, and put

𝔞_{j+k\sqrt{2}} = 𝔟^{j} · 𝔠^{k}

for some fixed ideals 𝔟, 𝔠 ⊆ R. Then the volumes of the two N-graded subsystems {𝔞_{j}} and \{𝔞_{k\sqrt{2}}\} will not in general coincide. In order to avoid this sort of pathology, we will henceforth adopt the following Convention 3.3. For the remainder of this section, we deal with the graded families arising from ideal filtrations; that is, we work with graded families {𝔞_{m}}_{m∈Φ} which satisfy the additional condition

(25)  𝔞_{m} ⊆ 𝔞_{m′} for any two indices m, m′ ∈ Φ with m ≥ m′.

Of course (25) is automatic for the graded families arising from valuations.

Remark 3.4. The volume of a graded family of 𝔪-primary ideals satisfying (25) is finite. Indeed, fix any m ∈ Φ. Since 𝔞_{m}^{ℓ} ⊂ 𝔞_{mℓ} for every positive integer ℓ, we see that

\frac{length(R/𝔞_{m}^{ℓ})}{(mℓ)^{n}} ≥ \frac{length(R/𝔞_{mℓ})}{(mℓ)^{n}}

for all ℓ ∈ N. So taking the limit as ℓ gets large, Lemma 3.8 below ensures that the volume of {𝔞_{•}} is bounded above by the rational number e(𝔞_{m})/m^{n}, where e(𝔞_{m}) denotes the multiplicity of the ideal 𝔞_{m}.

Example 3.5. (i). The volume of the trivial graded family {𝔞^{m}} of powers of a fixed ideal 𝔞 is equal to the multiplicity of the ideal. Likewise, the volume of \{\overline{𝔞^{m}}\} of integral closures of a fixed power of an ideal 𝔞 is also the multiplicity of 𝔞.

(ii). Let v be the valuation on k(x, y) given by ”order of vanishing at the origin” in Example 1.4(i). As we have seen, the valuation ideals of ν on k[x, y] are given by 𝔞_{m} = (x, y)^{m}. In this case, the volume is the multiplicity of the maximal ideal (x, y) in k[x, y], which is one.

(iii). Let v be the monomial valuation of Example 1.4(iii). Then each 𝔞_{m} ⊂ k[x, y] is generated by the monomials x^{a}y^{b}, where a + πb ≥ m. So the length of k[x, y]/𝔞_{m} is equal to the number of integer points in the first quadrant of the Cartesian plane inside the triangle bounded by the a-axis, the b-axis, and the line a + πb = m. The area of this triangle is roughly \frac{1}{2π}m^{2}. Taking the limit, the volume of ν on Speck[x, y] is \frac{1}{π}. An evident modification of this shows that any positive real number can occur as the volume of a graded family of ideals in k[x, y].

(iv). Let ν be the arc valuation of Example 1.4(iv). Then the quotients R/q_{m} are spanned by the residues of 1, x, x^{2}, …, x^{m−1}, so the length of R/q_{m} is m and the volume of ν is zero. However, the 1-volume is 1; Cf Remarks 3.7 and 3.17.

Remark 3.6. The volume of the family of ideals associated to a divisorial valuation was considered by Cutkosky and Srinivas in [3]. In the two-dimensional case, they show that this invariant is always a rational number: this essentially reflects the existence of Zariski decompositions. However Kuronya [15] gives an example of a four dimensional divisorial valuation with irrational volume. His construction makes use of Cutkosky’s curves in P^{3} having irrational Castelnuovo-Mumford regularity [2].

Remark 3.7. It is also possible to define the p-volume of a graded system 𝔞_{•} for any p ≤ n = dim R as

lim sup_{m∈Φ} \frac{length(R/𝔞_{m})}{m^{p}/p!}.

In the current paper, we will not pursue this further. See, however, Remark 3.17.

Lemma 3.8. Let {𝔞_{t}}_{t∈Φ} be a graded system of 𝔪-primary ideals in a Noetherian local ring (R, 𝔪) of dimension n satisfying (25). Then for any fixed positive m ∈ Φ,

lim sup_{t∈Φ} \frac{length(R/𝔞_{t})}{t^{n}} = lim sup_{ℓ∈N} \frac{length(R/𝔞_{mℓ})}{(mℓ)^{n}}.

In particular, the volume of the m-th Veronese graded subsystem {𝔞_{mℓ}}_{ℓ} is given by

vol({𝔞_{mℓ}}_{ℓ∈N}) = m^{n} vol({𝔞_{t}}_{t∈Φ}).

Remark 3.9. It follows from Lemma 3.8 that volume can be defined as

lim sup_{ℓ∈N} \frac{length(R/𝔞_{ℓm})}{(ℓm)^{n}/n!},

where m is any fixed non-zero real number. In particular, with the convention that 1 ∈ Φ, the volume is

lim sup_{m∈N} \frac{length(R/𝔞_{m})}{m^{n}/n!}.

Proof of Lemma 3.8. For each t ∈ Φ, we have

m · [\frac{t}{m}] ≤ t < m · [\frac{t}{m}] + m,

where [\frac{t}{m}] denotes, as usual, the greatest integer less than or equal to the real number \frac{t}{m}. Setting ℓ = [\frac{t}{m}], we have thanks to (25):

𝔞_{(ℓ+1)m} ⊂ 𝔞_{t} ⊆ 𝔞_{ℓm},

so that

\frac{length(R/𝔞_{(ℓ+1)m})}{t^{n}} ≥ \frac{length(R/𝔞_{t})}{t^{n}} ≥ \frac{length(R/𝔞_{ℓm})}{t^{n}}.

Since lim_{t→∞} \frac{t}{[ \frac{t}{m}]m} = 1, one has

lim sup_{ℓ→∞} \frac{length(R/𝔞_{ℓm})}{(ℓm)^{n}} = lim sup_{t→∞} \frac{length(R/𝔞_{ℓm})}{t^{n}},

and likewise with ℓ replaced by ℓ + 1. Thus

lim sup_{ℓ→∞} \frac{length(R/𝔞_{(ℓ+1)m})}{((ℓ + 1)m)^{n}} ≥ lim sup_{t→∞} \frac{length(R/𝔞_{t})}{t^{n}} ≥ lim sup_{ℓ→∞} \frac{length(R/𝔞_{ℓm})}{(ℓm)^{n}}.

Since the expression on the left here is equal to the expression on the right, the lemma is proved.

The next proposition shows that from the point of view of multiplier ideals, graded families with zero volume are trivial.

Proposition 3.10. Let 𝔞_{•} be a graded family of 𝔪-primary ideals in a local ring (R, 𝔪) essentially of finite type over a field of characteristic zero. Assume that 𝔞_{•} satisfies (25). If {𝔞_{•}} has volume zero, then each of its asymptotic multiplier ideals 𝔧_{m}(𝔞_{•}) is the unit ideal.

Proof. Fix any m ∈ Φ. Then for all ℓ ∈ N, we have

𝔞_{mℓ} ⊆ 𝔧_{m}^{ℓ},

whence e(𝔧_{m}) ≥ vol(𝔞_{•}) thanks to the previous Lemma. The assertion follows.

The following proposition, combined with the results of the previous section, proves Corollary C from the introduction.

Proposition 3.11. Let (R, 𝔪) be a regular local ring of dimension n, essentially of finite type over a field of characteristic zero. Let {𝔞_{m}}_{m∈Φ} be a graded family of 𝔪-primary ideals of R satisfying (25), and let {𝔧_{m}}_{m} be the associated sequence of asymptotic multiplier ideals. Assume that there is a fixed non-zero element δ ∈ R such that

(26)  δ · 𝔧_{m} ⊆ 𝔞_{m} for all m ∈ Φ.

Then

vol(𝔞_{•}) = lim sup_{m→∞} \frac{e(𝔞_{m})}{m^{n}} = lim sup_{m→∞} \frac{e(𝔧_{m})}{m^{n}},

where e(𝔞) denotes the multiplicity of the ideal 𝔞 in the local ring R.

(As in Lemma 3.8, the limits can be taken over all m ∈ Φ or just over all positive integer multiples of a fixed element in Φ. For graded systems indexed by the natural numbers N, one does not need to assume the filtration condition (25).)

Proof. Given any index m ∈ Φ, set 𝔡_{m} = (𝔞_{m} : 𝔧_{m}). This is an 𝔪-primary ideal, and since all the 𝔡_{m} contain the fixed element δ one verifies that

lim sup_{m→∞} \frac{e(𝔡_{m})}{m^{n}} = 0.

Indeed, since 𝔞_{1}^{m} ⊂ 𝔞_{m} ⊂ 𝔡_{m}, we see that (𝔞_{1}^{m} +δ) ⊂ 𝔡_{m} for all m. Thus e(𝔡_{m}) ≤ e(\overline{𝔞_{1}}^{m}) = m^{n−1}e(\overline{𝔞_{1}}), where \overline{𝔞_{1}} is the image of the ideal 𝔞_{1} in the (n − 1)-dimensional ring R/(δ).

Now fix a large index m ∈ Φ. Then for all ℓ we have

(𝔡_{m}𝔧_{m})^{ℓ} ⊆ 𝔞_{m}^{ℓ} ⊆ 𝔞_{mℓ} ⊆ 𝔧_{mℓ} ⊆ 𝔧_{m}^{ℓ},

and so

\frac{length(R/𝔧_{m}^{ℓ})}{ℓ^{n}} ≤ \frac{length(R/𝔞_{mℓ})}{ℓ^{n}} ≤ \frac{length(R/𝔞_{m}^{ℓ})}{ℓ^{n}} ≤ \frac{length(R/(𝔡_{m}𝔧_{m})^{ℓ})}{ℓ^{n}}

for all ℓ. Taking the limit as ℓ goes to infinity, we find that

(27)  e(𝔧_{m}) ≤ vol({𝔞_{mℓ}}_{ℓ}) ≤ e(𝔞_{m}) ≤ e(𝔡_{m}𝔧_{m}),

where {𝔞_{mℓ}}_{ℓ} is the m^{th} Veronese subgraded sequence of {𝔞_{•}}.

But now note that dividing by m^{n}, the expressions on the left and right here (namely \frac{e(𝔧_{m})}{m^{n}} and \frac{e(𝔡_{m}𝔧_{m})}{m^{n}}) have the same limit superior as m gets large. In fact, Teissier’s Minkowski Inequality [25, p. 39] implies that

e(𝔡_{m}𝔧_{m})^{1/n} ≤ e(𝔡_{m})^{1/n} + e(𝔧_{m})^{1/n}

for all m, whence

\frac{e(𝔡_{m}𝔧_{m})^{1/n}}{m} ≤ \frac{e(𝔡_{m})^{1/n}}{m} + \frac{e(𝔧_{m})^{1/n}}{m}

for all m. But lim_{m→∞} \frac{e(𝔡_{m})}{m^{n}} = 0, and so

lim sup_{m→∞} \frac{e(𝔡_{m}𝔧_{m})}{m^{n}} ≤ lim sup_{m→∞} \frac{e(𝔧_{m})}{m^{n}}.

On the other hand, since e(𝔧_{m}) ≤ e(δ_{m}𝔧_{m}), the reverse inequality always holds. Finally, using Lemma 3.8, we conclude from (27) that

lim sup_{m→∞} \frac{e(𝔧_{m})}{m^{n}} = vol(𝔞_{•}) = lim sup_{m→∞} \frac{e(𝔞_{m})}{m^{n}},

as claimed.

Remark 3.12. The proof shows also that the “volume” of {𝔧_{m}} is equal to the volume of the graded system {𝔞_{m}} (even though {𝔧_{m}} itself is not a graded system).

Remark 3.13. Another example of a graded family satisfying (26) is given by the base loci of the linear series of a big divisor. Specifically, fix a big divisor L on a smooth projective variety X, and let 𝔟′_{m} ⊂ O_{X} be the base ideal of the linear system |mL|. The components of the base locus stabilize as m → ∞, so choose one component and localize along it to get a graded family of ideals 𝔟_{m} in the local ring along the generic point of the component. One can show that there exists D such that O_{X}(−D) · 𝔧_{m} ⊂ 𝔟_{m} for all m (see [16], Chapter 10). So the conclusion of Proposition 3.11 holds for 𝔟_{•}.

Remark 3.14. The conclusion of Proposition 3.11 holds under the weaker assumption that there is a family of non-zero 𝔪-primary ideals 𝔡_{m}, with lim sup \frac{e(𝔡_{m})}{m^{n}} = 0, such that 𝔡_{m} · 𝔧_{m} ⊆ 𝔞_{m}. As far as we know, it is possible that every graded system 𝔞_{m} has this property, namely that the sequence 𝔞_{m} is “tightly bound” to the sequence 𝔧_{m} in the sense that lim sup \frac{e(a_{m}:j_{m})}{m^{n}} tends to zero as m goes to infinity. In particular, Proposition 3.11 may hold for a completely arbitrary graded system of 𝔪-primary ideals, so in particular, for an arbitrary valuation. Since we posed this question in an earlier version of this manuscript, Mircea Mustata has shed some light on the question of whether Proposition 3.11 holds more generally. Specifically, he shows that the volume of an arbitrary graded system of ideals is equal to the limit of the normalized multiplicities \frac{e(𝔞_{m})}{m^{n}} in general— that is, that the first equality in the conclusion of Proposition 3.11 holds without the assumption that there exists a δ such that δ𝔧_{m} ⊂ 𝔞_{m} for all m; see [20]. In particular, Mustata shows that Corollary C holds for any rank one valuation, Abhyankar or not. However, the relationship with the sequence of asymptotic multiplier ideals (as well as the second equality in Proposition 3.11) remains an open question in the general case.

Theorem 2.3 implies that the volume of an Abhyankar valuation on any model is positive, and we saw in Example 3.5 (iv) that the volume of a non-Abhyankar valuation can be zero. However, it is not the case that a valuation has positive volume if and only if it is Abhyankar, as the example below shows.

Example 3.15. In [27, pp. 102–104], there is a construction of a valuation on K = k(x, y) with value group an arbitrary additive subgroup of the rational numbers; see also [26, §10, Example 12]. Using this, we can construct a non-Abhyankar valuation of arbitrary volume (even normalizing so that ν(𝔪) = 1).

Let ν(y) = 1 and set ν(x) = β_{0} > 1, some rational number. Let c_{0} be the smallest positive integer such that c_{0}β_{0} ∈ Z. As in [27], there exists a valuation so that the polynomial

q_{1} = x^{c_{0}} − y^{β_{0}c_{0}}

has value β_{1} equal to any rational number greater than (or equal to) the ’expected value’ of β_{0}c_{0}. Let us choose this value so that β_{1} = \frac{d_{1}}{c_{1}} > β_{0}c_{0}, where d_{i} and c_{i} are relatively prime positive integers with c_{1} relatively prime to c_{0}.

This process can be repeated, so that we can construct a valuation having the values on x, y, and q_{1} as already specified, and having arbitrary rational value β_{2} ≥ β_{1}c_{1} on the polynomial

q_{2} = q_{1}^{c_{1}} + y^{β_{1}c_{1}}.

Again, we make this choice of β_{2} so that the smallest positive integer c_{2} such that c_{2}β_{2} ∈ Z is relatively prime each of the preceding c_{i}.

In this way, we inductively construct a sequence of polynomials q_{i}, rational numbers β_{i}, and positive integers c_{i} with the following properties:

q_{i+1} = q_{i}^{c_{i}} + y^{β_{i}c_{i}},

and

β_{i+1} > β_{i}c_{i},

where c_{i} is the smallest positive integer such that c_{i}β_{i} ∈ Z, and c_{i} is relatively prime to the product c_{0}c_{1} … c_{i−1}. As shown (even more generally) in [27], this uniquely defines a valuation ν on k(x, y), such that ν(q_{i}) = β_{i}. Indeed, using the Euclidean algorithm, and setting q_{−1} = y and q_{0} = x, every polynomial has a unique expression as a sum of ’monomials’ in the q_{i}:

q_{−1}^{a_{−1}}q_{0}^{a_{0}}q_{1}^{a_{1}} … q_{t}^{a_{t}}

where a_{−1} is arbitrary but the remaining exponents a_{i} satisfy a_{i} < c_{i}. Because each of these ’monomials’ has a distinct value, the valuation ideals on k[x, y] have the form

𝔞_{m} = ({q_{−1}^{a_{−1}}q_{0}^{a_{0}}q_{1}^{a_{1}} … q_{t}^{a_{t}} | ∑_{j=−1}^{t} β_{i}a_{i} ≥ m; a_{j} ≤ c_{j} − 1 for j ≥ 0}) .

In particular, the quotients k[x, y]/𝔞_{m} have vector space basis consisting of ’monomials’

q_{−1}^{a_{−1}}q_{0}^{a_{0}}q_{1}^{a_{1}} … q_{t}^{a_{t}} where ∑_{j=−1}^{t} β_{i}a_{i} < m and a_{j} ≤ c_{j} − 1 for j ≥ 0.

Although the number of products t here can be arbitrary, note that for each fixed m, we only need t up to the greatest integer such that β_{t} < m.

So computing the volume amounts to counting the number of monomials in this basis. A computation shows that the volume of ν is the limit of the following decreasing sequence of rational numbers

α_{i} = \frac{1}{β_{0}} (\frac{c_{0}β_{0}}{β_{1}} ) … (\frac{c_{i}β_{i}}{β_{i+1}} ) .

(Calculation hint: this is the limit of the subsequence \{2!\frac{λ(k[x,y]/𝔞_{m})}{m^{2}}\} indexed by m = c_{t}β_{t}, which bounds the volume below. On the other hand, we can approximate ν by a sequence of valuations ν_{i} which take the values β_{j} on q_{j} for j ≤ i, and the ’expected values’ on the remaining q_{i} (which is to say that the corresponding sequences of β_{j} and c_{j} satisfy β_{j} = c_{j−1}β_{j−1} for j > i). Then just observe that the volume of each ν_{i} bounds the volume of ν above and compute that the volume of ν_{i} is α_{i}.)

By choosing the values of β_{i} and c_{i} appropriately, one can make this limit be any nonnegative real number (note we’ve normalized so that β_{−1} = ν(y) = 1). For a completely explicit example, take c_{i} to be the standard enumeration of the prime numbers (c_{0} = 2, c_{1} = 3, …), and set β_{i+1} = c_{i}β_{i} + \frac{1}{c_{i+1}} . Then the volume turns out to be the reciprocal of the infinite sum

1 + \frac{1}{c_{0}} + \frac{1}{c_{0}c_{1}} + \frac{1}{c_{0}c_{1}c_{3}} + …,

which is approaches a real number between \frac{1}{2} and 1.

Discussion 3.16 (The Associated Graded Algebra of a Valuation). Fix a rank one valuation ν on a function field K/k centered a local domain (R, 𝔪) and let Φ be the corresponding value semigroup ν(R). The associated graded algebra of the valuation on R is

gr_{ν}R = ⨁_{m∈Φ} 𝔞_{m}/𝔞_{>m},

where 𝔞_{>m} denotes the valuation ideal {f | ν(f) > m}. It is easy to check that gr_{ν}R is a domain, but it is not finitely generated over R/𝔪 (its degree zero piece) in general. The transcendence degree of gr_{ν}R over R/𝔪 is equal to the rational rank of ν plus the transcendence degree of R_{ν}/𝔪_{ν} over R/𝔪, where 𝔪_{ν} denotes the maximal ideal of the valuation ring R_{ν}. If gr_{ν}R is finitely generated, therefore, its Krull dimension is equal to this sum. In this case, the associated graded ring has dimension equal to the dimension of R if and only if the valuation is Abhyankar.^{6}

When the associated graded ring of ν is finitely generated and N-graded, the volume of ν has a simple interpretation in terms of gr_{ν}R, namely it is equal to the Hilbert multiplicity of ν. Recall that if A = ⊕A_{m} is a finitely generated N-graded domain over a field A_{0} containing non-zero elements of every degree, then there exists a positive rational number e such that

(28)  dim A_{m} = e\frac{m^{n−1}}{(n − 1)!} + O(m^{n−2})

6Here we are using a slightly different, but equivalent (for function fields), form of Abhyankar’s inequality (17) which says that rat.rankν + trans.deg_{R/𝔪}(R_{ν}/𝔪_{ν}) ≤ R; this inequality differs from (17)

by addition of the same number, namely the transcendence degree of R/𝔪 over k, to both sides.

where n is the Krull dimension of A and dim denotes the dimension over the field A_{0}. This number e is called the Hilbert multiplicity of A. [To see this, note that for some r, the Veronese subalgebra A^{(r)} is generated in degree one (or r), so A decomposes as a direct sum of A^{(r)}-modules A_{(0)} ⊕ A_{(1)} ⊕ ⋯ ⊕ A_{(r−1)}, where A_{(i)} = ⊕_{j∈N}A_{jr+i}. Thus each of the modules A_{(i)} has some multiplicity e_{i} over the ring A^{(r)}. In general, these multiplicities can be different, but if A has elements of every degree, one shows that the e_{i} are all equal. Indeed, to see that e_{i} = e_{i′}, just take a non-zero element x ∈ A_{i−i′} and note that the cokernel of the injective map A_{(i′)} →^{x} A_{(i)} has dimension strictly less than the common dimension of A_{(i)} and A_{(i′)}. Thus A_{(i)} and A_{(i′)} necessarily have the same multiplicity, e. Thus the Hilbert polynomials of all the A_{(i)} have the same leading terms, leading to the formula for the dimension above (with e suitably normalized).]

Thus

length(R/𝔞_{m}) = ∑_{i=0}^{m−1} length(𝔞_{m}/𝔞_{m+1}) = \frac{e}{(n − 1)!} [∑_{i=0}^{m−1} i^{(n−1)} + O(m^{n−2})] ,

and since ∑_{i=0}^{m−1}[i^{(n−1)} + O(m^{n−2})] = \frac{m^{n}}{n} + O(m^{n−1}), we see that

lim_{m→∞} \frac{length(R/𝔞_{m})}{m^{n}/n!} = e.

Remark 3.17. When gr_{ν}R is finitely generated, the preceding discussion indicates that it is natural to consider the p-volume of ν on R, where p = rat.rk.ν +trans.deg._{R/𝔪}(R_{ν}/𝔪_{ν}); see Example 3.5 (iv). However, Example 3.15 indicates that the p-volume need not be finite in general, even when the graded ring has transcendence degree p.

4. Generalizations and Further Applications

Let D be an arbitrary effective divisor on a smooth variety X and let π : X ⟶ Y be a proper birational map to a smooth variety Y that collapses D to a point. The collection

(29)  {π_{∗}O_{X}(−mD)}_{m∈N}

forms a graded family of ideals in O_{Y}. Although this is not the graded family of a valuation, it can be handled by the methods developed here for families of valuation ideals because it is an intersection of graded families of valuation ideals.

Definition 4.1. Let {{a_{m}^{λ}}_{m∈Φ}}_{λ∈Λ} be an arbitrary collection of graded families, all indexed by the same semigroup Φ ⊂ R. The intersection graded family is defined by

⋂_{λ∈Λ} 𝔞_{•}^{λ} := { ⋂_{λ∈Λ} 𝔞_{m}^{λ}}_{m∈Φ} .

Note that if each 𝔞_{•}^{λ} satisfies (25), then so too does their intersection.

The asymptotic multiplier ideals of an intersection family satisfy

(30)  𝔧_{m} ( ⋂_{λ∈Λ} 𝔞_{•}^{λ}) ⊂ ⋂_{λ∈Λ} 𝔧_{m}({𝔞_{•}^{λ}})

Indeed, for each λ, we have {⋂_{λ∈Λ} 𝔞_{m}^{λ}} ⊂ 𝔞_{m}^{λ} for all m, so the corresponding inclusion holds for the asymptotic multiplier ideals.

Let S be any collection of rank one valuations centered on a domain R. For each m ∈ R_{≥0}, set

(31)  𝔞_{m} = {f ∈ R | ν(f) ≥ m for all ν ∈ S}.

The collection {𝔞_{m}}_{m∈R_{≥0}} forms a graded family of ideals in R indexed by the non-negative real numbers (or by considering only distinct such ideals, some subsemigroup of R.) This graded family is the intersection, over all ν ∈ S, of the graded families {q_{m}(ν)}_{m∈R_{≥0}} of ν-valuation ideals.

As a variant, one can also assign multiplicities to the valuations in S. Say for each ν in S, we assign some positive real number e_{ν}. Then for each m ∈ R, set

(32)  𝔞_{m} = \{f ∈ R | \frac{ν(f)}{e_{ν}} ≥ m for all ν ∈ S\}.

The collection {𝔞_{m}}_{m∈R} forms a graded family of ideals in R, indexed by (some subsemigroup of) the real numbers. For example, the graded family (29) above is of this form: If D = ∑ e_{i}D_{i} where the D_{i} are prime divisors, then the set S is the set of valuations ν_{i} given by order of vanishing along D_{i} and the multiplicities e_{ν_{i}} are the coefficients e_{i}. These graded families are also intersections: the graded family (32) is the intersection, over all ν in S, of the graded families {q_{e_{ν}m}(ν)}_{m} of the e_{ν}-th Veronese subfamily of the graded family of valuation ideals of ν. Alternatively, it can be interpreted as an intersection of graded families of valuation ideals: it is the intersection of the graded families of valuation ideals in the set S′, where the set S′ is obtained from S by replacing each valuation ν in S by the valuation \frac{1}{e_{ν}} ν. So this is really no more general than the intersection (31) discussed in the previous paragraph. Once again, condition (25) is satisfied by these families.

One can also define the product of two graded families {𝔞_{•}}_{m∈Φ} and {𝔟_{•}}_{m∈Φ} by

{𝔞_{m}𝔟_{m}}_{m∈Φ}.

Note that the asymptotic multiplier ideals of the product graded family satisfy

(33)  𝔧_{m}(𝔞_{•}𝔟_{•}) ⊆ 𝔧_{m}(𝔞_{•})𝔧_{m}(𝔟_{•}),

since, for all large p, we have the inclusion of ’usual’ multiplier ideals J (\frac{1}{p} · (𝔞_{mp}𝔟_{mp})) ⊂ J (\frac{1}{p} · 𝔞_{mp})J (\frac{1}{p} · 𝔟_{mp}) by subadditivity [6]. By induction, the product of any finite number of graded families indexed by the same semi-group is defined, and the same multiplicativity of the asymptotic multiplier ideals holds.

Our Main Theorem 2.3 and its ensuing corollaries can be extended to graded families arising as finite intersections or products of Abhyankar valuations. Explicitly:

Corollary 4.2. Let S be a finite collection of rank one Abhyankar valuations of a function field K/k of characteristic zero, all centered on some local ring R of a model of K/k. Let {𝔞_{•}} denote either the corresponding intersection or the corresponding product graded family of ideals. Then there exists a non-zero element δ ∈ R such that

δ 𝔧_{m} ⊂ 𝔞_{m}

for all m ∈ R, where {𝔧_{m}} is the associated sequence of asymptotic multiplier ideals. Furthermore, the conclusions of Theorem A and Corollary C hold for this graded family of ideals.

Proof. This is immediate from Theorem 2.3. For each ν ∈ S, let δ_{ν} be the non-zero element guaranteed by Theorem 2.3 and let δ be their product. The desired inclusion follows from (30) or from (33), respectively, for the intersection and the product case.

Example 4.3. For the graded system (29) of a divisor D, Corollary 4.2 immediately implies Corollary B from the Introduction. It also guarantees the existence of a non-zero δ such that

δ^{ℓ}(π_{∗}O_{X}(−mℓD)) ⊆ (π_{∗}O_{X}(−mD))^{ℓ}

ℓ, m ∈ N with m ≫ 0.

Finally, we point out the following Minkowski Inequality for intersections and products of graded families.

Corollary 4.4 (Minkowski Inequality). Let 𝔞_{•} and 𝔟_{•} be two graded families of 𝔪-primary ideals in a regular local ring essentially of finite type over a field of characteristic zero. Assume that the hypothesis of Proposition 3.11 holds for each of these systems. Then

vol(𝔞_{•} ⋂ 𝔟_{•})^{1/n} ≤ vol(𝔞_{•}𝔟_{•})^{1/n} ≤ vol(𝔞_{•})^{1/n} + vol(𝔟_{•})^{1/n}

Proof. Since 𝔞_{m}𝔟_{m} ⊆ 𝔞_{m} ⋂ 𝔟_{m} for all m, it is evident that vol(𝔞_{•} ⋂ 𝔟_{•}) ≤ vol(𝔞_{•}𝔟_{•}). So it is enough to prove the inequality for product families.

It follows from the subadditivity relation (33) that if the hypothesis of Proposition 3.11 holds for each of 𝔞_{•} and 𝔟_{•} then it holds also for the product system. This being said, the result follows from Proposition 3.11 and Teissier’s Minkowski inequality for multiplicities. Indeed,

vol(𝔞_{•}𝔟_{•})^{1/n} = [lim sup_{m} \frac{e(𝔞_{m}𝔟_{m})}{m^{n}}]^{1/n}

≤ lim sup_{m} \{[\frac{e(𝔞_{m})}{m^{n}}]^{1/n} + [\frac{e(𝔟_{m})}{m^{n}}]^{1/n}\}

≤ lim sup_{m} \{[\frac{e(𝔞_{m})}{m^{n}}]^{1/n}\} + lim sup_{m} \{[\frac{e(𝔟_{m})}{m^{n}}]^{1/n}\}

= vol(𝔞_{•})^{1/n} + vol(𝔟_{•})^{1/n},

as required.

Remark 4.5. Note that the hypothesis of Proposition 3.11 is satisfied in particular for graded families that are finite products or intersections of graded families of Abhyankar valuations. This is the content of Corollary 4.2. More generally, the product or intersection graded family of any finite collection of graded families that satisfy the hypothesis also satisfy the hypothesis. Indeed, we verified this in the proof of Corollary 4.4 for products using (33) and the same argument, using (30) instead of (33), works for intersections.

Example 4.6. As a special case of the Minkowski inequality, fix an 𝔪-primary ideal 𝔞 in a local ring R of a smooth complex variety and let ν_{1}, …, ν_{r} be the associated Rees valuations of 𝔞. (The definition of Rees valuations is recalled in the Introduction.) Set e_{i} = min_{f∈I} ν_{i}(f). Then the associated graded family

𝔞_{m} = \{f ∈ R | \frac{ν_{i}(f)}{e_{i}} ≥ m for i = 1, 2, …, r\}

is nothing more than the graded family of integral closures of powers of 𝔞 of Example 1.2 (ii). In particular, its volume is the multiplicity of 𝔞. This family is a finite intersection of graded families of Abhyankar valuations, so Corollary 4.4 can be applied. This produces the bound

e(𝔞)^{1/n} ≤ e_{1} vol(ν_{1})^{1/n} + ⋯ + e_{r} vol(ν_{r})^{1/n}.

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Department of Mathematics, University of Illinois at Chicago 851 South Morgan Street (M/C 249), Chicago, IL 60607-7045, USA

      E-mail address: ein@math.uic.edu

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

      E-mail address: rlaz@math.lsa.umich.edu

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

      E-mail address: kesmith@math.lsa.umich.edu