ANALYTIC SPREAD OF FILTRATIONS ON TWO-DIMENSIONAL NORMAL LOCAL RINGS

Abstract

In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.

1 Introduction

Divisorial filtrations on two-dimensional normal excellent local rings have excellent properties, as we show in this article.

1.1 Filtrations of powers of ideals and analytic spread

In this subsection, we give an outline of how the classical theory of the analytic spread of an ideal admits a simple geometric interpretation in the case of an ideal in a normal excellent local ring. The generalization of analytic spread to divisorial filtrations can then be seen as a natural extension of this theory.

Expositions of the theory of complete ideals, integral closure of ideals and their relation to valuation ideals, Rees valuations, analytic spread, and birational morphisms can be found, from different perspectives, in [23], [25], [33], and [36]. The book [33] and the article [25] contain references to original work in this subject. Concepts in this introduction which are not defined in this section or in these references can be found in §2 of this paper. A survey of recent work on symbolic algebras is given in [15]. A different notion of analytic spread for families of ideals is given in [16]. A recent paper exploring ideal theory in two-dimensional normal local domains using geometric methods is [31].

Let R be a normal excellent local ring with maximal ideal $m_R$ , and let I be an ideal in R. Let $\pi :X\rightarrow \mbox {Spec}(R)$ be projective and birational (so that $\pi $ is the blowup of an ideal) and such that X is normal and $I{\mathcal O}_X$ is an invertible sheaf. Let $I{\mathcal O}_X={\mathcal O}_X(-D)$ where D is an effective and anti-nef divisor (the intersection product $(D\cdot E)\le 0$ for all exceptional curves E of X). Then $\Gamma (X,{\mathcal O}_X(-nD))=\overline {I^n}$ , the integral closure of $I^n$ , for all $n\in {\mathbb N}$ . Write $D=a_1F_1+\cdots +a_sF_s$ where the $F_i$ are prime divisors. The local rings ${\mathcal O}_{X,F_i}$ are discrete (rank 1) valuation rings. Let $\nu _{F_i}$ be the associated valuations. We have that the integral closure of $I^n$ is

$$ \begin{align*}\overline{I^n}=\Gamma(X,{\mathcal O}_X(-nD))=I(\nu_{F_1})_{na_1}\cap \cdots\cap I(\nu_{F_s})_{na_s}, \end{align*} $$

where

$$ \begin{align*}I(\nu_{F_i})_b=\{f\in R\mid \nu_{F_i}(f)\ge b\} \end{align*} $$

are the valuation ideals in R associated with $\nu _{F_i}$ . The center of $\nu _{F_i}$ on R is the prime ideal $I(\nu _{F_i})_1$ . The Rees valuations of I are those $\nu _{F_j}$ such that $\overline {I^n}\neq \cap _{i\ne j}I(\nu _{F_i})_{na_i}$ . Let Y be the normalization of the blowup $B(I)$ of I, and let $I{\mathcal O}_Y={\mathcal O}_Y(-B)$ . Then $Y\rightarrow \mbox {Spec}(R)$ is projective (since R is universally Nagata). The divisor $-B$ is ample on Y and so the Rees valuations of I are exactly the prime components of B. By the universal property of blowing up, $\pi $ factors through $B(I)$ and since X is normal, $\pi $ factors through Y. Let $\phi :X\rightarrow Y$ be the induced morphism. Let F be a prime component of D, with associated valuation $\nu _F$ . Then $\nu _F$ is a Rees valuation of I if and only if $\phi $ does not contract F, in which case $\phi (F)=E$ is a prime component of B and we have that ${\mathcal O}_{X,F}={\mathcal O}_{Y,E}$ .

In the case that $\dim R=2$ , the prime divisor F is contracted by $\phi $ if and only if F is exceptional ( $\pi (F)=m_R$ ) and $(D\cdot F)=0$ . Thus, the Rees valuations of I are precisely the valuations associated with prime divisors F of X such that either $\nu _F$ has its center at a height 1 prime of R or F is exceptional for $\pi $ (the center of $\nu _F$ on R is $m_R$ ) and $(D\cdot F)<0$ .

Let us return to not having any restrictions on the dimension of R. We have an associated graded ring $R[It]=\sum _{n\ge 0}I^nt^n$ (the Rees algebra of I). The integral closure of $R[It]$ in $R[t]$ is the graded algebra $\overline {R[It]}=\sum _{n\ge 0}\overline {I^n}t^n$ , which is a finite extension of $R[It]$ (since R is universally Nagata). The blowup of I is $B(I)=\mbox {Proj}(R[It])$ , and $Y=\mbox {Proj}(\overline {R[It]})$ is the normalization of the blowup of I, which was introduced earlier. Let $\psi :B(I)\rightarrow \mbox {Spec}(R)$ be the projection.

The blowup $B(I)$ has the important subschemes

$$ \begin{align*}\psi^{-1}(V(I))=\mbox{Proj}(\mbox{gr}_I(R))\mbox{ and }\psi^{-1}(m_R)=\mbox{Proj}(R[It]/m_RR[It]). \end{align*} $$

The R-algebra $\mbox {gr}_I(R)=\sum _{n\ge 0}I^n/I^{n+1}t^n$ is the associated graded ring of I, and the R-algebra $R[It]/m_RR[It]$ is the fiber cone of I.

Since $\mbox {Proj}(R[It])\rightarrow \mbox {Spec}(R)$ and $\mbox {Proj}(\overline {R[It]})\rightarrow \mbox {Spec}(R)$ are birational, the dimensions of $\mbox {Proj}(R[It])$ and $\mbox {Proj}(\overline {R[It]})$ are the same as the dimension of R. Furthermore, since $\mbox {Proj}(\mbox {gr}_I(R))$ is a Cartier divisor on $\mbox {Proj}(R[It])$ , we have that $\dim (\mbox {Proj}(\mbox {gr}_I(R))=\dim R-1$ . Now, since $I\subset m_R$ , we have that $\mbox {Proj}(R[It]/m_RR[It])$ is a subscheme of $\mbox {Proj}(\mbox {gr}_I(R))$ , so we have $\dim (\mbox {Proj}(R[It]/m_RR[It]))\le \dim R-1$ .

Let $\psi _0:\mbox {Proj}(\mathrm {gr}_I(R))\rightarrow \mbox {Spec}(R/I)$ be the projective morphism induced by $\psi $ . Let P be a minimum prime of I. Then $\dim \psi _0^{-1}(P)=\dim R_P-1$ since $I_p$ is primary for the maximal ideal of $R_P$ . We have that $\dim \psi ^{-1}(m_R)=\dim \psi _0^{-1}(m_R)\ge \dim \psi _0^{-1}(P)$ by upper semicontinuity of fiber dimension [19, Cor. IV.13.1.5]. Thus,

$$ \begin{align*}\mbox{ht}(I)\le \dim \psi^{-1}(m_R)+1. \end{align*} $$

The analytic spread of I is defined to be

$$ \begin{align*}\ell(I)=\dim R[It]/m_RR[It]. \end{align*} $$

Since the dimension of the Proj of a graded ring is one less than the dimension of the ring, we have established in our case of normal excellent local rings the following theorems.

Theorem 1.1 ([33, Prop. 5.1.6 and Cor. 8.3.9]).

Let R be a Noetherian local ring, and let I be an ideal in R. Then

$$ \begin{align*}\mathrm{ht}(I)\le \ell(I)\le \dim \mathrm{gr}_I(R)=\dim R. \end{align*} $$

Theorem 1.2 ([33, Prop. 5.4.8]).

Let R be a Noetherian formally equidimensional local ring, and let I be an ideal in R. For every minimal prime ideal P of $\mathrm { gr}_I(R)$ , $\dim ({gr}_I(R)/P)=\dim R$ .

We return to the case that R is a normal excellent local ring of arbitrary dimension. We have that $\ell (I)=\dim R$ if and only if $\dim \psi ^{-1}(m_R)=\dim R-1$ . Since

$$ \begin{align*}Y=\mbox{Proj}(\overline{R[It]})\rightarrow B(I)=\mbox{Proj}(R[It]) \end{align*} $$

is finite, $\dim \psi ^{-1}(m_R)=\dim R-1$ if and only if there exists a prime divisor E on Y which contracts to $m_R$ ; that is, the center of $\nu _E$ on R is $m_R$ . Writing

$$ \begin{align*}I{\mathcal O}_Y={\mathcal O}_Y(-b_1F_1-\cdots - b_sF_s), \end{align*} $$

where $F_i$ are prime divisors and $b_i>0$ , we have that

$$ \begin{align*}\overline{I^n}=I(\nu_{F_1})_{nb_1}\cap \cdots \cap I(\nu_{F_s})_{nb_s}, \end{align*} $$

where $\nu _{F_i}$ is the discrete rank 1 valuation associated with the valuation ring ${\mathcal O}_{Y,F_i}$ . Since $-b_1F_1-\cdots -b_sF_s$ is ample on Y, we have that $\overline {I^n}\ne \cap _{i\ne j} I(\nu _{F_i})$ for all j and $n\gg 0$ (so that $\nu _{F_1},\ldots ,\nu _{F_s}$ are the Rees valuations of I). Thus, $\dim \psi ^{-1}(m_R)=\dim R-1$ holds if and only if $m_R\in \mbox {Ass}(R/\overline {I^n})$ for some n.

We have established the following theorem in our case of normal excellent local rings.

Theorem 1.3 ([27], [33, Th. 5.4.6]).

Let R be a formally equidimensional local ring, and let I be an ideal in R. Then $m_R\in \mbox {Ass}(R/\overline {I^n})$ for some n if and only if $\ell (I)=\dim (R)$ .

The assumption of being formally equidimensional is not required for the “if” direction of Theorem 1.3. (This is Burch’s theorem (see [6], [33, Prop. 5.4.7]).)

Let $k=R/m_R$ . Since $R[It]/m_RR[It]$ is a standard graded ring over k (finitely generated in degree 1), it has a Hilbert polynomial $P(n)$ which has degree $d=\ell (I)-1$ ; there exists a positive integer $n_0$ such that

(1) $$ \begin{align} \dim_k I^n/m_RI^n= P(n)\mbox{ for }n\ge n_0. \end{align} $$

As $\overline {R[It]}/m_R\overline {R[It]}$ is a finitely generated graded ring over k, there exists $e\in {\mathbb Z}_{>0}$ and polynomials $P_0,\ldots , P_{e-1}$ of degree $d=\ell (I)-1$ such that

(2) $$ \begin{align} \dim_k \overline{I^n}/m_R\overline{I^n}= P_i(n)\ \mbox{for} \ n\ge n_0 \ \mbox{where} \ i\equiv n \ \mbox{mod}\ e. \end{align} $$

1.2 Filtrations

Let $\mathcal I=\{I_n\}$ be a filtration on a local ring R. The Rees algebra of the filtration is $R[\mathcal I]=\oplus _{n\ge 0}I_n$ . Analogously to the case of ideals, we define the fiber cone of the filtration $\mathcal I$ to be $R[\mathcal I]/m_RR[\mathcal I]$ and the analytic spread of the filtration of $\mathcal I$ to be

(3) $$ \begin{align} \ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I]. \end{align} $$

We have that $\mathrm {ht}(I_n)=\mathrm {ht}(I_1)$ for all n [12, (7)], so it is natural to define $\mathrm {ht}(\mathcal I)=\mathrm {ht}(I_1)$ .

We always have [12, Lem. 3.6] that

$$ \begin{align*}\ell(\mathcal I)\le \dim R, \end{align*} $$

so the second inequality of Theorem 1.1 always holds. However, the first inequality of Theorem 1.1, $ \mbox {ht}(\mathcal I)\le \ell (\mathcal I), $ fails spectacularly, even attaining the condition that $\ell (\mathcal I)=0$ [12, Exams. 1.2, 6.1, and 6.6]. The last two of these examples are of symbolic algebras of space curves, which are divisorial filtrations. We give a further example where the inequality fails in Example 7.3 of this paper. Example 7.3 is of a symbolic algebra of an intersection of height 1 prime ideals in a two-dimensional excellent normal local ring. In the case that $\mathcal I$ is a Noetherian filtration ( $R[\mathcal I]$ is a finitely generated R-algebra), the lower bound $\mathrm {ht}(\mathcal I)\le \ell (\mathcal I)$ always holds [12, Prop. 3.7], so that the inequality of Theorem 1.1 for ideals continues to hold for Noetherian filtrations.

The condition that a filtration has analytic spread zero has a simple ideal theoretic interpretation (see [12, Lem. 3.8]). Suppose that $\mathcal I=\{I_n\}$ is a filtration in a local ring R. Then the analytic spread $\ell (\mathcal I)=0$ if and only if

$$ \begin{align*} \mbox{For all}\ n>0 \ \mathrm{and} \ f\in I_n, \mathrm{there \ exists} \ m>0 \ \mathrm{such \ that} \ f^m\in m_RI_{mn}. \end{align*} $$

1.3 Divisorial Filtrations

Let R be a local domain of dimension d with quotient field K. Let $\nu $ be a discrete valuation of K with valuation ring $V_{\nu }$ and maximal ideal $m_{\nu }$ . Suppose that $R\subset V_{\nu }$ . Then, for $n\in {\mathbb N}$ , define valuation ideals

$$ \begin{align*}I(\nu)_n=\{f\in R\mid \nu(f)\ge n\}=m_{\nu}^n\cap R. \end{align*} $$

A divisorial valuation of R (see [33, Def. 9.3.1]) is a valuation $\nu $ of K such that if $V_{\nu }$ is the valuation ring of $\nu $ with maximal ideal $m_{\nu }$ , then $R\subset V_{\nu }$ , and if $p=m_{\nu }\cap R$ , then $\mbox {trdeg}_{\kappa (p)}\kappa (\nu )=\mathrm {ht}(p)-1$ , where $\kappa (p)$ is the residue field of $R_{p}$ and $\kappa (\nu )$ is the residue field of $V_{\nu }$ . If $\nu $ is divisorial valuation of R such that $m_R= m_{\nu }\cap R$ , then $\nu $ is called an $m_R$ -valuation.

By [33, Th. 9.3.2], the valuation ring of every divisorial valuation $\nu $ is Noetherian, and hence is a discrete valuation. Suppose that R is an excellent local domain. Then a valuation $\nu $ of the quotient field K of R which is nonnegative on R is a divisorial valuation of R if and only if the valuation ring $V_{\nu }$ of $\nu $ is essentially of finite type over R (see [13, Lem. 5.1]).

In general, the filtration $\mathcal I(\nu )=\{I(\nu )_n\}$ is not Noetherian; that is, the graded R-algebra $\sum _{n\ge 0}I(\nu )_nt^n$ is not a finitely generated R-algebra. In a two-dimensional normal local ring R, the condition that the filtration of valuation ideals $\mathcal I(\nu )$ is Noetherian for all $m_R$ -valuations $\nu $ dominating R is the condition (N) of Muhly and Sakuma [29]. It is proved in [9] that a complete normal local ring of dimension 2 satisfies condition (N) if and only if its divisor class group is a torsion group.

An integral divisorial filtration of R (which we refer to as a divisorial filtration in this paper) is a filtration $\mathcal I=\{I_m\}$ such that there exist divisorial valuations $\nu _1,\ldots ,\nu _s$ and $a_1,\ldots ,a_s\in {\mathbb Z}_{\ge 0}$ such that for all $m\in {\mathbb N}$ ,

$$ \begin{align*}I_m=I(\nu_1)_{ma_1}\cap\cdots\cap I(\nu_s)_{ma_s}. \end{align*} $$

$\mathcal I$ is called an ${\mathbb R}$ -divisorial filtration if $a_1,\ldots ,a_s\in {\mathbb R}_{>0}$ and $\mathcal I$ is called a ${\mathbb Q}$ -divisorial filtration if $a_1,\ldots ,a_s\in {\mathbb Q}$ . If $a_i\in {\mathbb R}_{>0}$ , then

$$ \begin{align*}I(\nu_i)_{na_i}:=\{f\in R\mid \nu_i(f)\ge na_i\}=I(\nu_i)_{\lceil na_i\rceil}, \end{align*} $$

where $\lceil x\rceil $ is the roundup of a real number.

Given an ideal I in R, the filtration $\{\overline {I^n}\}$ is an example of a divisorial filtration of R. The filtration $\{\overline {I^n}\}$ is Noetherian if R is universally Nagata.

It is shown in [12, Th. 4.6] that the “if” statement of Theorem 1.3 is true for divisorial filtrations of a local domain R.

Theorem 1.4 [12, Th. 4.6].

Suppose that R is a local domain and $\mathcal I=\{I_n\}$ is a divisorial filtration on R such that $\ell (I)=\dim R$ . Then there exists a positive integer $n_0$ such that $m_R\in \mbox {Ass}(R/I_n)$ for all $n\ge n_0$ .

An interesting question is if the converse of Theorem 1.3 is also true for divisorial filtrations of a local ring R. We prove this for two-dimensional excellent normal local rings in this paper (Theorem 7.1, also stated in Theorem 1.5 of this introduction).

1.4 Divisorial filtrations on normal excellent local rings

Let R be a normal excellent local ring. Let $\mathcal I=\{I_m\}$ where

$$ \begin{align*}I_m=I(\nu_1)_{ma_1}\cap\cdots\cap I(\nu_s)_{ma_s} \end{align*} $$

for some divisorial valuations $\nu _1,\ldots ,\nu _s$ on R, be an ${\mathbb R}$ -divisorial filtration on a normal excellent local ring R, with $a_1,\ldots , a_s\in {\mathbb R}_{>0}$ . Then there exists a projective birational morphism $\phi :X\rightarrow \mbox {Spec}(R)$ such that there exist prime divisors $F_1, \ldots , F_s$ on X such that $V_{\nu _i}={\mathcal O}_{X,F_i}$ for $1\le i\le s$ . Let $D=a_1F_1+\cdots +a_sF_s$ , an effective ${\mathbb R}$ -divisor. Define $\lceil D\rceil =\lceil a_1\rceil F_1+\cdots +\lceil a_s\rceil F_s$ , an integral divisor. We have coherent sheaves ${\mathcal O}_X(-\lceil n D\rceil )$ on X such that

(4) $$ \begin{align} \Gamma(X,{\mathcal O}_X(-\lceil nD\rceil ))=I_n \end{align} $$

for $n\in {\mathbb N}$ . If X is nonsingular, then ${\mathcal O}_X(-\lceil nD\rceil )$ is invertible. The formula (4) is independent of choice of X. Furthermore, even on a particular X, there are generally many different choices of effective ${\mathbb R}$ -divisors G on X such that $\Gamma (X,{\mathcal O}_X(-\lceil nG\rceil ))=I_n$ for all $n\in {\mathbb N}$ . Any choice of a divisor G on such an X for which the formula $\Gamma (X,{\mathcal O}_X(-\lceil nG\rceil ))=I_n$ for all $n\in {\mathbb N}$ holds will be called a representation of the filtration $\mathcal I$ .

Given an ${\mathbb R}$ -divisor $D=a_1F_1+\cdots +a_sF_s$ on X, we have a divisorial filtration $\mathcal I(D)=\{I(D)_n\}$ where

$$ \begin{align*}I(D)_n=\Gamma(X,{\mathcal O}_X(-\lceil nD\rceil ))=I(\nu_1)_{\lceil na_1\rceil}\cap\cdots\cap I(\nu_s)_{\lceil na_s\rceil} =I(\nu_1)_{ma_1}\cap\cdots\cap I(\nu_s)_{ma_s}. \end{align*} $$

We write $R[D]=R[\mathcal I(D)]$ .

1.5 Summary of principal results in this paper

Let R be an excellent two-dimensional normal excellent local ring with maximal ideal  $m_R$ .

All possible analytic spreads $\ell (\mathcal I(D))=0,1,2$ can occur for ${\mathbb Q}$ -divisors D on R. An example where $\ell (\mathcal I(D))=0<\mathrm {ht}(\mathcal I(D))=1$ is given in Example 7.3. This example is of a symbolic filtration $\mathcal I(D)=\{Q_1^{(n)}\cap Q_2^{(n)}\cap Q_3^{(n)}\}$ where $Q_1,Q_2,Q_3$ are height 1 prime ideals in a two-dimensional normal excellent local ring R. In contrast, since the filtration $\mathcal I(D)$ is not Noetherian, we have (by [12, Cor. 1.9]) that for every $a\in {\mathbb Z}_{>0}$ , the analytic spread of the ideal $Q_1^{(a)}\cap Q_2^{(a)}\cap Q_3^{(a)}$ is $\ell (Q_1^{(a)}\cap Q_2^{(a)}\cap Q_3^{(a)})=2$ , the largest possible.

We prove that the conclusions of Theorem 1.3 hold for ${\mathbb Q}$ -divisorial filtrations on R in Theorem 7.1.

Theorem 1.5 (Theorem 7.1).

Let R be a two-dimensional normal excellent local ring. The following are equivalent for a ${\mathbb Q}$ -divisorial filtration $\mathcal I(D)$ on R.

1. (1) The analytic spread $\ell (\mathcal I(D))=\dim R[D]/m_RR[D]=2$ .

2. (2) $m_R\in \mbox {Ass}(R/I(nD))$ for some n.

3. (3) There exists $n_0\in {\mathbb Z}_{>0}$ such that $m_R\in \mbox {Ass}(R/I(nD))$ for all $n\ge n_0$ .

We generalize the formula on Hilbert functions of filtrations of powers of ideals in (1) and (2) to ${\mathbb Q}$ -divisorial filtrations on R in Theorem 8.1.

Theorem 1.6 (Theorem 8.1).

Suppose that R is a two-dimensional normal excellent local ring and $\mathcal I(D)$ is a ${\mathbb Q}$ -divisorial filtration on R. Then there exist a nonnegative rational number $\alpha $ and a bounded function $\sigma :{\mathbb N}\rightarrow {\mathbb Q}$ such that the length

$$ \begin{align*}\lambda_R(I(nD)/m_RI(nD))=\lambda_R((R[D]/m_RR[D])_n)=n\alpha+\sigma(n) \end{align*} $$

for $n\in {\mathbb N}$ . The constant $\alpha $ is positive if and only if $\dim (R[D]/m_RR[D])=2$ .

It is unlikely that the function $\sigma (n)$ will always be eventually periodic. It is shown in [14, Th. 9] that if D has exceptional support, then the Hilbert function of $\mathrm {gr}_{\mathcal I}(R)=\sum _{n\ge 0}I(nD)/I((n+1)D)t^n$ has an expression

$$ \begin{align*}\lambda_R(I(nD)/I((n+1)D))=n\beta+\tau(n), \end{align*} $$

where $\beta \in {\mathbb Q}$ and $\tau (n)$ is a bounded function. If R has equicharacteristic zero, then it is shown in [14, Th. 9] that $\tau (n)$ is eventually periodic, and [14, Exam. 5] gives an example where R has equicharacteristic $p>0$ and $\tau (n)$ is not eventually periodic.

Suppose that A is an excellent normal local ring of dimension 3. Let $Z\rightarrow \mbox {Spec}(A)$ be a resolution of singularities, and let D be an effective divisor on Z, all of whose components contract to the maximal ideal $m_A$ . Then the Hilbert polynomial $ h(n)=\lambda _A(I(nD)/ I((n+1)D))$ may be far from being polynomial like. The examples ([14, Exam. 6] and [11, Th. 1.4]) have the property that

$$ \begin{align*}\lim_{n\rightarrow \infty}\frac{h(n)}{n^2} \end{align*} $$

is an irrational number. These examples are in three-dimensional equicharacteristic rings A of any characteristic. The reason for this irrational behavior in dimension 3 is because of the lack of existence of Zariski decompositions in dimension 3.

We now give an outline of the proof of Theorem 7.1. Let $\pi :X\rightarrow \mbox {Spec}(R)$ be a resolution of singularities such that D is represented on X. Let $E_1,\ldots ,E_r$ be the prime exceptional divisors of $\pi $ . An ${\mathbb R}$ -divisor $\Delta $ on X is anti-nef if $(E\cdot \Delta )\le 0$ for all prime exceptional divisors E on X. Since X has dimension 2, D has a Zariski decomposition, $\Delta =D+B$ where $\Delta $ is an anti-nef divisor and B is an effective divisor with exceptional support such that

$$ \begin{align*}I(nD)=\Gamma(X,{\mathcal O}_X(-\lceil nD\rceil))=\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))=I(n\Delta) \end{align*} $$

for all $n\in {\mathbb N}$ . This decomposition does not exist in higher dimensions, even after blowing up ([8], [30, §IV.2.10], [21, §2.3]).

Proposition 1.7 (Corollary 6.5).

Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then the following are equivalent.

1. (1) There exists n such that $m_R\in \mbox {Ass}(R/I(n\Delta ))$ .

2. (2) There exists $n_0$ such that $m_R\in \mbox {Ass}(R/I(n\Delta ))$ for all $n\ge n_0$ .

3. (3) There exists j such that $E_j$ is exceptional and $(\Delta \cdot E_j)<0$ .

Let $E_j$ be an exceptional divisor of $\pi $ and

$$ \begin{align*}P_j=\bigoplus_{n\ge 0}\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-E_j)) \end{align*} $$

for $1\le j\le r$ . $P_j$ is a prime ideal in $R[\Delta ]=R[D]$ . In Proposition 6.7, it is shown that

$$ \begin{align*}\sqrt{m_RR[\Delta]}=\cap_{i=1}^rP_i. \end{align*} $$

The following proposition computes the dimension of $R[\Delta ]/P_j$ in terms of the intersection theory of X.

Proposition 1.8 (Proposition 6.9).

Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X and $E_j$ is a prime exceptional divisor for $\pi :X\rightarrow \mbox {Spec}(R)$ . Then:

1. (1) $\dim R[\Delta ]/P_j=2$ if $(\Delta \cdot E_j)<0$ .

2. (2) $\dim R[\Delta ]/P_j\le 1$ if $(\Delta \cdot E_j)=0$ .

Since $\sqrt {m_RR[\Delta ]}=\cap _{i=1}^rP_i$ , we deduce Theorem 7.1 from Propositions 6.5 and 6.9.

The theory of Zariski decomposition was created and developed by Zariski in [35] for projective surfaces over an algebraically closed field. In §4, we give the relative version of this theory, over a two-dimensional excellent normal local ring, and in §5, we extend some results in [35] for numerically effective divisors on a nonsingular projective surface to our situation of a resolution of singularities of a two-dimensional normal excellent local ring. We prove the main results of this paper on asymptotic properties of divisors on a resolution of singularities of a two-dimensional normal excellent local ring in §6. We prove Theorem 7.1 in §7 and Theorem 8.1 in §8.

1.6 Notation

We will denote the nonnegative integers by ${\mathbb N}$ and the positive integers by ${\mathbb Z}_{>0}$ , and the set of nonnegative rational numbers by ${\mathbb Q}_{\ge 0}$ and the positive rational numbers by ${\mathbb Q}_{>0}$ . We will denote the set of nonnegative real numbers by ${\mathbb R}_{\ge 0}$ and the positive real numbers by ${\mathbb R}_{>0}$ . If $x\in {\mathbb R}$ , then $\lceil x\rceil $ is the smallest integer, which is greater than or equal to x.

The maximal ideal of a local ring R will be denoted by $m_R$ . We will denote the length of an R-module M by $\lambda _R(M)$ . Scholie IV.7.8.3 of [18] gives a list of good properties of excellent local rings which we will assume.

2 Divisors on a resolution of singularities of a two-dimensional local ring

Throughout this paper, R is a two-dimensional excellent normal local ring with quotient field K, maximal ideal $m_R$ , and residue field $k=R/m_R$ .

From this section to §6, $\pi :X\rightarrow \mbox {Spec}(R)$ is a resolution of singularities such that $\pi $ is projective and all exceptional prime divisors of $\pi $ are nonsingular. Such a resolution of singularities exists by [24] or [7]. Let $E_1,\ldots ,E_r$ be the exceptional prime divisors for $\pi $ . A divisor is exceptional if all its prime components map to $m_R$ by $\pi $ . We will further assume that $\pi $ is not an isomorphism.

Remark 2.1. Suppose that $\mathcal F$ is a coherent sheaf on X. Then $H^0(X,\mathcal F)$ is a finitely generated R-module, $H^1(X,\mathcal F)$ is an R module of finite length, and $H^2(X,\mathcal F)=0$ .

Proof. By [20, Th. III.5.2], $H^0(X,\mathcal F)$ is a finitely generated R-module. By [20, Th. III.5.2 and Cor. III.11.2], $H^1(X,\mathcal F)$ is an R-module of finite length and by [20, Cor. III.11.2], $H^2(X,\mathcal F)=0$ since $\dim \pi ^{-1}(m_R)=1$ .

An element of the free abelian group $\mbox {Div}(X)$ on the prime divisors of X is called a divisor. Elements of $\mbox {Div}(X)\otimes {\mathbb Q}$ are called ${\mathbb Q}$ -divisors, and elements of $\mbox {Div}(X)\otimes {\mathbb R}$ are called ${\mathbb R}$ -divisors. We will sometimes refer to a divisor as an integral divisor if we want to emphasize this fact. If $D_1$ and $D_2$ are ${\mathbb R}$ -divisors, then write $D_2\ge D_1$ if $D_2-D_1$ is an effective divisor. The degree $\deg (\mathcal L)$ for $\mathcal L$ an invertible sheaf on a projective curve is defined in §3.

We use the intersection theory on X developed in [23, §§12 and 13]. The intersection theory on X is determined by the formula $(D\cdot E)=\deg ({\mathcal O}_X(D)\otimes {\mathcal O}_E)$ if D is a divisor on X and E is a prime exceptional divisor on X.

An ${\mathbb R}$ -divisor D is numerically effective (nef) if $(E\cdot D)\ge 0$ for all prime exceptional divisors E of X. An ${\mathbb R}$ -divisor D on X is anti-effective or anti-nef if $-D$ is, respectively, effective or nef. A ${\mathbb Q}$ -divisor D is anti-ample if $-D$ is ample and an (integral) divisor D is anti-very ample if $-D$ is very ample.

Let F be a prime divisor on X. Then ${\mathcal O}_{X,F}$ is a (rank 1) discrete valuation ring. Let $\nu _F$ be the associated valuation. For $0\ne f\in K$ , the divisor of f on X is $(f)=\sum \nu _F(f)F$ where the sum is over all the prime divisors F of X. Two divisors $D_1$ and $D_2$ are linearly equivalent, written $D_1\sim D_2$ if there exists $f\in K$ such that $(f)=D_2-D_1$ . Two divisors $D_1$ and $D_2$ which are linearly equivalent are also numerically equivalent; that is, $(E\cdot D_2)=(E\cdot D_1)$ for all prime exceptional divisors E of $\pi $ .

Let $D=\sum b_iF_i$ be an integral divisor on X. There is an associated invertible sheaf ${\mathcal O}_X(D)$ on X which is determined by the property that if U is an affine open subset of X and $h\in K$ is such that $h=0$ is a local equation of D in U, then ${\mathcal O}_X(D)\mid U=\frac {1}{h}{\mathcal O}_U$ . Thus,

$$ \begin{align*}\Gamma(X,{\mathcal O}_X(D))=\{f\in R\mid (f)+D\ge 0\}. \end{align*} $$

Since R is a subset of $\Gamma (X,{\mathcal O}_X)$ in K and R is normal, we have that $\Gamma (X,{\mathcal O}_X)=R$ by Remark 2.1, and so if D is an effective divisor, then $\Gamma (X,{\mathcal O}_X(-D))$ is an ideal in R.

If $D=\sum _{i=1}^s a_iF_i$ with $a_i\in {\mathbb R}$ is an ${\mathbb R}$ -divisor, let $\lceil D\rceil = \sum \lceil a_i\rceil F_i$ .

Let F be a prime divisor on X. For $\alpha \in {\mathbb R}_{\ge 0}$ , define valuation ideals in R by

$$ \begin{align*}I(\nu_F)_{\alpha}=\{f\in R\mid \nu_F(f)\ge\alpha\}. \end{align*} $$

We necessarily have that $I(\nu _F)_{\alpha }=I(\nu _F)_{\lceil \alpha \rceil }$ .

For an effective ${\mathbb R}$ -divisor $D=a_1F_1+\cdots +a_sF_s$ , where $F_1,\ldots ,F_s$ are prime divisors on X and $a_i\in {\mathbb R}_{\ge 0}$ , we have an associated ideal in R

$$ \begin{align*}I(D):=I(\nu_{F_1})_{a_1}\cap \cdots \cap I(\nu_{F_s})_{a_s}=I(\nu_{F_1})_{\lceil a_1\rceil}\cap \cdots \cap I(\nu_{F_s})_{\lceil a_s\rceil}=\Gamma(X,{\mathcal O}_X(-\lceil D\rceil)). \end{align*} $$

Let D be a divisor on X. Then $\Gamma (X,{\mathcal O}_X(D))\ne 0$ . The fixed component of D is the largest effective divisor F on X such that

$$ \begin{align*}\Gamma(X,\mathcal O_X(D))=\Gamma(X,{\mathcal O}_X(D-F)). \end{align*} $$

For $n\in {\mathbb N}$ , let $B_n$ be the fixed component of $nD$ and let

$$ \begin{align*}M_i=\{n\in {\mathbb N}\mid E_i \ \mbox{is not a component of} \ B_n\}. \end{align*} $$

$M_i$ is a numerical semigroup, so if $M_i$ is nonzero, there exists $h_i\in {\mathbb Z}_{>0}$ such that for $n\gg 0$ , $n\in M_i$ if and only if $h_i$ divides n.

The global sections $\Gamma (X,{\mathcal O}_X(D))$ of ${\mathcal O}_X(D)$ generate ${\mathcal O}_X(D)$ at a point $q\in X$ if ${\mathcal O}_X(D)_q=\Gamma (X,{\mathcal O}_X(D)){\mathcal O}_{X,q}$ . The points $q\in X$ where ${\mathcal O}_X(D)$ is generated by global sections are necessarily disjoint from the support of the fixed component of D.

Lemma 2.2. Let D be an effective divisor on X, and let F be a prime divisor in the support of the fixed component of $-D$ . Then the support of F is exceptional.

Proof. Write $D=\sum _{i=1}^r a_iF_i$ where the $F_i$ are distinct prime divisors on X and $a_i\in {\mathbb N}$ . Suppose that $F_j$ is not exceptional for $\pi $ . Let $q_j=\pi (F_j)$ , a height 1 prime ideal in R. Since $\pi $ is an isomorphism over $\mbox {Spec}(R)\setminus m_R$ , we have that $R_{q_j}={\mathcal O}_{X,F_j}$ , so

$$ \begin{align*}\begin{array}{lll} {\mathcal O}_X(-D)_{F_j}&=&(q_j^{a_j})_{q_j}=(I(\nu_j)_{a_j})_{q_j}=\Gamma(X,{\mathcal O}_X(-D))_{q_j}\\ &=&\Gamma(X,{\mathcal O}_X(-D )){\mathcal O}_{X,F_j}. \end{array} \end{align*} $$

Thus, $F_j$ is not in the support of F.

The intersection matrix of the exceptional curves of $\pi $ is the $r\times r$ matrix $\left ((E_i\cdot E_j)\right )$ , which is negative-definite (see [23, Lem. 14.1]).

Proposition 2.3. Let D be a ${\mathbb Q}$ -divisor on X. Then D is ample if and only if $(D\cdot E)>0$ for all prime exceptional divisors E on X.

This is proved in [23, Th. 12.1]. As commented in the proof of [23, Th. 12.1], the additional assumption there that $H^1(X,{\mathcal O}_X)=0$ is not necessary for this conclusion.

Lemma 2.4. The support of a nonzero effective anti-nef ${\mathbb R}$ -divisor D on X contains all exceptional prime divisors.

Proof. Let S be the set of exceptional prime divisors which are in the support of D. Write $D=B+\sum _{i=1}^ra_iE_i$ where B is an effective divisor which contains no exceptional prime divisors in its support and all $a_i\ge 0$ . For all $E_j$ , we have that

$$ \begin{align*}0\ge (D\cdot E_j)=(B\cdot E_j)+\sum_{i\ne j}a_i(E_i\cdot E_j)+a_j(E_j^2), \end{align*} $$

and so

(5) $$ \begin{align} -a_j(E_j^2)\ge (B\cdot E_j)+\sum_{i\ne j}a_i(E_i\cdot E_j)\ge 0. \end{align} $$

If B is nonzero, then there exists $E_{j}$ such that $(E_{j}\cdot B)>0$ and thus $a_{j}>0$ and so $E_j\in S$ . If $B=0$ , then there exists $E_j$ such that $(E_j\cdot D)<0$ since $D\ne 0$ and the intersection matrix $\left ((E_i\cdot E_j)\right )$ is nonsingular. Thus, S is nonempty. If $E_{j'}\in S$ and $E_j$ is such that $(E_j\cdot E_{j'})>0$ , then $E_j\in S$ by (5). The exceptional fiber $\pi ^{-1}(m_R)$ is connected as R is normal and $\pi $ is birational (by [20, Cor. III.11.4]). Thus, S is the set of all exceptional prime divisors of X.

Lemma 2.5. X is the blowup of an $m_R$ -primary ideal.

Proof. Since the intersection matrix $\left ((E_i\cdot E_j)\right )$ is negative definite, there exists an effective anti-ample ${\mathbb Q}$ -divisor A on X with exceptional support (by Proposition 2.3). Thus, $-dA$ is very ample for some $d\in {\mathbb Z}_{>0}$ . Let $I=\Gamma (X,{\mathcal O}_X(-dA))$ . The ideal I is $m_R$ -primary since the support of A is exceptional. The integral closure of $\sum _{n\ge 0}I^nt^n$ in $R[t]$ is

$$ \begin{align*}\sum_{n\ge 0}\overline{I^n}t^n=\sum_{n\ge 0}\Gamma(X,{\mathcal O}_X(-ndA))t^n. \end{align*} $$

Since R is excellent, $\sum _{n\ge 0}\overline {I^n}t^n$ is a finitely generated graded R-algebra. Thus, after replacing d with a higher power of d, we may assume that $I^n=\overline {I^n}=\Gamma (X,{\mathcal O}_X(-ndA))$ for all $n\in {\mathbb Z}_{>0}$ (as follows from [4, Props. III.3.2 and III.3.3 on pages 158 and 159]).

Let $Y=\mbox {Proj}(\oplus _{n\ge 0}I^n)$ , which is normal since $\oplus _{n\ge 0}I^n$ is integrally closed. Since ${\mathcal O}_X(-dA)$ is generated by global sections, we have that $I{\mathcal O}_X={\mathcal O}_X(-dA)$ . By the universal property of blowing up (see [20, Prop. II.7.14]), there exists a unique R-morphism $\phi :X\rightarrow Y$ such that $\phi ^*{\mathcal O}_Y(1)\cong {\mathcal O}_X(-dA)$ . $\phi $ is a birational morphism which is an isomorphism away from the preimage of $m_R$ . $\phi $ is of finite type since $X\rightarrow \mbox {Spec}(R)$ is. Since $(-A\cdot E)>0$ for all exceptional curves of X, we have that $\phi $ does not contract any curves of X and thus $\phi $ is quasi-finite. Let $p\in X$ and $q=\phi (p)$ . Let $A={\mathcal O}_{Y,q}$ and $B={\mathcal O}_{X,p}$ . The birational extension $A\rightarrow B$ satisfies $m_AB$ is $m_B$ -primary since $\phi $ is quasi-finite. Since A is normal and excellent, it is analytically irreducible by [18, Scholie IV.7.8.3(vii)]. Thus, by Zariski’s main theorem [1, (10.7), p. 240] or [10, Prop. 21.53], we have that $A=B$ and so $\phi $ is an isomorphism and X is the blowup of the $m_R$ -primary ideal I.

Lemma 2.6. Let A be a universally Nagata domain, and let I be an ideal in A. Let $Y=\mbox {Proj}(\bigoplus _{n\ge 0}I^n)$ . Then the graded ring $\bigoplus _{n\ge 0}\Gamma (Y,I^n{\mathcal O}_Y)$ is a finite $\bigoplus _{n\ge 0}I^n$ -module and there exists $n_0\in {\mathbb Z}_{>0}$ such that $\Gamma (Y,I^n{\mathcal O}_Y)=I^n$ for $n\ge n_0$ .

Proof. This follows from the proof on the last two lines of page 122 to the first half of page 123 of [20, Th. II.5.19], along with the fact (observed in [20, Rem. 5.19.2]) that the integral closure of a Nagata domain in its quotient field is a finite extension (by [26, Prop. 31.B]).

3 Riemann–Roch theorems for curves

We summarize the famous Riemann–Roch theorems for curves. The following theorems are standard over algebraically closed fields. A reference where they are proved over an arbitrary field k is [22, §7.3]. The results that we need are stated in [22, Rem. 7.3.33].

Let E be an integral regular projective curve over a field k. For $\mathcal F$ a coherent sheaf on E, define $h^i(\mathcal F)=\dim _kH^i(E,\mathcal F)$ .

Let $D=\sum a_ip_i$ be a divisor on E, where $p_i$ are prime divisors on E (closed points) and $a_i\in {\mathbb Z}$ . We have an associated invertible sheaf ${\mathcal O}_X(D)$ . Define

$$ \begin{align*}\deg(D)=\deg({\mathcal O}_E(D))=\sum a_i[{\mathcal O}_{E_i,p_i}/m_{p_i}:k]. \end{align*} $$

The Riemann–Roch formula is

(6) $$ \begin{align} \chi({\mathcal O}_{E}(D)):=h^0({\mathcal O}_{E}(D))-h^1({\mathcal O}_{E}(D))=\mbox{deg}(D)+1-p_a(E), \end{align} $$

where $p_a(E)$ is the arithmetic genus of E.

We further have Serre duality,

(7) $$ \begin{align} H^1(E,{\mathcal O}_{E}(D))\cong H^0(E,{\mathcal O}_{E}(K-D)), \end{align} $$

where $K=K_E$ is a canonical divisor on E. As a consequence, we have

(8) $$ \begin{align} \deg D>2p_a(E)-2=\deg(K)\mbox{ implies }H^1(E,{\mathcal O}_{E}(D))=0. \end{align} $$

We have the following well-known consequence of these formulas, which we record for future reference.

Lemma 3.1. Let E be an integral regular projective curve over a field k. Let $\{D_n\}_{n\ge 0}$ be an infinite sequence of divisors on E such that $\deg (D_n)$ is bounded from below, and let Z be a divisor on E. Then there exists $s\in {\mathbb Z}_{>0}$ such that

$$ \begin{align*}h^1({\mathcal O}_{E}(D_n+Z))\le s \ \mbox{for all} \ n\in {\mathbb N}. \end{align*} $$

Proof. There exists an integer c such that $\deg (D_n)\ge c$ for all n. Let U be an effective divisor on E of degree larger than $2p_a(E)-2+c$ . By Serre duality (7),

$$ \begin{align*}h^1({\mathcal O}_{E}(D_n+Z))=h^0({\mathcal O}_{E}(K-(D_n+Z)), \end{align*} $$

where K is a cononical divisor on E. We have

$$ \begin{align*} \mbox{deg}(K-(Z+D_n))\le \deg(K-Z)-c. \end{align*} $$

If $\mbox {deg}(K-Z)-c<0$ , then certainly $h^0({\mathcal O}_{E}(K-(D_n+Z))=0$ . If $\mbox {deg}(K-Z)-c\ge 0$ , then $h^1({\mathcal O}_{E}(K-(D_n+Z)+U)=0$ by (8) and so

$$ \begin{align*}\begin{array}{lll} h^0({\mathcal O}_{E}(K-(D_n+Z))&\le& h^0({\mathcal O}_{E}(K-(D_n+Z)+U)\\ &=&\deg(K-(D_n+Z))+\deg(U)+1-p_a(E)\\ &\le& \deg(K-Z)-c+\mbox{deg}(U)+1-p_a(E). \end{array}\\[-40pt] \end{align*} $$

If $\mathcal L$ is an invertible sheaf on E, then $\mathcal L\cong {\mathcal O}_E(D)$ for some divisor D on E, and we may define $\deg (\mathcal L)=\deg ({\mathcal O}_X(D))=\deg (D)$ .

We will apply the above formulas in the case that E is a prime exceptional divisor for a resolution of singularities $\pi :X\rightarrow \mbox {Spec}(R)$ as in §2. We take $k=R/m_R$ . We have that E is projective over $k=R/m_R$ , and E is a nonsingular (by assumption) integral curve. Let D be a divisor on X. Then $\deg ({\mathcal O}_X(D)\otimes {\mathcal O}_E)=(D\cdot E)$ .

4 Zariski decomposition

In this section, we present a relative form of the Zariski decomposition defined for projective surfaces over an algebraically closed field in [35]. Lemma 4.1 in the case that D is exceptional follows directly from [35] or [3, Th. 3.3].

We continue with our ongoing assumptions that R is a two-dimensional excellent normal local ring with quotient field K, maximal ideal $m_R$ , and residue field $k=R/m_R$ and $\pi :X\rightarrow \mbox {Spec}(R)$ is a resolution of singularities such that the exceptional prime divisors $E_1,\ldots ,E_r$ are nonsingular.

The proof of the following lemma is a modification of the proof of [3, Th. 3.3].

Lemma 4.1. Let D be an effective ${\mathbb R}$ -divisor on X. Then there exist unique effective ${\mathbb R}$ -divisors $\Delta $ and B on X such that the following (1) and (2) hold.

1. (1) $\Delta =D+B$ is anti-nef and B has exceptional support.

2. (2) $(\Delta \cdot E)=0$ if E is a component of B.

Further,

1. (3) $\Delta $ is the unique minimal effective anti-nef ${\mathbb R}$ -divisor such that $\Delta -D$ is effective with exceptional support.

2. (4) If D is a ${\mathbb Q}$ -divisor, then $\Delta $ and B are ${\mathbb Q}$ -divisors.

The decomposition $\Delta =D+B$ of the conclusions of Lemma 4.1 is called the Zariski decomposition of D.

Proof. For $x=(x_1,\ldots ,x_r)\in {\mathbb R}^r$ , consider the inequalities

(9) $$ \begin{align} 0\le x_i \ \mbox{for} \ 1\le i\le r \end{align} $$

and

(10) $$ \begin{align} \left((D+\sum_{i=1}^rx_iE_i)\cdot E_j\right)\le 0 \ \mbox{for} \ 1\le j\le r. \end{align} $$

Since the matrix $((E_i\cdot E_j))$ is negative-definite and by Proposition 2.3, there exists an anti-ample, effective divisor $A=\sum _{i=1}^ra_iE_i$ on X. Thus, $a_i>0$ for all i (by Lemma 2.4) and after possibly replacing A with a positive multiple of A, $x=a=(a_1,\ldots ,a_r)$ satisfies (9) and (10). Let

(11) $$ \begin{align} S=\{x\in {\mathbb R}^r\mid x_i\le a_i \ \mbox{for all} \ i \ \mbox{and the} \ 2r \ \mbox{inequalities} \ (9) \ \mbox{and} \ (10)\ \mbox{are satisfied}\}. \end{align} $$

The set S is nonempty and compact. Thus, there is at least one point in S such that $\sum _{i=1}^rx_i$ is minimized on S. Let $b=(b_1,\ldots ,b_r)$ be such a point. Let $B=b_1E_1+\cdots +b_rE_r$ and $\Delta =D+B$ . Then $\Delta $ is an effective, anti-nef ${\mathbb R}$ -divisor and B is an effective ${\mathbb R}$ -divisor with exceptional support. Let $E_j$ be a component of B. Since b minimizes $\sum x_i$ , $B-\epsilon E_j$ is effective and $\Delta -\epsilon E_j$ is not anti-nef for all $\epsilon>0$ sufficiently small. However, $\left ((\Delta -\epsilon E_j)\cdot E_i\right )\le 0$ for all $i\ne j$ , so we must have that $\left ((\Delta -\epsilon E_j)\cdot E_j\right )>0$ for all positive $\epsilon $ and thus $(\Delta \cdot E_j)=0$ since $\Delta $ is anti-nef. Thus, the decomposition $\Delta =D+B$ satisfies (1) and (2).

For $b=(b_1,\ldots ,b_r)$ and $b'=(b_1',\ldots ,b_r')\in {\mathbb R}^r$ , define

$$ \begin{align*}\mbox{min}(b,b')=(\mbox{min}(b_1,b_1'),\ldots,\mbox{min}(b_r,b_r')). \end{align*} $$

If b and $b'$ satisfy (9) and (10), then $\mbox {min}(b,b')$ also satisfies (9) and (10), as we now show. For a fixed j, we may assume that $\mbox {min}(b_j,b_j')=b_j$ (after possibly interchanging b and $b'$ ). Then, since $(E_i\cdot E_j)\ge 0$ if $i\ne j$ , we have that

$$ \begin{align*}((D+\sum_i\mbox{min}(b_i,b_i')E_i)\cdot E_j)\le ((D+\sum_ib_iE_i)\cdot E_j)\le 0. \end{align*} $$

Suppose that $B=\sum b_iE_i$ and $B'=\sum b_i'E_i$ are effective ${\mathbb R}$ -divisors such that $\Delta =D+B$ and $\Delta '=D=B'$ satisfy both (1) and (2). We will show that $B=B'$ and so $\Delta =\Delta '$ . Let $\mbox {min}(B,B')=\sum _i\mbox {min}(b_i,b^{\prime }_i)E_i$ . There exist $x_i\ge 0$ such that $\mbox {min}(B,B')=B-\sum _i x_iE_i$ . Since $D+\mbox {min}(B,B')$ is anti-nef, for each element $E_j$ of the support of B, we have

$$ \begin{align*}0\ge \left((D+\mbox{min}(B,B'))\cdot E_j\right)=\left((\Delta-\sum_i x_i E_i)\cdot E_j\right)=-\sum_i x_i(E_i\cdot E_j). \end{align*} $$

Thus, $\sum _i x_i(E_i\cdot E_j)\ge 0$ and so

$$ \begin{align*}\left((\sum_i x_iE_i)\cdot (\sum_j x_jE_j)\right)=\sum_i\sum_j x_ix_j(E_i\cdot E_j)\ge 0. \end{align*} $$

Since the matrix $\left ((E_i\cdot E_j)\right )$ is negative-definite, we have that $x_i=0$ for all i. Thus, $B=\mbox {min}(B,B')$ . Similarly, $B'=\mbox {min}(B,B')$ and so $B=B'$ . Thus, there is a unique effective ${\mathbb R}$ -divisor B with exceptional support such that B and $\Delta =D+B$ satisfy (1) and (2).

We now show that $\Delta $ is the unique minimal effective and anti-nef ${\mathbb R}$ -divisor on X such that $\Delta -D$ is effective with exceptional support. Let U be an effective anti-nef ${\mathbb R}$ -divisor on X such that $U-D$ is effective with exceptional support. Let $U'=D+\mbox {min}(\Delta -D,U-D)$ . As shown earlier in the proof, $U'\ge D$ is effective and anti-nef. Write $U'-D=\sum u_iE_i$ and $B=\Delta -D=\sum b_iE_i$ . We have $\sum u_i\le \sum b_i\le \sum a_i$ , so $U'-D\in S$ (defined in (11)). Since $\sum b_i$ is the minimum of $\sum x_i$ on S, we have that $u_i=b_i$ for all i and so $U'=\Delta $ . Thus, $\Delta \le U$ .

Now, suppose that D is an effective ${\mathbb Q}$ -divisor on X. Let $\Delta =D+B$ be the Zariski decomposition of D. After possibly reindexing the $E_1,\ldots ,E_r$ , we may assume that the support of B is $E_1\cup \cdots \cup E_s$ for some s with $1\le s\le r$ . Expand $D=F+\sum _{i=1}^rc_iE_i$ where F is an effective ${\mathbb Q}$ -divisor whose support does not contain any prime exceptional divisor and $c_1,\ldots ,c_r\in {\mathbb Q}_{\ge 0}$ . Then $\Delta =F+\sum _{i=1}^rd_iE_i$ with $c_i\le d_i$ for all i and $d_i=c_i$ for $s+1\le i\le r$ . Furthermore, for $1\le j\le s$ , we have $0=(\Delta \cdot E_j)=\sum _{i=1}^sd_i(E_i\cdot E_j)+g_j$ where $g_j=(F\cdot E_j)+\sum _{i=s+1}^rc_i(E_i\cdot E_j)\in {\mathbb Q}$ . Since the $s\times s$ matrix $\left ((E_i\cdot E_j)\right )_{1\le i,j\le s}$ is negative-definite, and thus is nonsingular, we have that $d_1,\ldots ,d_s\in {\mathbb Q}$ . Thus, $\Delta $ and B are ${\mathbb Q}$ -divisors.

Remark 4.2. From (3) of the conclusions of Lemma 4.1, we deduce that if $D_1\le D_2$ are effective ${\mathbb R}$ -divisors such that $D_2-D_1$ has exceptional support and the respective anti-nef parts of their Zariski decompositions are $\Delta _1$ and $\Delta _2$ , then $\Delta _1\le \Delta _2$ .

Lemma 4.3. Suppose that D is an effective ${\mathbb R}$ -divisor on X and $\Delta =D+B$ is the Zariski decomposition of D. Then, for all $n\in {\mathbb N}$ ,

$$ \begin{align*}\Gamma(X,{\mathcal O}_X(-\lceil nD\rceil))=\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil)). \end{align*} $$

Proof. Suppose that $f\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ . Then $(f)-\lceil n\Delta \rceil \ge 0$ . Writing $n\Delta =\lceil n\Delta \rceil -G$ with $G\ge 0$ , we have $-n\Delta =G-\lceil n\Delta \rceil $ . From

$$ \begin{align*}-nD=-n\Delta+nB=-\lceil n\Delta\rceil+(G+nB) \end{align*} $$

and the fact that $G+nB\ge 0$ , we have that $(f)-nD\ge 0$ so that $f\in \Gamma (X,{\mathcal O}_X(-\lceil nD\rceil ))$ .

Let S be the set of prime divisors in the support of B. Suppose that

$$ \begin{align*}f\in \Gamma(X,{\mathcal O}_X(-\lceil nD\rceil )). \end{align*} $$

Then $(f)-nD\ge 0$ . Write $(f)-nD=A+C$ where A and C are effective ${\mathbb R}$ -divisors on X, no components of A are in S and all components of C are in S. We have that $(f)-n\Delta =A+(C-nB)$ . If $E\in S$ , then

$$ \begin{align*}(E\cdot(A+(C-nB)))=(E\cdot((f)-n\Delta))=0, \end{align*} $$

which implies $(E\cdot (C-nB))=-(E\cdot A)\le 0$ . The intersection matrix of the curves in S is negative-definite since it is so for the set of all exceptional curves, so $C-nB\ge 0$ (by [35, Lem. 7.1]). Thus, $(f)-n\Delta \ge 0$ , which implies $(f)-\lceil n\Delta \rceil \ge 0$ since $(f)$ is an integral divisor. Thus, $f\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ .

5 Nef divisors

In this section, we extend to our relative situation $X\rightarrow \mbox {Spec}(R)$ some theorems proved by Zariski in [35] for projective surfaces over an algebraically closed field. We stay as close as possible to Zariski’s original proof, although some parts require modification. In [21], and the references in that book, a theory of nef divisors on nonsingular projective varieties of arbitrary dimension over an algebraically closed field of characteristic zero is derived. Much of this theory can be extended to the relative situation, over $\mbox {Spec}(A)$ , where the local ring A is normal and essentially of finite type over an algebraically closed field of characteristic zero, or even of positive characteristic.

We continue with our ongoing assumptions that R is a two-dimensional excellent normal local ring with quotient field K, maximal ideal $m_R$ , and residue field k, and that $\pi :X\rightarrow \mbox {Spec}(R)$ is a resolution of singularities such that the exceptional prime divisors $E_1,\ldots ,E_r$ of $\pi $ are all nonsingular.

Proposition 5.1. Let $\Delta $ be an effective anti-nef divisor on X. For $n\ge 0$ , let $B_n$ be the fixed component of $-n\Delta $ . Suppose that E is a prime divisor which is in the support of the fixed component $B_n$ of $-n\Delta $ for infinitely many n. Then E is exceptional for $\pi $ and $(\Delta \cdot E)=0$ .

Proof. By Lemma 2.2, E is exceptional. We will assume that $(\Delta \cdot E)<0$ and derive a contradiction. Since $\Gamma (X,{\mathcal O}_X(-\Delta ))\ne 0$ , there exists an effective divisor D on X such that $D\sim -\Delta $ . Write $D = U+F_1+\cdots +F_s$ where U is an effective divisor with no exceptional divisors in its support and $F_1=E,F_2,\ldots ,F_s$ are prime exceptional divisors. Let $\Delta _i=U+ F_1+\cdots +F_i$ for $0\le i\le s$ .

We have short exact sequences

$$ \begin{align*}0\rightarrow {\mathcal O}_X(nD-\Delta_0)\rightarrow {\mathcal O}_X(nD)\rightarrow {\mathcal O}_X(nD)\otimes{\mathcal O}_{\Delta_0}\rightarrow 0. \end{align*} $$

There exists a very ample effective divisor H on X which contains no exceptional prime divisors in its support and whose support is disjoint from $\Delta _0$ by [20, Th. III.5.2] since $\Delta _0$ intersects $\pi ^{-1}(m_R)$ in only a finite number of closed points and so $\Delta _0$ is a closed subscheme of the affine scheme $X\setminus V(H)$ and thus $\Delta _0$ is an affine scheme. We thus have that $H^1(\Delta _0,{\mathcal O}_X(-nD)\otimes {\mathcal O}_{\Delta _0})=0$ for all n and so

(12) $$ \begin{align} h^1({\mathcal O}_X(nD))\le h^1({\mathcal O}_X(nD-\Delta_0)) \end{align} $$

for all $n\in {\mathbb N}$ .

For $i<s$ and $n\in {\mathbb N}$ , we have short exact sequences

$$ \begin{align*}0\rightarrow {\mathcal O}_X(nD-\Delta_i-F_{i+1})\rightarrow {\mathcal O}_X(nD-\Delta_i)\rightarrow {\mathcal O}_X(nD-\Delta_i)\otimes {\mathcal O}_{F_{i+1}}\rightarrow 0. \end{align*} $$

Thus,

$$ \begin{align*}h^1({\mathcal O}_X(nD-\Delta_i))\le h^1({\mathcal O}_X(nD-\Delta_{i+1})+h^1(F_{i+1},O_X(nD-\Delta_i)\otimes {\mathcal O}_{F_{i+1}}). \end{align*} $$

$(D\cdot F_{i+1})=(-\Delta \cdot F_{i+1})\ge 0$ implies that there exists $\sigma _i>0$ such that $h^1(F_{i+1},O_X(nD-\Delta _i)\otimes {\mathcal O}_{F_{i+1}})\le \sigma _i$ for all $n\in {\mathbb N}$ by Lemma 3.1, so

(13) $$ \begin{align} h^1({\mathcal O}_X(nD-\Delta_i))\le h^1({\mathcal O}_X(nD-\Delta_{i+1}))+\sigma_i \end{align} $$

for all $i\ge 0$ and $n\in {\mathbb N}$ .

Now, consider the exact sequences

$$ \begin{align*}0\rightarrow {\mathcal O}_X(nD-\Delta_0-F_1)\rightarrow {\mathcal O}_X(nD-\Delta_0)\rightarrow {\mathcal O}_X(nD-\Delta_0)\otimes {\mathcal O}_{F_1}\rightarrow 0 \end{align*} $$

for $n\in {\mathbb N}$ . Since $(F_1\cdot D)=(F_1\cdot -\Delta )>0$ , we have that $H^1(F_1,{\mathcal O}_X(nD-\Delta _i)\otimes {\mathcal O}_{F_1})=0$ for $n\gg 0$ by (8). From the natural inclusion ${\mathcal O}_X(nD-\Delta _0)\rightarrow {\mathcal O}_X(nD)$ , we deduce that $F_1$ is in the support of the fixed locus of $nD-\Delta _0$ if $F_1$ is in the support of the fixed locus of $-n\Delta $ . Thus, for n such that $F_1$ is a component of the base locus $B_n$ of $-n\Delta $ , the image of $H^0(X,{\mathcal O}_X(nD-\Delta _0))$ in $H^0(F_1,{\mathcal O}_X(nD-\Delta _i)\otimes {\mathcal O}_{F_1})$ is zero. Thus,

$$ \begin{align*}h^1({\mathcal O}_X(nD-\Delta_0))=h^1({\mathcal O}_X(nD-\Delta_0-F_1))-\chi({\mathcal O}_{F_1}(nD-\Delta_0)\otimes{\mathcal O}_{F_1}) \end{align*} $$

so that by the Riemann Roch theorem (6),

(14) $$ \begin{align} h^1({\mathcal O}_X(nD-\Delta_0))=h^1({\mathcal O}_X(nD-\Delta_0-F_1))+n(\Delta\cdot F_1)+(\Delta_0\cdot F_1)+p_a(F_1)-1. \end{align} $$

As explained before the statement of Lemma 2.2, there exists a positive integer h such that for $n\gg 0$ , $F_1$ is a component of $B_n$ if $h\nmid n$ .

By (12) and (13), there exists a constant $c>0$ such that

$$ \begin{align*}h^1({\mathcal O}_X(nD))\le h^1({\mathcal O}_X((n-1)D))+c \end{align*} $$

for all $n\in {\mathbb Z}_{>0}$ and for all $n\gg 0$ such that $h\nmid n$ we have by (12)–(14) that

$$ \begin{align*}h^1({\mathcal O}_X(nD))\le h^1({\mathcal O}_X((n-1)D)+n(\Delta\cdot F_1)+c. \end{align*} $$

Thus, we have $h^1({\mathcal O}_X(nD))<0$ for $n\gg 0$ since we have assumed that $(\Delta \cdot F_1)<0$ . However, this is impossible, giving a contradiction and so $(\Delta \cdot F_1)=0$ .

Proposition 5.2. Let $\Gamma $ be an effective divisor on X such that $-\Gamma $ has no fixed component. Then:

1. (1) ${\mathcal O}_X(-n\Gamma )$ is generated by global sections for all $n\gg 0$ .

2. (2) There exists $s\in {\mathbb Z}_{>0}$ such that $h^1(X,{\mathcal O}_X(-n\Gamma ))<s$ for all $n\in {\mathbb N}$ .

Proof. The set of base points

$$ \begin{align*}\Omega=\{p\in X\mid {\mathcal O}_X(-\Gamma)_p \ \mbox{is not generated by global sections}\} \end{align*} $$

of $\Gamma (X,{\mathcal O}_X(-\Gamma ))$ is a finite set of closed points, which are necessarily contained in the exceptional fiber of $\pi $ . Let $C\ge 0$ be an effective divisor on X such that $-C$ is very ample for $\pi $ . There exists an integer $m>0$ such that there exists an effective divisor $H\sim -mC$ with no exceptional components in its support and such that $\Omega $ is disjoint from its support (by [20, Th. III.5.2]). After replacing C with this multiple $mC$ , we may assume that $H\sim -C$ . Let $f\in K$ , the quotient field of R, be such that $(f)-C=H$ . We may regard the effective divisor H as a closed subscheme of X.

We have a short exact sequence

$$ \begin{align*}0\rightarrow {\mathcal O}_X(C)\stackrel{f}{\rightarrow} {\mathcal O}_X\rightarrow {\mathcal O}_H\rightarrow 0, \end{align*} $$

and tensoring with ${\mathcal O}_X(-i\Gamma -jC)$ , we have short exact sequences

(15) $$ \begin{align} 0\rightarrow {\mathcal O}_X(-i\Gamma-(j-1)C)\stackrel{f}{\rightarrow}{\mathcal O}_X(-i\Gamma-jC)\rightarrow {\mathcal O}_X(-i\Gamma-jC)\otimes{\mathcal O}_H\rightarrow 0. \end{align} $$

For $i,j\ge 0$ , let $A_{i,j}$ be the natural image of $\Gamma (X,\mathcal O_X(-i\Gamma -jC))$ in

$$ \begin{align*}\Gamma(H,{\mathcal O}_X(-i\Gamma-jC)\otimes {\mathcal O}_H), \end{align*} $$

upon taking global sections of (15). Since the base points of $\Gamma (X,\mathcal O_X(-i\Gamma -jC))$ are a subset of $\Omega $ and so are disjoint from H, we have that, for all $i,j\ge 0$ , $A_{i,j}{\mathcal O}_{H,q}={\mathcal O}_X(-i\Gamma -jC)_q$ for all $q\in H$ .

There exists $n\in {\mathbb Z}_{>0}$ such that there exists an effective divisor G on X such that $G\sim -nC$ , the support of G contains no exceptional components of $\pi $ and $\mbox {sup}(H)\cap \mbox {sup}(G)\cap \mbox {sup}(\pi ^{-1}(m_R))=\emptyset $ (by [20, Th. III.5.2]). We may regard G as a closed subscheme of X. Thus, H is a closed subscheme of the affine scheme $X\setminus G$ and so H is affine, say $H=\mbox {Spec}(S)$ . The restriction of $\pi $ to H is determined by a ring homomorphism $R\rightarrow S$ . Now, $S=\Gamma (H,{\mathcal O}_H)$ is a finitely generated R-module since $\pi $ is a projective morphism (by [20, Cor. II.5.20]). As explained in [20, Cor. II.5.5], since S is Noetherian, the functor $M\rightarrow \tilde M$ gives an equivalence of categories between the category of finitely generated S-modules and the category of coherent ${\mathcal O}_{\mbox {Spec}(S)}$ -modules, with inverse $\mathcal F\mapsto \Gamma (\mbox {Spec}(S),\mathcal F)$ .

In particular, letting $B_{i,j}=\Gamma (H,{\mathcal O}_X(-i\Gamma -jC)\otimes {\mathcal O}_H)$ for $i,j\ge 0$ , we have that ${\mathcal O}_X(-i\Gamma -jC)\otimes {\mathcal O}_H=\widetilde {B_{i,j}}$ . We also have that $B_{i,j}$ is the tensor product over S of i copies of $B_{1,0}$ and j copies of $B_{0,1}$ (see [20, Prop. II.5.2]).

We have that the ring $A_{0,0}$ is a quotient of $\Gamma (X,{\mathcal O}_X)=R$ since $\pi $ is proper birational and R is normal. Let $A_{0,0}[t_1,t_2]$ be a polynomial ring over $A_{0,0}$ , which is bigraded by specifying that $\deg (a)=(0,0)$ if $a\in A_{0,0}$ , $\deg (t_1)=(1,0)$ , and $\deg (t_2)=(0,1)$ . Let M be the bigraded $A_{0,0}$ -subalgebra $M:=\sum _{i,j\ge 0} A_{i,j}t_1^it_2^j$ of $A_{0,0}[t_1,t_2]$ . Similarly, let B be the bigraded S-subalgebra $B:=\bigoplus _{i,j\ge 0}B_{i,j}t_1^it_2^j$ of $S[t_1,t_2]$ .

We have a natural inclusion of graded rings $M\rightarrow B$ .

Since H is disjoint from $\Omega $ , we have that

$$ \begin{align*}A_{1,0}^iA_{0,1}^jS_q=A_{ij}S_q={\mathcal O}_X(-i\Gamma-jA)\otimes{\mathcal O}_{H,q}=(B_{i,j})_q \end{align*} $$

for all $q\in H$ and $i,j\ge 0$ . Thus,

(16) $$ \begin{align} A_{1,0}^iA_{0,1}^jS=B_{i,j} \ \mbox{for all} \ i,j\ge 0. \end{align} $$

Let A be the bigraded $A_{0,0}$ -subalgebra $A:= A_{0,0}[A_{1,0}t_1,A_{0,1}t_2]$ of M. Now, we have a natural surjection $A_{1,0}^iA_{0,1}^j\otimes _RS\rightarrow B_{i,j}$ for all $i,j\ge 0$ by (16). Thus, the natural homomorphism $A\otimes _RS\rightarrow B$ is surjective. Since S is a finitely generated R-module, we have that B is a finitely generated bigraded A-module. Since $A\subset M\subset B$ and A is Noetherian, we have that M is also a finitely generated A-module.

By [35, Lem. 4.3], since A is generated in bidegrees $(1,0)$ and $(0,1)$ , and M is a finitely generated bigraded R-module, there exists $N\in {\mathbb Z}_{>0}$ such that

(17) $$ \begin{align} A_{i,j}=A_{i,j-1}A_{0,1} \ \mbox{whenever} \ j\ge N\ \mbox{and} \ i\ge 0 \ \mbox{is arbitrary} \end{align} $$

and

(18) $$ \begin{align} A_{i,j}=A_{i-1,j}A_{1,0} \ \mbox{whenever} \ i\ge N \ \mbox{and} \ j\ge 0 \ \mbox{is arbitrary.} \end{align} $$

Thus, taking global sections in the short exact sequences (15), and applying (18), we have that if $i\ge N$ and $j\ge 0$ , then

(19) $$ \begin{align} &\Gamma(X,{\mathcal O}_X(-i\Gamma-jC))\nonumber\\&\quad=\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C))f+\Gamma(X,{\mathcal O}_X(-(i-1)\Gamma-jC))\Gamma(X,{\mathcal O}_X(-\Gamma)). \end{align} $$

Since $-C$ is ample, for fixed i, ${\mathcal O}_X(-i\Gamma -jC)$ is generated by global sections for all $j\gg 0$ (by [20, Th. II.5.17]). Let i be a fixed integer $\ge N$ and let $j>0$ be such that ${\mathcal O}_X(-i\Gamma -jC)$ is generated by global sections.

The only points $q\in X$ where it is possible for ${\mathcal O}_X(-i\Gamma -(j-1)C)_q$ to not be generated by global sections are the points of $\Omega $ . Suppose that $q\in \Omega $ . Thus, q is not in the support of $H=(f)-C$ , and so $f=0$ is a local equation of C at q and $f{\mathcal O}_{X,q}={\mathcal O}_X(-C)_q$ . Furthermore, since $q\in \Omega $ , $\Gamma (X,{\mathcal O}_X(-\Gamma ))O_{X,q}\subset m_q{\mathcal O}_{X}(-\Gamma )$ where $m_q$ is the maximal ideal of ${\mathcal O}_{X,q}$ , equation (19), and Nakayama’s lemma show that

$$ \begin{align*}\begin{array}{lll} {\mathcal O}_X(-i\Gamma-jC)_q&=&\Gamma(X,{\mathcal O}_X(-i\Gamma-jC)){\mathcal O}_{X,q}\\ &=&\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C))f{\mathcal O}_{X,q}\\ &&+\Gamma(X,{\mathcal O}_X(-(i-1)\Gamma-jC){\mathcal O}_X(-\Gamma)m_q\\ &=&\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C)){\mathcal O}_X(-C)_q. \end{array} \end{align*} $$

Thus, $\Gamma (X,{\mathcal O}_X(-i\Gamma -(j-1)C)){\mathcal O}_{X,q}= {\mathcal O}_X(-i\Gamma -(j-1)C)_q$ , and since this is true for all $q\in \Omega $ , ${\mathcal O}_X(-i\Gamma -(j-1)C)$ is generated by global sections.

By descending induction on j, we obtain that ${\mathcal O}_X(-i\Gamma )$ is generated by global sections for all $i\ge N$ .

We now prove the second statement of the proposition. Let $g_0,\ldots ,g_r\in \Gamma (X,{\mathcal O}_X(-N\Gamma ))$ generate $\Gamma (X,{\mathcal O}_X(-N\Gamma ))$ as an R-module. Then $g_0,\ldots ,g_r$ induce a proper R-morphism $\phi :X\rightarrow {\mathbb P}_R^r$ such that $\phi ^*{\mathcal O}_{{\mathbb P}_R^r}(1)\cong {\mathcal O}_X(-N\Gamma )$ (by [20, Th. II.7.1 and Cor. II.4.8]). In fact, $\phi $ is projective, by [17, Prop. II.5.5(v)] or [34, Lem. 29.43.15 and Tag 01W7] and [34, Lem. 29.43.16(1) and Tag 01W7]. Let Z be the image of $\phi $ in ${\mathbb P}_R^r$ (which is closed since $\phi $ is proper), and let ${\mathcal O}_Z(1)={\mathcal O}_{{\mathbb P}^r_R}(1)\otimes {\mathcal O}_Z$ . Let $\overline \phi :X\rightarrow Z$ be the induced projective R-morphism. By [20, Cor. III.11.2], for $s\in {\mathbb Z}$ , the support of $R^1\overline \phi *{\mathcal O}_X(-s\Gamma )$ is contained in the finite set of closed points of Z which are the images of curves contracted by $\overline \phi $ (the prime exceptional divisors E of $\pi $ such that $(E\cdot -\Gamma )=0$ ). By [20, Th. II.5.19], $\Gamma (Z,R^1\overline \phi *{\mathcal O}_X(-s\Gamma ))$ is a finitely generated R-module. Since its support is the maximal ideal of R, the length of $\Gamma (Z,R^1\overline \phi *{\mathcal O}_X(-s\Gamma ))$ as an R-module is finite.

From the Leray spectral sequence, we obtain exact sequences (see [32, Th. 11.2]) for $m\in {\mathbb Z}$ ,

$$ \begin{align*}0\rightarrow H^1(Z,\overline \phi_*{\mathcal O}_X(-m\Gamma))\rightarrow H^1(X,{\mathcal O}_X(-m\Gamma)) \rightarrow H^0(Z,R^1\overline\phi_*{\mathcal O}_X(-m\Gamma)). \end{align*} $$

For $m\in {\mathbb N}$ , write $m=nN+s$ with $0\le s<N$ . Then ${\mathcal O}_X(-m\Gamma )\cong \overline \phi ^*{\mathcal O}_Z(n)\otimes {\mathcal O}_X(-s\Gamma )$ . Then, by the projection formula (see [20, Exer. III.8.3]), we obtain exact sequences for $n,s\in {\mathbb Z}$

(20) $$ \begin{align} \begin{array}{l} 0\rightarrow H^1(Z,{\mathcal O}_Z(n)\otimes \overline\phi_*{\mathcal O}_X(-s\Gamma)) \rightarrow H^1(X,\overline\phi^*{\mathcal O}_Z(n)\otimes{\mathcal O}_X(-s\Gamma))\\ \rightarrow H^0(Z,(R^1\overline\phi_*{\mathcal O}_X(-s\Gamma))\otimes{\mathcal O}_Z(n)). \end{array} \end{align} $$

Let

$$ \begin{align*}s_1=\max\{\lambda_R\Gamma(Z,R^1\overline\phi_*{\mathcal O}_X(-s\Gamma))\mid 0\le s<N\}. \end{align*} $$

We have that $H^1(Z,{\mathcal O}_Z(n)\otimes \overline \phi _*{\mathcal O}_X(-s\Gamma ))=0$ for all $0\le s<N$ and $n\gg 0$ (see [20, Th. III.5.2]). Let

$$ \begin{align*}s_2=\max\{\lambda_RH^1(Z,{\mathcal O}_Z(n) \otimes \overline\phi_*{\mathcal O}_X(-s\Gamma)) \mid 0\le s<N \mbox{ and }n\in {\mathbb N}\} \end{align*} $$

$s_2$ is finite by [20, Prop. III.8.5, Th. III.8.8, and Cor. III.11.2]. By (20), we have that $\lambda _RH^1(X,{\mathcal O}_X(-m\Gamma ))\le s_1+s_2$ for all $m\in {\mathbb N}$ .

Proposition 5.3. Let $\Delta $ be an effective anti-nef divisor on X. For $n\ge 0$ , let $B_n$ be the fixed component of $-n\Delta $ . Then there exists an effective exceptional divisor G on X such that $B_n\le G$ for all $n\in {\mathbb Z}_{>0}$ .

Proof. To prove the proposition, it suffices to prove it for it for some positive multiple d of $\Delta $ , since for $n\in {\mathbb N}$ , writing $n=md+s$ with $0\le s<d$ , we have $B_n\le B_{md}+B_s$ .

Write $-\Delta =\sum _{i=1}^t a_iF_i$ . Let

$$ \begin{align*}M_i=\{n\in {\mathbb N}\mid F_i\mbox{ is not a component of }B_n\}. \end{align*} $$

$M_i$ is a numerical semigroup, so if $M_i$ is nonzero, there exists $h_i\in {\mathbb Z}_{>0}$ such that for $n\gg 0$ , $n\in M_i$ if and only if $h_i$ divides n. Let

$$ \begin{align*}\mathcal B(D)=\{F_i\mid F_i\mbox{ is a component of } \ B_n \ \mbox{for infinitely many} \ n\}. \end{align*} $$

By Proposition 5.1, $F_i\in \mathcal B(D)$ implies $(F_i\cdot \Delta )=0$ and $F_i$ is exceptional for $\pi $ . After possibly reindexing the $F_i$ , we may assume that the support of $\mathcal B(D)$ is $\cup _{i=1}^sF_i$ , for some $s\le t$ . We have that $M_i=0$ or $h_i>1$ for $1\le i\le s$ . Thus, the support of $B_n$ is $\cup _{i=1}^sF_i$ if $n\gg 0$ and $h_i\nmid n$ for all i such that $1\le i\le s$ and $M_i$ is nonzero.

If we replace $\Delta $ with $n_0\Delta $ for some $n_0\gg 0$ , we have that the support of $B_1$ is $\mathcal B(D)$ . By Proposition 5.2, there exists $s_0\in {\mathbb N}$ such that the effective divisor $\Gamma =\Delta +B_1$ satisfies the condition that $h^1({\mathcal O}_X(-n\Gamma ))\le s_0$ for all $n\ge 1$ since $-\Gamma $ has no fixed component.

For a given $n\in {\mathbb Z}_{>0}$ , consider the following conditions on a divisor $Z_n$ .

1. (a) $n\Gamma \ge Z_n\ge n\Delta $ .

2. (b) $-Z_n$ has no fixed component.

3. (c) $h^1({\mathcal O}_X(-Z_n))\le s_0$ .

Let $C_n$ be a minimal element in the set of divisors satisfying (a), (b), and (c). Let $B_n'=C_n-n\Delta $ . Then $nB_1\ge B_n'\ge B_n$ (since $-n\Delta =-n\Gamma +nB_1=-C_n+B_n'$ and $C_n\le n\Gamma $ ). Thus, it suffices to show that the $B_n'$ are bounded from above.

For $1\le i\le s$ , we have short exact sequences

$$ \begin{align*}0\rightarrow O_X(-C_n)\rightarrow {\mathcal O}_X(-C_n+F_i)\rightarrow {\mathcal O}_X(-C_n+F_i)\otimes {\mathcal O}_{F_i}\rightarrow 0, \end{align*} $$

giving exact sequences

$$ \begin{align*}\begin{array}{l} 0\rightarrow H^0(X,{\mathcal O}_X(-C_n))\rightarrow H^0(X,{\mathcal O}_X(-C_n+F_i))\rightarrow H^0(F_i,{\mathcal O}_X(-C_n+F_i)\otimes {\mathcal O}_{F_i})\\ \rightarrow H^1(X,{\mathcal O}_X(-C_n))\rightarrow H^1(X,{\mathcal O}_X(-C_n+F_i)) \rightarrow H^1(F_i,{\mathcal O}_X(-C_n+F_i)\otimes {\mathcal O}_{F_i})\rightarrow 0. \end{array} \end{align*} $$

We will show that

(21) $$ \begin{align} -(C_n\cdot F_i)\le \mbox{max}\{s_0-(F_i^2)-1+p_a(F_i),2p_a(F_i)-2-(F_i^2),0\} \end{align} $$

for all n and $1\le i\le s$ .

First assume that $F_i$ is not a component of $B_n'$ . Then $(B_n'\cdot F_i)\ge 0$ . Since $(F_i\cdot \Delta )=0$ by Proposition 5.1, we have that $(C_n\cdot F_i)\ge 0$ and so (21) holds.

Now, assume that $F_i$ is a component of $B_n'$ . We have that either

(22) $$ \begin{align} H^0(X,{\mathcal O}_X(-C_n+F_i))=H^0(X,{\mathcal O}_X(-C_n)) \end{align} $$

or

(23) $$ \begin{align} h^1({\mathcal O}_X(-C_n+F_i))>s_0. \end{align} $$

If (22) holds, then $h^0({\mathcal O}_X(-C_n+F_i)\otimes O_{F_i})\le s_0$ . Thus,

$$ \begin{align*}s_0\ge h^0({\mathcal O}_X(-C_n+F_i)\otimes{\mathcal O}_{F_i})\ge ((-C_n+F_i)\cdot F_i)+1-p_a(F_i) \end{align*} $$

by the Riemann–Roch formula (6), and so (21) holds.

Suppose that (23) holds. Then $h^1(F_i,{\mathcal O}_X(-C_n+F_i)\otimes {\mathcal O}_{F_i})>0$ , and so

$$ \begin{align*}((-C_n+F_i)\cdot F_i)<2p_a(F_i)-2 \end{align*} $$

by (8). Thus, (21) holds.

For i with $1\le i\le s$ , let $\sigma _i=\mbox {max}\{s_0-(F_i^2)-1+p_a(F_i),2p_a(F_i)-2-(F_i^2),0\}$ . Since $(F_i\cdot \Delta )=0$ for $1\le i\le s$ by Proposition 5.1, and by (21), we have that

$$ \begin{align*}(B_n'\cdot F_i)=((C_n-n\Delta)\cdot F_i)=(C_n\cdot F_i)\ge -\sigma_i. \end{align*} $$

In particular, $\sigma _i\ge -(B_n'\cdot F_i)$ .

Since the intersection matrix $((F_i\cdot F_j))$ for $1\le i,j\le s$ is negative-definite, and thus is nonsingular, there exists a ${\mathbb Q}$ -divisor $\mathcal E=c_1F_1+\cdots +c_s F_s$ such that $(\mathcal E\cdot F_i)=-\sigma _i$ for $1\le i\le s$ . Then

$$ \begin{align*}((\mathcal E-B_n')\cdot F_i)=-\sigma_i-(B_n'\cdot F_i)\le 0 \end{align*} $$

for all i implies $\mathcal E\ge B_n'$ by [35, Lem. 7.1], since the intersection matrix is negative-definite. Thus, the $B_n'$ are bounded from above.

Corollary 5.4. Let $\Delta $ be an effective anti-nef ${\mathbb Q}$ -divisor on X. Let $B_n$ be the fixed component of $-\lceil n\Delta \rceil $ ; that is, the largest effective divisor on X such that

$$ \begin{align*}\Gamma(X,\mathcal O_X(-\lceil n\Delta\rceil ))=\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil -B_n)). \end{align*} $$

Then:

1. (1) The integral divisor $B_n$ has exceptional support for all $n\in {\mathbb N}$ .

2. (2) There exists an effective integral divisor G with exceptional support such that $B_n\le G$ for all $n\in {\mathbb Z}_{>0}$ .

Proof. Statement (1) follows from Lemma 2.2. If $\Delta $ is an integral divisor, then Statement (2) follows from Proposition 5.3.

Now, assume that $\Delta $ is a ${\mathbb Q}$ -divisor. Write $\Delta =\sum \frac {b_i}{d}F_i$ with $d\in {\mathbb Z}_{>0}$ and $b_i\in {\mathbb N}$ , where the $F_i$ are distinct prime divisors on X. Since $d\Delta $ is an integral divisor, there exists an effective integral divisor C with exceptional support such that $B_{nd}\le C$ for all $n\in {\mathbb N}$ . Let $n\in {\mathbb N}$ , and write $n=md-c$ with $m\in {\mathbb N}$ and $0\le c<d$ . Then ${\mathcal O}_X(-\lceil n\Delta \rceil )={\mathcal O}_X(-md\Delta +\lfloor c\Delta \rfloor )$ . Thus, $B_n\le B_{md}+\lfloor c\Delta \rceil \le C+d\Delta $ .

Lemma 5.5. Let $\{D_n\}$ with $n\ge 0$ be an infinite sequence of divisors on X, and let Z be an effective divisor on X. If the sequence $h^1({\mathcal O}_X(D_n))$ is bounded from above and if for each prime exceptional component E of $Z, \ (D_n\cdot E)$ is bounded from below, then $h^1({\mathcal O}_X(D_n+Z))$ is bounded from above.

Proof. By induction on the number of components of Z, we may assume that $h^1({\mathcal O}_X(D_n+Z-F))$ is bounded where F is a prime component of Z. We have a short exact sequence

$$ \begin{align*}0\rightarrow {\mathcal O}_X(-F)\rightarrow {\mathcal O}_X\rightarrow {\mathcal O}_{F}\rightarrow 0, \end{align*} $$

giving exact sequences

$$ \begin{align*}H^1(X,{\mathcal O}_X(D_n+Z-F))\rightarrow H^1(X,{\mathcal O}_X(D_n+Z))\rightarrow H^1(F,{\mathcal O}_{X}(D_n+Z)\otimes{\mathcal O}_{F}). \end{align*} $$

If F is exceptional, there exists $s\in {\mathbb Z}_{>0}$ such that $h^1(F,{\mathcal O}_X(D_n+Z)\otimes {\mathcal O}_{F})\le s$ for all $n\ge 0$ by Lemma 3.1, so $h^1({\mathcal O}_X(D_n+Z))$ is bounded from above. If F is not exceptional, then F is affine and so $H^1(F,{\mathcal O}_X(D_m+Z))\otimes {\mathcal O}_{F})=0$ for all m, so again $h^1({\mathcal O}_X(D_n+Z))$ is bounded from above.

Proposition 5.6. Let $\Delta $ be an effective anti-nef divisor on X. Then $h^1({\mathcal O}_X(-n\Delta ))$ is bounded for $n\in {\mathbb N}$ .

Proof. Let $C_n$ be the effective divisors of the proof of Proposition 5.3, so that $B_n'=C_n-n\Delta $ are effective divisors and there exists an effective divisor G with exceptional support such that $B_n'\le G$ for all $n\in {\mathbb N}$ . Since $-\Delta $ is nef, we have that $(-C_n\cdot E)$ is bounded from below for each prime exceptional component E of G. Furthermore, we have (by the proof of Proposition 5.3) that $h^1({\mathcal O}_X(-C_n))\le s_0$ for all $n\in {\mathbb N}$ . For each effective divisor $Z\le G$ , Proposition 5.5 gives us an upper bound for $h^1({\mathcal O}_X(-C_n+Z))$ over $n\in {\mathbb N}$ . The maximum of these bounds is an upper bound for $h^1({\mathcal O}_X(-n\Delta ))$ over $n\in {\mathbb N}$ .

Corollary 5.7. Let $\Delta $ be an effective anti-nef divisor on X, and let $\mathcal F$ be a coherent sheaf on X. Then $h^1({\mathcal O}_X(-n\Delta )\otimes \mathcal F)$ is bounded for $n\in {\mathbb N}$ .

Proof. There exists an effective anti-ample divisor A on X with exceptional support by Proposition 2.3. There exists $n_0\in {\mathbb Z}_{>0}$ such that $\mathcal F\otimes {\mathcal O}(-n_0A)$ is generated by global sections, so there is a surjection ${\mathcal O}_X^s\rightarrow \mathcal F\otimes {\mathcal O}_X(-n_0A)$ for some s, giving a short exact sequence of coherent sheaves

$$ \begin{align*}0\rightarrow \mathcal K\rightarrow {\mathcal O}_X(n_0A)^s\rightarrow \mathcal F\rightarrow 0 \end{align*} $$

and surjections

$$ \begin{align*} H^1(X,{\mathcal O}_X(-n\Delta+n_0A))^s\rightarrow H^1(X,{\mathcal O}_X(-n\Delta)\otimes\mathcal F). \end{align*} $$

Thus, $h^1({\mathcal O}_X(-n\Delta )\otimes \mathcal F)$ is bounded above for $n\in {\mathbb N}$ since $-\Delta $ is nef, and by Lemma 5.5 and Proposition 5.6.

6 Asymptotic properties of divisors on a resolution of singularities

We continue with the notation introduced in the introduction and in §2. We assume that R is a two-dimensional excellent normal local ring with quotient field K, maximal ideal $m_R$ , and residue field k, and that $\pi :X\rightarrow \mbox {Spec}(R)$ is a resolution of singularities such that the exceptional prime divisors $E_1,\ldots ,E_r$ of $\pi $ are all nonsingular.

As explained in the introduction, if F is prime divisor on X and $\alpha \in {\mathbb R}_{\ge 0}$ , then there is a valuation ideal $I(\nu _F)_{\alpha }=\{f\in R\mid\ \nu _F(f)\ge \alpha \}$ of R, where $\nu _F$ is the valuation of the discrete (rank 1) valuation ring ${\mathcal O}_{X,F}$ .

Proposition 6.1. Suppose that $\Delta _1\subset \Delta _2$ are effective anti-nef ${\mathbb Q}$ -divisors on X such that $\Delta _1\ne \Delta _2$ . Then there exists $n_0\in {\mathbb Z}_{>0}$ such that $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))\ne \Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))$ for all $n\ge n_0$ .

Proof. Write $\Delta _1=\sum a_iF_i$ and $\Delta _2=\sum b_iF_i$ where the $F_i$ are distinct prime divisors on X. We have $b_i\ge a_i$ for all i and $b_j>a_j$ for some j. If $F_j$ is not exceptional, then certainly $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))\ne \Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))$ for n sufficiently large by Lemma 2.2.

Now, suppose that $F_j$ is exceptional. By (2) of Lemma 5.4, there exists an effective exceptional divisor $H=\sum c_iF_i$ such that the fixed component $B_n$ of $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))$ satisfies $B_n\le H$ for all $n\in {\mathbb N}$ . Observe that $g\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))$ implies $\nu _j(g)\ge \lceil n b_j\rceil $ . By definition of $B_n$ , for $n\in {\mathbb Z}_{>0}$ , there exists $f_n\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))$ such that $(f_n)-\lceil n\Delta _1\rceil =A_n+B_n$ where $F_j$ is not a component of the effective divisor $A_n$ . Thus, $\nu _j(f_n)=\lceil na_j\rceil +\delta $ with $\delta \le c_j$ . We have that $n>\frac {c_j+1}{b_j-a_j}$ implies $\lceil na_j\rceil +\delta <\lceil nb_j\rceil $ . Thus, $\nu _j(f_n)<\lceil nb_j\rceil $ so that $f_n\not \in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))$ .

Corollary 6.2. Suppose that $\Delta _1\subset \Delta _2$ are effective anti-nef ${\mathbb Q}$ -divisors on X. Then the following are equivalent.

1. (1) $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))=\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))$ for infinitely many $n\in {\mathbb Z}_{>0}$ .

2. (2) $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))=\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _2\rceil ))$ for all $n\gg 0$ .

3. (3) $\Delta _1=\Delta _2$ .

Proof. Proposition 6.1 proves the essential implication (1) implies (3). The directions (3) implies (2) and (2) implies (1) are immediate.

Proposition 6.3. Let $\Delta =\sum _{i=1}^s a_iF_i$ be an effective anti-nef ${\mathbb Q}$ -divisor on X, and let E be a prime exceptional divisor on X. Then $E=F_j$ for some j with $a_j>0$ . The following are equivalent:

1. (1) There exists $n\in {\mathbb Z}_{>0}$ such that

   $$ \begin{align*}I(n\Delta)=\cap_{i=1}^s I(\nu_{F_i})_{na_i}\ne \cap_{i\ne j}I(\nu_{F_i})_{na_i}. \end{align*} $$

2. (2) There exists $n_0\in {\mathbb Z}_{>0}$ such that

   $$ \begin{align*}I(n\Delta)=\cap_{i=1}^s I(\nu_{F_i})_{na_i}\ne \cap_{i\ne j}I(\nu_{F_i})_{na_i} \end{align*} $$

   for all $n\ge n_0$ .

3. (3) $(\Delta \cdot F_j)<0$ .

Proof. It follows from Lemma 2.4 that $E=F_j$ for some j with $a_j>0$ .

Let $D_1=\sum _{i\ne j}a_iF_i$ , so that $D_1\le \Delta $ . Let $\Delta _1=D_1+B_1$ be the Zariski decomposition of $D_1$ . We have that $\Delta _1\le \Delta $ by Remark 4.2, and so $0\le \Delta -\Delta _1=a_jF_j-B_1$ so that $0\le B_1\le a_jF_j$ . Thus, $\Delta _1=\Delta -\lambda F_j$ with $0\le \lambda \le a_j$ .

If $\Delta _1\ne \Delta $ , then $\lambda>0$ , and so

(24) $$ \begin{align} (F_j\cdot\Delta)=(F_j\cdot\Delta_1)+\lambda(F_j^2)<0. \end{align} $$

If $\Delta _1=\Delta $ , then $B_1=a_jF_j$ . Since $a_j>0$ , we have that

(25) $$ \begin{align} 0=(\Delta_1\cdot F_j)=(\Delta\cdot F_j) \end{align} $$

by (2) of Lemma 4.1.

Suppose that (1) holds. Then $\Delta _1\ne \Delta $ so that $(F_j\cdot \Delta )<0$ by (24), so that (1) implies (3) holds. Certainly, (2) implies (1) is true, so we are reduced to proving (3) implies (2). Now, (3) implies $\Delta _1\ne \Delta $ by (24) and (25). If (2) does not hold, then there exist infinitely many $n\in {\mathbb Z}_{>0}$ such that $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))=\Gamma (X,{\mathcal O}_X(-\lceil n\Delta _1\rceil ))$ so that $\Delta _1=\Delta _2$ by Corollary 6.2, giving a contradiction.

Corollary 6.4. Let $\Delta =\sum _{i=1}^s a_iF_i$ be an effective anti-nef ${\mathbb Q}$ -divisor on X, and let E be a prime exceptional divisor on X so that $E=F_j$ for some j with $a_j>0$ . The following are equivalent:

1. (1) $I(n\Delta )=\cap _{i=1}^r I(\nu _{F_i})_{na_i}= \cap _{i\ne j}I(\nu _{F_i})_{na_i}$ for all $n\in {\mathbb Z}_{>0}$ .

2. (2) $(\Delta \cdot F_j)=0$ .

Corollary 6.5. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then the following are equivalent.

1. (1) There exists n such that $m_R\in \mbox {Ass}(R/I(n\Delta ))$ .

2. (2) There exists $n_0$ such that $m_R\in \mbox {Ass}(R/I(n\Delta ))$ for all $n\ge n_0$ .

3. (3) There exists a prime exceptional divisor E for $\pi $ such that $(\Delta \cdot E)<0$ .

Proof. Write $\Delta =\sum _{i=1}^sa_iF_i$ , so that $I(n\Delta )=\cap _{i=1}^sI(\nu _{F_i})_{na_i}$ . For a fixed n, we have that $m_R\in \mbox {Ass}(R/\cap _{i=1}^sI(\nu _{F_i})_{na_i})$ if and only if

$$ \begin{align*}\cap_{i=1}^sI(\nu_{F_i})_{na_i}\ne \cap_{F_i}I(\nu_{F_i})_{na_i}, \end{align*} $$

where the second intersection is over the $F_i$ which are not exceptional. This condition occurs if and only if there exists j such that $F_j$ is exceptional and

$$ \begin{align*}\cap_{i=1}^sI(\nu_{F_i})_{na_i}\ne \cap_{i\ne j}I(\nu_{F_i})_{na_i}. \end{align*} $$

Thus, by Proposition 6.3, the three conditions of the corollary are equivalent.

Let $\Delta =\sum _{i=1}^sa:iF_i$ be an effective and anti-nef ${\mathbb Q}$ -divisor on X. By Lemma 2.4, all prime exceptional divisors $E_1,\ldots ,E_r$ are in the support of $\Delta $ . After permuting the $F_i$ , we may assume that $F_i=E_i$ and $a_i>0$ for $1\le i\le r$ . We have that

$$ \begin{align*}R[\Delta]:= \bigoplus_{n\ge 0}\Gamma(X,{\mathcal O}_X(-\lceil n\Delta \rceil))=\bigoplus_{n\ge 0}\cap_{i=1}^sI(\nu_{F_i})_{na_i}. \end{align*} $$

Let $P_j=\bigoplus _{n\ge 0}\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil -E_j))$ for $1\le j\le r$ . We have that

(26) $$ \begin{align} \Gamma(X,{\mathcal O}_X(-E_j))=\{f\in R\mid \nu_{E_j}(f)>0\}=m_R \end{align} $$

for $1\le j\le r$ . Suppose that $f\in \Gamma (X,{\mathcal O}_X(-\lceil m\Delta \rceil ))$ and $g\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ are such that $fg\in \Gamma (X,{\mathcal O}_X(-\lceil (m+n)\Delta \rceil -E_j)$ . Then

$$ \begin{align*}\nu_{E_j}(f)+\nu_{E_j}(g)=\nu_{E_j}(fg)\ge (m+n)a_j+1 \end{align*} $$

implies $\nu _{E_j}(f)\ge ma_j+1$ or $\nu _{E_j}(g)\ge na_j+1$ so that $f\in \Gamma (X,{\mathcal O}_X(-\lceil m\Delta \rceil -E_j)$ or $g\in \Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil -E_j)$ . Thus, $P_j$ is a prime ideal in $R[\Delta ]$ .

If $f\in m_R$ , then $\nu _{E_j}(f)\ge 1$ for $1\le j\le r$ so that

(27) $$ \begin{align} m_RR[\Delta]\subset P_j. \end{align} $$

We have exact sequences

$$ \begin{align*}0\rightarrow P_j\rightarrow R[\Delta]\rightarrow \bigoplus_{n\ge 0}\Gamma(E_j,{\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_{E_j}). \end{align*} $$

Remark 6.6. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then $\dim R[\Delta ]/P_j=0$ if and only if $R[\Delta ]/P_j=R/m_R$ .

Proof. Suppose that for some $m>0$ there exists $f\in \Gamma (X,{\mathcal O}_X(-\lceil m\Delta \rceil ))$ such that its class $\overline f$ in $\Gamma (X,{\mathcal O}_X(-\lceil m\Delta \rceil ))/\Gamma (X,{\mathcal O}_X(-\lceil m\Delta \rceil -E_j))$ is nonzero. Then

$$ \begin{align*}\overline ft^m\in \sum_{n=0}^{\infty}\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))/\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-E_j))t^n=R[\Delta]/P_j \end{align*} $$

is nonzero. The element $\overline ft^m$ is not a unit since it is homogeneous of positive degree and it is not nilpotent since $R[\Delta ]/P_j$ is an integral domain. Thus, $\dim R[\Delta ]/P_j>0$ . Thus, by (26), $\dim R[\Delta ]/P_j=0$ implies $R[\Delta ]/P_j=R/m_R$ .

Proposition 6.7. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then

$$ \begin{align*}\sqrt{m_RR[\Delta]}=\cap_{i=1}^rP_i. \end{align*} $$

Proof. We have that $\sqrt {m_RR[\Delta ]}\subset \cap _{i=1}^rP_i$ by (27).

Let $h\in \cap _{i=1}^rP_i$ . We will show that $h^n\in m_RR[\Delta ]$ for some $n\in {\mathbb Z}_{>0}$ , which will establish the proposition. We may assume that h is homogeneous, so that

$$ \begin{align*}h\in \cap_{i=1}^r\Gamma(X,{\mathcal O}_X(-\lceil a\Delta\rceil-E_i))= \Gamma(X,{\mathcal O}_X(-\lceil a \Delta\rceil-E_1-\cdots-E_r) \end{align*} $$

for some $a\in {\mathbb N}$ . We must show that $h^n\in m_R\Gamma (X,{\mathcal O}_X(-\lceil an\Delta \rceil ))$ for some $n\in {\mathbb Z}_{>0}$ .

First, suppose that $a=0$ . We have that $\Gamma (X,{\mathcal O}_X(-E_1-\cdots -E_r)=m_R$ , so we already have that $h\in m_R\Gamma (X,{\mathcal O}_X)=m_R$ .

Now, suppose that $a>0$ . After replacing $\Delta $ with a positive multiple of $\Delta $ and h with a power of h, we may assume that $\Delta $ is an integral divisor and that $h\in \Gamma (X,{\mathcal O}_X(-\Delta -\sum _{i=1}^rE_i))$ . By Lemma 2.5, there exists an $m_R$ -primary ideal I in R such that X is the blowup of I, so that $X=\mbox {Proj}(\bigoplus _{n\ge 0}I^n)$ and $I{\mathcal O}_X={\mathcal O}_X(-C)$ is very ample, where C is an effective divisor whose support is the union of all exceptional prime divisors $E_1,\ldots , E_r$ . The graded ring $\bigoplus _{n\ge 0}\Gamma (X,I^n{\mathcal O}_X)$ is a finite $\bigoplus _{n\ge 0}I^n$ -module and there exists $n_0\in {\mathbb Z}_{>0}$ such that the R-ideal $\Gamma (X,I^n{\mathcal O}_X)=I^n$ for $n\ge n_0$ by Lemma 2.6. Since R and X are normal, $\Gamma (X,I^n{\mathcal O}_X)=\overline {I^n}$ for all $n\ge 0$ .

After possibly replacing I with a positive power of I, we may assume that $\Gamma (X,I^n{\mathcal O}_X)=I^n$ for all $n\in {\mathbb N}$ and that there exists an effective divisor $H\sim -C$ on X with no exceptional prime divisors in its support. Let $f\in \Gamma (X,{\mathcal O}_X(-C))=I$ be such that $(f)-C=H$ . We have a short exact sequence

$$ \begin{align*}0\rightarrow {\mathcal O}_X(C)\stackrel{f}{\rightarrow}{\mathcal O}_X\rightarrow {\mathcal O}_H\rightarrow 0. \end{align*} $$

There exists $\alpha \in {\mathbb Q}_{>0}$ such that $F:=\sum _{i=1}^rE_i-\alpha C\ge 0$ . There exists $e\in {\mathbb Z}_{>0}$ such that $e\alpha C$ is an integral divisor and so $eF$ is an integral divisor. Thus, for $n\in {\mathbb Z}_{>0}$ , we have that

$$ \begin{align*}\begin{array}{lll} h^{n2e}&\in& \Gamma(X,{\mathcal O}_X(-n2e\Delta-n2e(\sum_{i=1}^rE_i))=\Gamma(X,{\mathcal O}_X(-n2e\Delta-n2e\alpha C-n2eF)\\ &\subset& \Gamma(X,{\mathcal O}_X(-n2e\Delta-n2e\alpha C))=\Gamma(X,{\mathcal O}_X(-n2e(\Delta+\frac{\alpha}{2}C)-n2e\frac{\alpha}{2}C)). \end{array} \end{align*} $$

Now, the effective integral divisor $2e(\Delta +\frac {\alpha }{2}C)$ is anti-ample by Proposition 2.3, since $\Delta $ is anti-nef. Thus, there exists $n_0\in {\mathbb Z}_{>0}$ such that ${\mathcal O}_X(-n2e(\Delta +\frac {\alpha }{2}C))$ is generated by global sections for all $n\ge n_0$ . Let $\Gamma =n_02e(\Delta +\frac {\alpha }{2}C)$ . By the argument of the proof of Proposition 5.2, applying (17), there exists $N>0$ such that

$$ \begin{align*}&\Gamma(X,{\mathcal O}_X(-i\Gamma-jC))\\&\quad=\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C)\Gamma(X,{\mathcal O}_X(-C))+f\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C) \end{align*} $$

whenever $j\ge N$ and $i\ge 0$ . Since $f\in \Gamma (X,{\mathcal O}_X(-C))$ , we have that

$$ \begin{align*}\begin{array}{lll} \Gamma(X,{\mathcal O}_X(-i\Gamma-jC))&=&\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C))\Gamma(X,{\mathcal O}_X(-C))\\ &=&I\Gamma(X,{\mathcal O}_X(-i\Gamma-(j-1)C))\subset I\Gamma(X,{\mathcal O}_X(-in_02e\Delta)). \end{array} \end{align*} $$

Thus,

$$ \begin{align*}h^{nn_02e}\in \Gamma(X,{\mathcal O}_X(-n\Gamma-\frac{nn_02e\alpha}{2}C))\subset I\Gamma(X,{\mathcal O}_X(-nn_02e\Delta)) \subset m_R\Gamma(X,{\mathcal O}_X(-nn_02e\Delta))\end{align*} $$

whenever n is so large that $n\ge \frac {N}{n_0e\alpha }$ .

Corollary 6.8. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then

$$ \begin{align*}\dim R[\Delta]/m_RR[\Delta]=0 \end{align*} $$

if and only if the image of $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ in $\Gamma (E_j,{\mathcal O}_X(-\lceil n\Delta \rceil )\otimes {\mathcal O}_{E_j})$ is zero for $1\le j\le r$ and for all $n>0$ .

Proof. By Proposition 6.7, we have that $\dim R[\Delta ]/m_RR[\Delta ]=0$ if and only if $\dim R[\Delta ]/P_j=0$ for all j, and this second condition holds if and only if $R[\Delta ]/P_j=R/m_R$ for all j by Remark 6.6.

Proposition 6.9. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X and $E_j$ is a prime exceptional divisor for $\pi :X\rightarrow \mbox {Spec}(R)$ . Then:

1. (1) $\dim R[\Delta ]/P_j=2$ if $(\Delta \cdot E_j)<0$ .

2. (2) $\dim R[\Delta ]/P_j\le 1$ if $(\Delta \cdot E_j)=0$ .

Proof. Suppose that $(\Delta \cdot E_j)<0$ . We have short exact sequences

$$ \begin{align*}0\rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil-E_j)\rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil )\rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_{E_j}\rightarrow 0. \end{align*} $$

Taking global sections, we have short exact sequences

$$ \begin{align*}\begin{array}{l} 0\rightarrow \Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-E_j)) \rightarrow \Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))\\\rightarrow \Gamma(E_j,O_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_{E_j}) \rightarrow H^1(X,{\mathcal O}_X(-\lceil n\Delta\rceil-E_j)). \end{array} \end{align*} $$

There exists $d\in {\mathbb Z}_{>0}$ such that $d\Delta $ is an integral divisor. By Corollary 5.7, applied to $d\Delta $ and the coherent sheaves ${\mathcal O}_X(-\lceil s\Delta \rceil -E_j)$ for $0\le s<d$ , we have that

$$ \begin{align*}h^1(X,{\mathcal O}_X(-\lceil n\Delta\rceil -E_j)) \end{align*} $$

is bounded for positive n. Since $(-\Delta \cdot E_j)>0$ , we have (by the Riemann–Roch theorem (6)) that there exists $c'>0$ such that

$$ \begin{align*}h^0({\mathcal O}_X(-\lceil n\Delta\rceil )\otimes{\mathcal O}_{E_j})>c'n \end{align*} $$

for $n\gg 0$ . Thus, there exists $c>0$ such that the image $A_n:=\mbox {Im}(\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ in $B_n:=\Gamma (E_j,{\mathcal O}_X(-\lceil n\Delta \rceil )\otimes {\mathcal O}_{E_j})$ satisfies

(28) $$ \begin{align} \lambda_R(\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))/\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil -E_j))= \lambda_R(A_n)=\dim_kA_n\ge cn \end{align} $$

for $n\gg 0$ .

Let $A=\oplus _{n\ge 0}A_n$ . We have that $B_{0}$ is a finite field extension of $k=R/m_R=A_0$ . Now, ${\mathcal O}_X(-d\Delta )\otimes {\mathcal O}_{E_j}$ is ample on the projective curve $E_j$ , so there exists $e\in {\mathbb Z}_{>0}$ which is divisible by d such that ${\mathcal O}_X(-e\Delta )\otimes {\mathcal O}_{E_j}$ is very ample and $\overline B=\oplus _{m\ge 0}B_{me}$ is a finitely generated $B_{0}$ -algebra which is generated by its terms of the lowest positive degree $me$ (see [20, Th. II.5.19 and Exer. II.5.14]). Thus, $\overline B$ is the coordinate ring of a projective embedding of the curve $E_j$ in a projective space over $B_0$ , determined by a $B_0$ -basis of $\Gamma (E_j,{\mathcal O}_X(-e\Delta )\otimes {\mathcal O}_{E_j})$ . Thus, $\overline B$ has dimension 2. Let $\overline A=\bigoplus _{m\ge 0}A_{me}$ .

By (28), for $n\gg 0$ , there exists $F\in A_{ne}$ such that $0\ne F$ . The ring $\overline B_{(F)}$ of elements of degree zero in the localization $\overline B_F$ is such that $\mbox {Spec}(\overline B_{(F)})$ is the affine variety $E_j\setminus V(F)$ , with maximal ideals in $\overline B_{(F)}$ corresponding to height 1 homogeneous prime ideals in $\mbox {Proj}(\overline B)$ which do not contain F (by [20, Prop. II.2.5]). Thus, there exists a homogeneous height 1 prime ideal $Q=\oplus _{n> 0}Q_{ne}$ in $\overline B$ which does not contain F.

Let $P=\overline A\cap Q$ , where $P=\oplus _{n> 0}P_{ne}$ with $P_{ne}=Q_{ne}\cap A_{ne}$ . $\dim \overline B/Q=1$ implies that there exists $d\in {\mathbb Z}_{>0}$ such that $\dim _k(B_{ne}/Q_{ne})<d$ for all n (by [5, Th. 4.1.3]). Thus, by (28), we have that $P\ne 0$ . P is not the graded maximal ideal $\oplus _{n\ge 0}A_{ne}$ of $\overline A$ since $F\not \in P$ .

We have constructed a chain of distinct homogeneous prime ideals $0\subset P\subset \bigoplus _{n>0}A_{ne}$ in $\overline A$ and thus $\overline A$ has dimension $\ge \ 2$ . The extension $\overline A\rightarrow A$ is integral, so $\dim A\ge 2$ by the going up theorem (see [2, Th. 5.11]). We have that $m_R\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))\subset \Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil -E_j))$ for all $n\ge 0$ by (27). We thus have a surjection $R[\Delta ]/m_RR[\Delta ]\rightarrow A$ and so $\dim A\le \dim R[\Delta ]/m_RR[\Delta ]$ . However, $\dim R[\Delta ]/m_RR[\Delta ]\le 2$ by [14, Lem. 3.6], so that $\dim A=2$ .

Now, suppose that $(\Delta \cdot E_j)=0$ . Let $B_n=\Gamma (E_j,{\mathcal O}_X(-\lceil n\Delta \rceil )\otimes {\mathcal O}_{E_j})$ , and let $A_n$ be the natural image of $\Gamma (X,{\mathcal O}_X(-\lceil n\Delta \rceil ))$ in $B_n$ . We have that $A_0\cong R/m_R=k$ and $B_0$ is a finite field extension of k. Let $A=\sum _{n\ge 0}A_nt^n$ where t is an indeterminate. We have that $A\cong R[\Delta ]/P_j$ .

By the Riemann–Roch theorem (6) and Lemma 3.1, there exists $d>0$ such that $\dim _k(B_n)<d$ for all $n\in {\mathbb N}$ .

For $a\in {\mathbb Z}_{>0}$ , define ${}_aA=\sum _{n\ge 0}{}_aA_nt^n$ to be the graded subring of A defined by ${}_aA=k[A_1t,A_2t^2,\ldots ,A_at^a]$ . The ring ${}_aA$ is a finitely generated graded k-algebra. For fixed a, there exists $e\in {\mathbb Z}_{>0}$ such that ${}_aA^{(e)}=\sum _{n\ge 0}{}_aA_{en}t^{en}$ is generated in degree e (as follows from [4, Props. III.3.2 and III.3.3 on pages 158 and 159]). Since ${}_aA$ is a finitely generated ${}_aA^{(e)}$ -module, we have that $\dim {}_aA=\dim {}_aA^{(e)}$ . Since $\dim _k {}_aA^{(e)}_n<d$ for all $n\in {\mathbb N}$ , we have that $\dim {}_aA\le 1$ for all $a\in {\mathbb Z}_{>0}$ by [5, Th. 4.1.3]. Suppose that $Q_0\subset Q_1\subset \cdots \subset Q_s$ is a chain of distinct prime ideals in A. Since $\cup _{a\ge 0}({}_aA)=A$ , for all $a\gg 0$ , $Q_0\cap {}_aA\subset Q_1\cap {}_aA\subset \cdots \subset Q_s\cap {}_aA$ is a chain of distinct prime ideals in A. Thus, $\dim A\le 1$ .

Corollary 6.10. Suppose that $\Delta $ is an effective anti-nef ${\mathbb Q}$ -divisor on X. Then

$$ \begin{align*}\dim R[\Delta]/m_RR[\Delta]=2 \end{align*} $$

if and only if there exists an exceptional prime divisor E of $\pi $ such that $(\Delta \cdot E)<0$ .

Proof. This follows from Propositions 6.7 and 6.9.

7 Analytic spread of divisorial filtrations

Theorem 7.1 is a generalization to (not necessarily Noetherian) divisorial filtrations on a two-dimensional normal local ring of a theorem of McAdam, for filtrations of powers of ideals, in [27] and [33, Th. 5.4.6]. We recall the exact statement of McAdam’s theorem in Theorem 1.3 of the introduction. The concept of a divisorial filtration $\mathcal I(D)=\{I(nD)\}$ is defined in the introduction.

Theorem 7.1. Let R be a two-dimensional normal excellent local ring. The following are equivalent for a ${\mathbb Q}$ -divisorial filtration $\mathcal I(D)$ on R.

1. (1) The analytic spread $\ell (\mathcal I(D))=\dim R[D]/m_RR[D]=2$ .

2. (2) $m_R\in \mbox {Ass}(R/I(nD))$ for some n.

3. (3) There exists $n_0\in {\mathbb Z}_{>0}$ such that $m_R\in \mbox {Ass}(R/I(nD))$ for all $n\ge n_0$ .

Proof. Let $\pi :X\rightarrow \mbox {Spec}(R)$ be a resolution of singularities such that $D=\sum _{i=1}^s a_iF_i$ for some prime divisors $F_i$ on X and the exceptional divisors $E_1,\ldots , E_r$ of $\pi $ are nonsingular. Let $\Delta =D+B$ be the Zariski decomposition of D on X, so that $\mathcal I(D)=\mathcal I(\Delta )$ and $R[D]=R[\Delta ]$ (by Lemma 4.3). Then this theorem follows from Corollaries 6.5 and 6.10.

Corollary 7.2. Let R be a two-dimensional normal excellent local ring, and let $\mathcal I(D)$ be a ${\mathbb Q}$ -divisorial filtration on R. Then $\dim R[D]/m_RR[D]\le 1$ if and only if there exist height 1 prime ideals $Q_1,\ldots , Q_s$ in R and $b_1,\ldots ,b_s\in {\mathbb Q}_{>0}$ such that $I(nD)= Q_1^{(\lceil nb_1\rceil )}\cap \cdots \cap Q_s^{(\lceil nb_s\rceil )}$ for all $n\in {\mathbb N}$ .

Proof. We have that $I(nD)= Q_1^{(\lceil nb_1\rceil )}\cap \cdots \cap Q_s^{(\lceil nb_s\rceil )}$ for all $n\in {\mathbb N}$ if and only if $m_R\not \in \mbox {Ass}(R/I(nD))$ for all n which holds if and only if $\dim R[D]/m_RR[D]\le 1$ by Theorem 7.1.

Example 7.3. There exists a ${\mathbb Q}$ -divisorial filtration $\mathcal I(D)$ on a two-dimensional normal excellent local ring R such that the analytic spread $\ell (\mathcal I(D))=0$ and height

$$ \begin{align*}\mathrm{ht}(\mathcal I(D))=\mathrm{ht}(I(D))=1, \end{align*} $$

giving an example where $\mathrm {ht}(\mathcal I(D))>\ell (\mathcal I(D))$ . The Rees algebra of the example is a Non-Noetherian symbolic algebra $R[D]=\sum _{n\ge 0}Q_1^{(n)}\cap Q_2^{(n)}\cap Q_3^{(n)}$ where $Q_1, Q_2, Q_3$ are height 1 prime ideals in R.

Proof. Let k be an algebraically closed field, and let F be an irreducible cubic form in the polynomial ring $k[x,y,z]$ such that $E=\mbox {Proj}(k[x,y,z]/(F))$ is an elliptic curve. Let $R=k[[x,y,z]]/(F)$ , a complete, normal excellent local ring of dimension 2 with maximal ideal $m_R=(x,y,z)$ . Let $\pi :X\rightarrow \mbox {Spec}(R)$ be the blowup of the maximal ideal $m_R$ of R. X is nonsingular with $\pi ^{-1}(m_R)\cong E$ , $m_R{\mathcal O}_X={\mathcal O}_X(-E)$ , ${\mathcal O}_X(-E)\otimes {\mathcal O}_E\cong {\mathcal O}_E(1)$ and ${(E^2)=-3}$ . We have that ${\mathcal O}_X(-E)\otimes {\mathcal O}_E\cong {\mathcal O}_E(q_1+q_2+q_3)$ for some closed points $q_1,q_2,q_3\in E$ . Let $p_1,p_2,p_3 \in E$ be distinct closed points on E such that the degree 0 invertible sheaf $\mathcal L={\mathcal O}_E(q_1+q_2+q_3-p_1-p_2-p_3)$ has infinite order in $\mbox {Pic}^0(X)$ . Then $h^0(\mathcal L^n)=0$ for all $n\in {\mathbb Z}$ . In each regular local ring ${\mathcal O}_{X,p_i}$ , let $u_i,v_i$ be a regular system of parameters such that $u_i=0$ is a local equation of E at $p_i$ . Let $F_i$ be the Zariski closure of $v_i=0$ in X, which is an integral curve. Let $\pi (F_i)=Q_i\in \mbox {Spec}(R)$ . $R/Q_i$ is Henselian since it is complete, so by [28, Th. 4.2, p. 32], we have that E intersects the integral curve $F_i$ only at the point $p_i$ . $F_i$ intersects E transversally at $p_i$ so that $(E\cdot F_i)=1$ . Let $D=F_1+F_2+F_3$ . The Zariski decomposition of D is $\Delta =D+E$ . We have that ${\mathcal O}_X(-n\Delta )\otimes {\mathcal O}_E\cong \mathcal L^n$ for all n. Thus, $\Gamma (X,{\mathcal O}_X(-n\Delta -E))=\Gamma (X,{\mathcal O}_X(-n\Delta ))$ for all $n\in {\mathbb Z}_{>0}$ , and so by Proposition 6.7,

$$ \begin{align*}R[\Delta]/\sqrt{m_RR[\Delta]}=\bigoplus_{n\ge 0}\Gamma(X,{\mathcal O}_X(-n\Delta))/\Gamma(X,{\mathcal O}_X(-n\Delta-E)) =R/m_R=k. \end{align*} $$

Thus,

$$ \begin{align*}\dim R[\Delta]/m_RR[\Delta]=\dim R[\Delta]/\sqrt{m_RR[\Delta]}=0. \end{align*} $$

Since $0=\ell (\mathcal I(D))<1=\mbox {ht}(\mathcal I(D))$ , we have that $R[D]$ is Non-Noetherian (by [12, Prop. 3.7]).

8 The Hilbert function of $R[D]/m_RR[D]$

Theorem 8.1. Suppose that R is a two-dimensional normal excellent local ring and $\mathcal I(D)$ is a ${\mathbb Q}$ -divisorial filtration on R. Then there exist a nonnegative rational number $\alpha $ and a bounded function $\sigma :{\mathbb N}\rightarrow {\mathbb Q}$ such that

$$ \begin{align*}\lambda_R(I(nD)/m_RI(nD))=\lambda_R((R[D]/m_RR[D])_n)=n\alpha+\sigma(n) \end{align*} $$

for $n\in {\mathbb N}$ . The constant $\alpha $ is positive if and only if $\dim (R[D]/m_RR[D])=2$ .

The function $\sigma $ is bounded from both above and below. The proof gives an explicit calculation of the constant $\alpha $ in terms of the intersection theory of a suitable resolution of singularities in equation (35). The constant $\alpha $ is a nonnegative integer if $\Delta $ is an integral divisor in the Zariski decomposition $D=\Delta +B$ .

Proof. There exists a resolution of singularities $\pi :X\rightarrow \mbox {Spec}(R)$ such that D is an effective ${\mathbb Q}$ -divisor on X, $m_R{\mathcal O}_X$ is invertible, and the prime exceptional divisors $E_1,\ldots , E_r$ of X are all nonsingular. Let G be the effective exceptional divisor such that $m_R{\mathcal O}_X={\mathcal O}_X(-G)$ . Let $\Delta =D+B$ be the Zariski decomposition of D on X. There exists $d\in {\mathbb Z}_{>0}$ such that $d\Delta $ is an integral divisor.

Suppose that the ideal $m_R$ is generated by $f_1,\ldots ,f_b$ . We have an induced short exact sequence of coherent sheaves on X

$$ \begin{align*}0\rightarrow \mathcal K\rightarrow {\mathcal O}_X^b\rightarrow m_R{\mathcal O}_X\rightarrow 0. \end{align*} $$

Tensoring with ${\mathcal O}_X(-\lceil n\Delta \rceil )$ and taking global sections, we have short exact sequences

$$ \begin{align*}0\rightarrow m_R\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil)\rightarrow \Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-G))\rightarrow H^1(X,\mathcal K\otimes {\mathcal O}_X(-\lceil n\Delta\rceil)). \end{align*} $$

Thus, there exists $c_1\in {\mathbb Z}_{>0}$ such that

(29) $$ \begin{align} \lambda_R(\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-G))/m_R\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))\le c_1 \end{align} $$

for all $n\in {\mathbb N}$ by Corollary 5.7, applied to the effective anti-nef divisor $d\Delta $ and the coherent sheaves $\mathcal F=\mathcal K\otimes {\mathcal O}_X(-\lceil s\Delta \rceil )$ for $0\le s<d$ . From the short exact sequences

$$ \begin{align*}0\rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil-G)\rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil) \rightarrow {\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_G\rightarrow 0, \end{align*} $$

we have inclusions for $n\in {\mathbb N}$

$$ \begin{align*}\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil))/\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil-G))\rightarrow \Gamma(G,{\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_G), \end{align*} $$

and by Corollary 5.7, there exists $c_2\in {\mathbb Z}_{>0}$ such that

(30) $$ \begin{align} |\lambda_R(\Gamma(G,{\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_G)) -\lambda_R(\Gamma(X,{\mathcal O})X(-\lceil n\Delta\rceil))/\Gamma(X,{\mathcal O}_X(-\lceil n\Delta\rceil)))| \le c_2. \end{align} $$

We are reduced to computing $h^0({\mathcal O}_X(-\lceil n\Delta \rceil )\otimes {\mathcal O}_G)$ for $n\in {\mathbb N}$ . Write $G=\sum _{i=1}^r a_iE_i$ with $a_i\in {\mathbb Z}_{>0}$ .

Let $e=\sum _{i=1}^ra_i$ . There exists a function $\tau :\{1,\ldots ,e\}\rightarrow \{1,\ldots ,r\}$ such that letting $C_1=E_{\tau (1)}$ and $C_{j+1}=C_j+E_{\tau (j+1)}$ for $1\le j<e$ , we have that $C_e=G$ . We have short exact sequences

(31) $$ \begin{align} 0\rightarrow {\mathcal O}_X(-C_j)\otimes{\mathcal O}_{E_{\tau(j+1)}}\rightarrow {\mathcal O}_{C_{j+1}}\rightarrow {\mathcal O}_{C_j}\rightarrow 0 \end{align} $$

for $1\le j<e$ . The cohomology groups $h^1({\mathcal O}_X(-\lceil n\Delta \rceil -mE_j)\otimes {\mathcal O}_{E_{\tau (j+1)}})$ are bounded for $1\le j<e$ and $n\in {\mathbb N}$ by Lemma 3.1. Let

$$ \begin{align*}f=\max\{h^1({\mathcal O}_X(-\lceil n\Delta\rceil -mE_j)\otimes{\mathcal O}_{E_{\tau(j+1)}}) \mid 1\le j<e\mbox{ and }n\in {\mathbb N}\}. \end{align*} $$

Tensoring the sequences (31) with ${\mathcal O}_X(-\lceil n\Delta \rceil )$ and taking cohomology, we find that

(32) $$ \begin{align} |h^0({\mathcal O}_X(-\lceil n\Delta\rceil)\otimes {\mathcal O}_{C_{j+1}}) -h^0({\mathcal O}_X(-\lceil n\Delta\rceil)\otimes {\mathcal O}_{C_j})-h^0({\mathcal O}_X(-\lceil n\Delta\rceil -C_j)\otimes{\mathcal O}_{E_{\tau(j+1)}})|\le f \end{align} $$

for $1\le j<e$ and $n\in {\mathbb N}$ . Setting $C_0=0$ , we have that there exists $\lambda \in {\mathbb Z}_{>0}$ such that

(33) $$ \begin{align} |h^0(X,{\mathcal O}_X(-\lceil n\Delta\rceil)\otimes{\mathcal O}_G)-\sum_{i=0}^{e-1}h^0(X,{\mathcal O}_X(-\lceil n\Delta\rceil-C_i)\otimes{\mathcal O}_{E_{\tau(i+1)}})|<\lambda \end{align} $$

for all $n\in {\mathbb N}$ . Writing $n=md+s$ with $0\le s<d$ , we have

$$ \begin{align*} h^0({\mathcal O}_X(-\lceil n\Delta\rceil-C_j)\otimes {\mathcal O}_{E_{\tau(j+1)}}) =h^0({\mathcal O}_X(-md\Delta-\lceil s\Delta\rceil-C_j)\otimes{\mathcal O}_{E_{\tau(j+1)}}). \end{align*} $$

By Lemma 3.1 and the Riemann–Roch theorem (6), there exists $g\in {\mathbb Z}_{>0}$ such that

(34) $$ \begin{align} |h^0({\mathcal O}_X(-md\Delta-\lceil s\Delta\rceil -C_j)\otimes {\mathcal O}_{E_{\tau(j+1)}})-md(-\Delta\cdot E_{\tau(j+1)})| \le g \end{align} $$

for $1\le j <e$ and $m\in {\mathbb N}$ . Thus, the theorem holds with

(35) $$ \begin{align} \alpha=(-\Delta\cdot G). \end{align} $$