Good ideals and p g -ideals in two-dimensional normal singularities
aa r X i v : . [ m a t h . A C ] A ug GOOD IDEALS AND p g -IDEALS IN TWO-DIMENSIONALNORMAL SINGULARITIES TOMOHIRO OKUMA, KEI-ICHI WATANABE, AND KEN-ICHI YOSHIDA
Abstract.
In this paper, we introduce the notion of p g -ideals and p g -cycles,which inherits nice properties of integrally closed ideals on rational singularities.As an application, we prove an existence of good ideals for two-dimensional Goren-stein normal local rings. Moreover, we classify all Ulrich ideals for two-dimensionalsimple elliptic singularities. Introduction
In a two-dimensional rational singularity, Lipman showed in [14] that every in-tegrally closed ideal is “stable” in the sense that I = IQ holds for every minimalreduction Q of I . He also shows that if I, J are integrally closed ideals, then theproduct IJ is also integrally closed. (Later, Cutkosky [3] showed that in a two-dimensional normal local ring A , if I is integrally closed for every integrally closedideal I , then A is a rational singularity. ) These facts play very important role tostudy ideal theory on a two-dimensional rational singularity.On the other hand, as far as the authors know, almost nothing was done concern-ing ideal theory of non-rational singularities.Let ( A, m ) be a normal local ring of dimension 2 and f : X → Spec A be aresolution of singularity of A . Then p g ( A ) = ℓ A ( H ( X, O X )) is an important in-variant of A (here we denote by ℓ A ( M ) the length of an A module M with finitelength) and called the “geometric genus” of A . A rational singularity is character-ized by p g ( A ) = 0 and A is a “minimally elliptic singularity” if A is Gorenstein and p g ( A ) = 1.Now, take an integrally closed m -primary ideal I . Then I has a resolution f : X → Spec A with I O X invertible. In this case, I O X = O X ( − Z ) for some “anti-nef”cycle Z and we denote I = I Z . The important fact is that ℓ A ( H ( X, O X ( − Z ))) playsan important role for the property of I Z . We can show that ℓ A ( H ( X, O X ( − Z ))) ≤ p g ( A ) for every anti-nef cycle Z such that O X ( − Z ) has no fixed component. Wecall Z a p g -cycle and I Z a p g -ideal if we have ℓ A ( H ( X, O X ( − Z ))) = p g ( A ).A surprising fact is that the class of p g -ideal inherits nice properties of integrallyclosed ideals of rational singularities (in a rational singularity, every integrally closedideal is a p g -ideal by our definition). Namely, if I Z is a p g -ideal, then I Z is stableand if Z, Z ′ are p g -cycles, then I Z I Z ′ is integrally closed and also a p g -ideal. Theidea of this paper is to develop ideal theory for normal two-dimensional local ring Mathematics Subject Classification.
Primary 13A35; Secondary 14B05, 14J17.
Key words and phrases. good ideal, Ulrich ideal, p g -cycle, Gorenstein ring.This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) GrantNumbers 23540068, 23540059, 25400050. f given p g and to investigate what difference causes the difference of p g to the idealtheory of the ring.We apply the notion of p g -ideals to show existence of good ideals on any two-dimensional normal Gorenstein ring. The notion of good ideals was defined by S.Goto and S. Iai in [5]. Definition 1.1 (Goto–Iai–Watanabe [5]) . Let ( A, m ) be a Cohen-Macaulay localring, I be an m -primary ideal of A and Q a minimal reduction of I . We say I is agood ideal if it satisfies the following conditions:(1) I = IQ .(2) Q : I = I .Let us explain the organization of the paper. In this paper, the ring is a two-dimensional normal local ring containing an algebraically closed field. In Section 2,we prepare the notions and terminologies which we need later (e.g. minimally ellipticsingularity, good ideals, Ulrich ideals and so on). Furthermore, we give fundamentaltools in this paper: Propositions 2.5, 2.6 and Vanishing theorem (Theorem 2.7),Kato’s Riemann-Roch Theorem (Theorem 2.8).In Section 3, we introduce the notion of p g -cycles and p g -ideals; see Theorem 3.1,Definition 3.2 and Lemma 3.4. Note that every anti-nef cycle in a rational singularityis a p g -cycle in our sense. Moreover, p g -cycles enjoy nice properties: e.g. the sum of p g -cycles is always a p g -cycle (see Theorem 3.5).In Section 4, we prove an existence of p g -ideals for any two-dimensional normallocal ring. As an application, we prove the existence of good ideals as the maintheorem in this paper. Theorem 1.2 (See Theorem 4.1) . Let ( A, m ) be a two-dimensional normal localring. Then :(1) There exists a resolution on which p g -cycles exist. (2) If A is non-regular Gorenstein, then it has a good ideal. In Section 5, we prove that an m -primary ideal I of a two-dimensional rationalsingularity A is a good ideal if and only if it is an integrally closed ideal representedon the minimal resolution, which is a generalization of [5, Theorem 7.8] for non-Gorenstein case.In Section 6, we evaluate the number of minimal generators of integrally closedideals which is represented by some anti-nef cycle in terms of intersection numbersof related cycles.In Section 7, we investigate Ulrich ideals (3-generated good ideals) of minimallyelliptic singularities. For instance, we prove that there exist no Ulrich ideals forany minimally elliptic singularity of degree e ≥
5; see Theorem 7.7. Moreover, weclassify all Ulrich ideals for simple elliptic singularities (Theorem 7.10).2.
Preliminaries
Throughout this paper, let A be an excellent normal local ring of dimension2 with the unique maximal ideal m such that A contains an algebraically closedfield k ∼ = A/ m unless otherwise specified. Let f : X → Spec A be a resolution ofsingularities with exceptional divisor E = f − ( m ). .1. Cycle.
A divisor supported in E is called a cycle . Let E = S ri =1 E i be thedecomposition into irreducible components of E . A divisor D is said to be nef if theintersection numbers DE i are nonnegative for all E i ; D is said to be anti-nef if − D is nef. If DE i = 0 for all E i , then we say that D is numerically trivial and write D ≡
0. Since the intersection matrix ( E i E j ) is negative definite, if a cycle Z = 0is anti-nef, then Z ≥ E . The resolution f : X → Spec A is said to be minimal if X contains no ( − C (i.e., C ∼ = P , C = − Reduction, Multiplicity.
Let I be an m -primary ideal of A . Then the Hilbertfunction ℓ A ( A/I n +1 ) is a polynomial for sufficiently large n . That is, there exists apolynomial P I ( n ) of the form e ( I ) (cid:18) n + 22 (cid:19) − e ( I ) (cid:18) n + 11 (cid:19) + e ( I )such that ℓ A ( A/I n +1 ) = P I ( n ) for n ≫
0. Then e ( I ), e ( I ) and e ( I ) are integersand e ( I ) is called the multiplicity of I . On the other hand, we can take a parameterideal Q = ( a, b ) so that I r +1 = QI r for some integer r ≥
0. Such an ideal Q is calleda minimal reduction of I . Then we have e ( I ) = e ( Q ) = ℓ A ( A/Q ).2.3.
Integrally closed ideal.
Let I denote the integral closure of I , that is, I isan ideal which consists of all solutions z for some equation with coefficients c i ∈ I i : Z n + c Z n − + · · · + c n − Z + c n = 0. Then I ⊆ I ⊆ √ I .For any cycle Z on X , we write(2.1) I Z = H ( O X ( − Z )) . Since A = H ( O X ), I Z is an m -primary integrally closed ideal if Z >
0. An m -primary ideal I is said to be represented on X if the ideal sheaf I O X is invertibleand I = H ( I O X ). If I is represented on X , there exists an anti-nef cycle Z such that I O X = O X ( − Z ); I is also said to be represented by Z . Note that I is representedon some resolution if and only if it is integrally closed (cf. [14]).Note that if I = I Z and O X ( − Z ) is generated, then e ( I ) = − Z and I n = I nZ .2.4. Geometric genus, Singularity.
When the cohomology group H i ( F ) is an A -module, we denote by h i ( F ) the length ℓ A ( H i ( F )). It is known that h ( O X ) isindependent of the choice of the resolution. The invariant p g ( A ) := h ( O X ) is calledthe geometric genus of A . Definition 2.1 ( Rational singularity, Elliptic singularity ) . A ring A is said tobe a rational singularity (resp. a minimally elliptic singularity ) if p g ( A ) = 0 (resp. A is Gorenstein and p g ( A ) = 1).Assume that A is a minimally elliptic singularity, and let Z f be the fundamentalcycle. Then e = − Z f is called the degree of A . It is known that e ( m ) = max { , e } , ℓ A ( m / m ) = max { e, } and that A is a complete intersection if and only if e ≤ A is said to be a simple elliptic singularity of degree e if the exceptional set E of the minimal resolution of Spec A is a nonsingular elliptic curve with E = − e . very toric singularity and quotient singularity are rational singularities. Forinstance, k [[ x, y, z ]] / ( x a + y b + z c ) is rational if and only if 1 /a + 1 /b + 1 /c >
1. Notethat any simple elliptic singularity is a minimally elliptic singularity.
Example 2.2.
Let k be an algebraically closed field of characteristic zero or p ≥ k [[ x, y, z ]] / ( x + y + z ) is a simple elliptic singularity ofdegree 1.(2) A hypersurface k [[ x, y, z ]] / ( x + y + z ) is a simple elliptic singularity ofdegree 2.(3) A hypersurface k [[ x, y, z ]] / ( x + y + z ) is a simple elliptic singularity ofdegree 3.(4) A complete intersection ring k [[ x, y, z, w ]] / ( y − xz, w − yz − x ) is a simpleelliptic singularity of degree 4.2.5. Good ideal, Ulrich ideal.
In this subsection, let A be a Cohen-Macaulaylocal ring of any dimension d . Definition 2.3 ( Good ideal, Ulrich ideal ) . Let I be an m -primary ideal. Then:(1) I is called a stable ideal if I = QI for some minimal reduction Q of I .(2) I is called a good ideal if I is stable and Q : I = I for some minimal reduction Q of I .(3) I is called an Ulrich ideal if I is stable and I/I is a free A/I -module.Now assume that I is stable. Then I ⊂ Q : I and I ⊂ Q for any minimalreduction Q of I . By the characterization of core of ideals, a goodness of I isequivalent to the condition core( I ) = I (see [2, Example 3.1]). Recall that core( I )is the intersection of all minimal reductions of I .If we assume that A is a Gorenstein ring, then by duality theorem, we have ℓ A ( A/I ) = ℓ A ( Q : I/Q ) . Hence in this case, under the condition I = QI , I is a good ideal if and only if2 · ℓ A ( A/I ) = e ( I ) (see [5]). Moreover, I is an Ulrich ideal if and only if I is a goodideal with µ A ( I ) = d + 1, where µ A ( I ) denotes the cardinality of a minimal set ofgenerators of I (see [6]). So, in this case, Ulrich ideals are typical examples of goodideals.We note a simple but useful lemma for good ideals. Lemma 2.4.
Let I ′ be an ideal containing I and integral over I and assume that I ′ = QI ′ holds. Then if I is a good ideal, then I = I ′ . In particular, if A is atwo-dimensional rational singularity and I is a good ideal of A , then I is integrallyclosed.Proof. Since I ′ ⊂ Q : I ′ , we have I ′ ⊂ Q : I ′ ⊂ Q : I = I . Hence I = I ′ . (cid:3) Fundamental short exact sequences.
We say that O X ( − Z ) has no fixedcomponent if H ( O X ( − Z )) = H ( O X ( − Z − E i )) for every E i ⊂ E , i.e., the baselocus of the linear system H ( O X ( − Z )) does not contain any component of E .Suppose that O X ( − Z ) has no fixed component and h ∈ I Z a general element.Then we obtain the following exact sequence:(2.2) 0 → O X × h −→ O Z ( − Z ) → C → , here C is supported on the strict transform of the curve Spec A/ ( h ). Note that thebase points of H ( O X ( − Z )) is contained in Supp C . Proposition 2.5.
Let A be a two-dimensional normal local ring as above. Let Z , Z ′ be anti-nef cycles on some resolution X → Spec A . Suppose that O X ( − Z ) hasno fixed components. Then we have h ( O X ( − Z ′ )) ≥ h ( O X ( − Z − Z ′ )) . Proof.
Let h be a general element of I Z . Then the short exact sequence (2.2) impliesthat H ( O X ( − Z ′ )) → H ( O X ( − Z − Z ′ )) → H ( C ⊗ O X ( − Z ′ )) = 0since C is a coherent sheaf on an affine space. (cid:3) Let Z , Z be anti-nef cycles on the resolution X → Spec A so that O X ( − Z ) and O X ( − Z ) are generated. Take general elements f i ∈ I Z i for each i = 1 ,
2, so thatthere exists the following exact sequence:(2.3) 0 → O X ( f ,f ) −→ O X ( − Z ) ⊕ O X ( − Z ) ( − f f ) −→ O X ( − Z − Z ) → . Taking a cohomology yields0 → A → I Z ⊕ I Z ( − f f ) −→ I Z + Z → H ( O X ) . Hence we have the following.
Proposition 2.6.
Under the notation as above, if we put (2.4) ε ( Z , Z ) := ε ( I Z , I Z ) := ℓ A ( I Z + Z /f I Z + f I Z ) , then we have (1) 0 ≤ ε ( Z , Z ) ≤ p g ( A ) . (2) ε ( Z , Z ) = p g ( A ) − h ( O X ( − Z )) − h ( O X ( − Z )) + h ( O X ( − Z − Z )) .In particular, ε ( Z , Z ) is independent on the choice of general elements f ∈ I , f ∈ I .If Z = Z = Z , then Q = ( f , f ) is a minimal reduction of I = I = I and ε ( Z, Z ) = ε ( I, I ) = ℓ A ( I /QI ) . Canonical divisor, Vanishing theorem.
Let K X denote the canonical di-visor on X . Since the intersection matrix ( E i E j ) is negative-definite, there exists a Q -divisor Z K X supported in E such that K X + Z K X ≡
0. It is known that: Z K X ≥ X is the minimal resolution; Z K X = 0 if and only if A is rational Gorenstein and X is the minimal resolution; K X is linearly equivalent to − Z K X if and only if A isGorenstein.The following theorem is a generalization of Grauert–Riemenschneider vanishingtheorem in two dimensional case. Theorem 2.7 (Laufer [12, Theorem 3.2], cf.[18, Ch. 4, Exe.15]) . For any nef divisor D on X , we have H ( X, O X ( K X + D )) = 0 . .8. Riemann-Roch formula.
Let us recall Kato’s Riemann-Roch formula whichis very useful in order to calculate colength. For any invertible sheaf L on X , wedefine χ ( L ) by χ ( L ) = ℓ A (cid:0) H ( X \ E, L ) /H ( X, L ) (cid:1) + h ( L ) . Note that χ ( O X ) = p g ( A ) since A is normal. Theorem 2.8 ( Kato’s Riemann-Roch formula [11]) . For a cycle
Z > , we have χ ( L ( − Z )) − χ ( L ) = − Z + K X Z L Z. In particular, ℓ A ( A/I Z ) + h ( O X ( − Z )) = − Z + K X Z p g ( A ) . p g -cycles and p g -ideals The main aim of this section is to introduce the notion of p g -cycles and p g -ideals.We first show the following theorem, which is the key result in this paper. Theorem 3.1.
Let
Z > be a cycle. Suppose that O X ( − Z ) has no fixed component.Then we have the following. (1) h ( O X ( − Z )) ≤ p g ( A ) . (2) If h ( O X ( − Z )) = p g ( A ) , then O X ( − Z ) is generated ( by global sections ) .Proof. We use the exact sequence (2.2).(1) It follows from Proposition 2.5 because h ( O X ) = p g ( A ).(2) If h ( O X ( − Z )) = p g ( A ), then the restriction H ( O Z ( − Z )) → H ( C ) is surjec-tive. This implies that H ( O X ( − Z )) has no base points. (cid:3) Definition 3.2 ( p g -cycle, p g -ideal ) . A cycle
Z > p g -cycle if O X ( − Z )is generated and h ( O X ( − Z )) = p g ( A ). An m -primary ideal I is called a p g -ideal if I is represented by a p g -cycle on some resolution. The definition of p g -ideal isindependent of the representation of the ideal by Lemma 3.4. Example 3.3. If A is rational, then every anti-nef cycle is a p g -cycle. In fact,Lipman [14] proved that if A is rational and Z > X , then O X ( − Z ) is generated and H ( O X ( − Z )) = 0.A birational morphism φ : Y → Spec A is called a partial resolution if Y is normaland φ induces an isomorphism Y \ φ − ( m ) ∼ = Spec A \ { m } . Lemma 3.4.
Let I be an m -primary ideal, and let f : X → Spec( A ) and f : X → Spec( A ) be partial resolutions with only rational singularities. Assume that I isrepresented by a cycle Z i on X i for i = 1 , . Then h ( O X ( − Z )) = h ( O X ( − Z )) .Proof. Take a resolution f : X → Spec A which factors through f and f as follows: X φ −→ X φ ↓ ↓ f X f −→ Spec A hen φ i are resolution of singularities of X i , and φ ∗ Z = φ ∗ Z because they aredetermined by the invertible sheaf I O X . Let Z = φ ∗ Z . From the Leray spectralsequence, we obtain the following exact sequence:0 → H ( φ i ∗ O X ( − Z )) → H ( O X ( − Z )) → H ( R φ i ∗ O X ( − Z )) . By projection formula, R j φ i ∗ O X ( − Z ) = O X i ( − Z i ) ⊗ R j φ i ∗ O X . Since X i has onlyrational singularities, we have R φ i ∗ O X ( − Z ) = 0 and φ i ∗ O X ( − Z ) = O X i ( − Z i ).Thus we obtain that h ( O X i ( − Z i )) = h ( O X ( − Z )) for i = 1 , (cid:3) Any p g -ideal is an integrally closed m -primary ideal by definition. Indeed, allpowers of p g -ideals is p g -ideals and thus integrally closed. Theorem 3.5.
Assume that Z is a p g -cycle on the resolution X . Then for anycycle Z ′ on X such that O X ( − Z ′ ) is generated, ε ( Z, Z ′ ) = 0 . In particular, Z ′ is a p g -cycle if and only if so is Z + Z ′ .When this is the case, if f ∈ I Z , f ′ ∈ I Z ′ are general elements, then I Z + Z ′ = f I Z ′ + f ′ I Z . Proof.
Consider ε ( Z, Z ′ ) = p g ( A ) − h ( O X ( − Z )) − h ( O X ( − Z ′ )) + h ( O X ( − Z − Z ′ )) . Assume Z is a p g -cycle. Then ε ( Z, Z ′ ) = − h ( O X ( − Z ′ )) + h ( O X ( − Z − Z ′ )) ≤ ε ( Z, Z ′ ) ≥
0, we obtain that ε ( Z, Z ′ ) = 0, thatis, h ( O X ( − Z ′ )) = h ( O X ( − Z − Z ′ )). Hence Z ′ is a p g -cycle if and only if Z + Z ′ is a p g -cycle. (cid:3) Corollary 3.6.
Let Z be a p g -cycle on X and Q a minimal reduction of I := I Z .Then I n is integrally closed for all n ≥ , I = IQ , and I ⊂ Q : I .Proof. We can apply the previous theorem as Z = Z = Z . (cid:3) Remark . In our upcoming paper, we will prove that for an m primary ideal I ina two-dimensional normal local ring A , the Rees algebra R ( I ) = ⊕ n ≥ I n t n is normaland Cohen-Macaulay if and only if I is a p g -ideal.In the rest of this section, we give a characterization of p g -cycles.For any cycle D > X , the restriction O X → O D implies the surjection H ( O X ) → H ( O D ). Thus h ( O D ) ≤ p g ( A ). Theorem 3.8 (Reid [18, § . Assume that p g ( A ) > . There exists a smallestcycle C X > on X such that h ( O C X ) = p g ( A ) . If A is Gorenstein and f isminimal, then C X = Z K X . The cycle C X is called the cohomological cycle on X . Definition 3.9.
For any cycle D on X , let D ⊥ = P DE i =0 E i . Proposition 3.10.
Assume that p g ( A ) > . Let Z > be a cycle such that O X ( − Z ) has no fixed component. Then Z is a p g -cycle if and only if O C X ( − Z ) ∼ = O C X . roof. If O C X ( − Z ) ∼ = O C X , then h ( O X ( − Z )) ≥ h ( O C X ) = p g ( A ). By Theo-rem 3.1, h ( O X ( − Z )) = p g ( A ) and O X ( − Z ) is generated.Conversely, assume that h ( O X ( − Z )) = p g ( A ). Then h ( O X ( − mZ )) = p g ( A )and O X ( − mZ ) is generated for every m ∈ N by Theorem 3.5. . If Z ⊥ = 0, then itfollows from Theorem 2.7 that H ( O X ( − mZ )) = 0 for sufficiently large m ∈ N ; itcontradicts that p g ( A ) >
0. Let
D > Z ⊥ and DE i < E i ≤ Z ⊥ . There exist m ∈ N such that ( mZ + D ) E i < E i ⊂ E .By Theorem 2.7 again, H ( O X ( − nmZ − nD )) = 0 for some n ∈ N . Then p g ( A ) = h ( O X ( − nmZ )) = h ( O nD ( − nmZ )). Since O X ( − Z ) is generated and Z ≡ D ,we have O nD ( − Z ) ∼ = O nD . It follows that h ( O nD ) = h ( O nD ( − nmZ )) = p g ( A ).By the definition of C X , we have nD ≥ C X . Hence O C X ( − Z ) ∼ = O C X . (cid:3) It follows from Theorem 3.8 and Proposition 3.10 that if A is Gorenstein, p g ( A ) >
0, and f : X → Spec A is minimal, then there exist no p g -cycles on X . Therefore, ingeneral, p g -ideals are represented on non-minimal resolutions. In the next proposi-tion, we discuss the minimality of representation. Proposition 3.11.
Let I be a p g -ideal represented by a cycle Z on X . Then thereexist the minimum X → Spec A of the resolutions on which I is represented and anatural morphism X → X . We call X the minimal resolution with respect to I .The resolution X is the minimum with respect to I if and only if ZC < for every ( − -curve C on X .Proof. Let X → Spec A be a partial resolution obtained by normalizing the blowing-up by the ideal I . Since I O X = O X ( − Z ), X is also obtained by contracting allcurves E i ⊂ E with ZE i = 0; let ψ : X → X denote the contraction. If I isrepresented on a resolution X ′ → Spec A , then I O X ′ is invertible and thus thereexists a unique morphism X ′ → X by universal property of blowing-ups. Hencethe minimal resolution with respect to I is obtained as the minimal resolution ofsingularities of X .Let C be a ( − X with ZC = 0 and let X → X ′ be the contractionof C . Then X ′ → Spec A is a resolution, and I is represented on X ′ since we havea morphism X ′ → X . Conversely assume that the natural morphism ψ : X → X to the minimal resolution X with respect to I is not trivial. Then the exceptionalset of ψ contains a ( − C , and the invertible sheaf O X ( − Z ) is trivial on C ,since O X ( − Z ) = I O X = ψ ∗ I O X . (cid:3) Existence of good ideals in two-dimensional normal Gorensteinsingularities
The aim of this section is to prove the following, which is the main theorem inthis paper.
Theorem 4.1.
Let ( A, m ) be a two-dimensional normal local ring. Then :(1) There exists a resolution on which p g -cycles exist. (2) If A is non-regular Gorenstein, then it has a good ideal. This theorem follows from Propositions 4.2, 4.5 below. We use the notation ofthe preceding sections. roposition 4.2. Assume that A is Gorenstein. Let Z > be a p g -cycle on X .Then I := I Z is a good ideal if and only if K X Z = 0 .Proof. Applying Theorem 2.8, we have 2 · ℓ A ( A/I ) = − Z − K X Z . As noted insubsection 2.5, I is good if and only if I = IQ and 2 · ℓ A ( A/I ) = e ( I ) for someminimal reduction Q of I . However the condition I = IQ is always satisfied for p g -ideals by Corollary 3.6. Since O X ( − Z ) is generated, e ( I ) = − Z . This completesthe proof. (cid:3) Lemma 4.3.
Suppose that
C > is a cycle on X such that h ( O C ) = p g ( A ) . Let b : Y → X be the blowing-up of a finite subset B ⊂ Supp( C ) . Let F = b − ( B ) and e C = b ∗ C − F . Then h ( O e C ) = p g ( A ) .Proof. It suffices to show that h ( O e C ) ≥ p g ( A ). We have that R b ∗ O Y ( − b ∗ C ) = 0and H i ( O F ( − b ∗ C + F )) = 0 for i = 0 ,
1, since O F ( − b ∗ C ) ∼ = O F and F is the sumof ( − → O F ( − b ∗ C + F ) → O b ∗ C → O e C → , we obtain that h ( O b ∗ C ) = h ( O e C ). On the other hand, it follows from the exactsequence 0 → b ∗ O Y ( − b ∗ C ) → b ∗ O Y → b ∗ O b ∗ C → b ∗ O b ∗ C = O C , since b ∗ O Y ( − b ∗ C ) = O X ( − C ) and b ∗ O Y = O X . Therefore, p g ( A ) = h ( O C ) = h ( b ∗ O b ∗ C ) ≤ h ( O b ∗ C ) = h ( O e C ) . (cid:3) Remark . In the situation above, if C ′ ⊂ Y denote the strict transform of C , theequality h ( O C ′ ) = h ( O C ) does not hold in general. Let A = C [[ x, y, z ]] / ( x + y + z ) and X the minimal good resolution, i.e., E is simple normal crossing and any( − E = E + · · · + E is star-shaped, where E denotes the central curve, and Z K X = E + E . If b : Y → X is the blowing-up of a point of E \ ( E ∪ E ∪ E ), then the strict transform of E contracts to a rational singularity. If C = Z K X , then h ( O C ) = 1 but h ( O C ′ ) = 0. Proposition 4.5.
There exist a resolution g : Y → Spec A and a p g -cycle Z on Y .Furthermore, such a resolution can be obtained from X by blowing-ups of smoothpoints of the exceptional set. If Z K X is a cycle ( i.e., all the coefficients are integers ) and Z K X > , then Z can be taken as a p g -cycle satisfying ZK Y = 0 .Proof. Since the intersection matrix is negative definite, there exists an anti-nef cycle
W > E i , − W E j ≥ max { K X E j , ( K X + E i ) E j , · g ( E j ) } for every E j ⊂ E . Consider the following exact sequence:0 → O X ( − W − E i ) → O X ( − W ) → O E i ( − W ) → . Since − W − ( K X + E i ) is nef, it follows from Theorem 2.7 that H ( O X ( − W − E i )) =0. Therefore the map H ( O X ( − W )) → H ( O E i ( − W ))is surjective. If O X ( − W ) has a base point p ∈ E i , then p should also be a basepoint of H ( O E i ( − W )). On the other hand, since deg O E i ( − W ) ≥ g ( E i ), the inear system H ( O E i ( − W )) has no base points. Thus we obtain that O X ( − W ) isgenerated. Since − W − K X is nef, we have H ( O X ( − W )) = 0 from Theorem 2.7.Hence the exact sequence0 → O X ( − W ) → O X → O W → h ( O W ) = p g ( A ). Since O X ( − W ) is generated, there exists a function h ∈ H ( O X ( − W )) such that div( h ) = W + H , where H is a reduced divisor includingno component of E , and that E + H is normal crossing at E ∩ H . Let C > X such that h ( O C ) = p g ( A ); at least the cycle W satisfies this property(but O X ( − C ) need not be generated). Let Y = X and B = Supp( C ) ∩ H . For i ≥
0, if B i = ∅ , take the blowing-up b i +1 : Y i +1 → Y i of B i , and let H i +1 be thestrict transform of H i by b i +1 , C i +1 = b ∗ i +1 C i − F i +1 , where F i +1 = b − i +1 ( B i ), and B i +1 = Supp( C i +1 ) ∩ H i +1 . Note that there exists n such that B n = ∅ ; in fact,such n is not more than the maximal coefficient of C . If B n = ∅ , let Y = Y n and Z = div Y ( h ) − H n . Then O C n ( − Z ) ∼ = O C n ( − div Y ( h )) ∼ = O C n . By Lemma 4.3, h ( O C n ( − Z )) = p g ( A ). From the surjection H ( O Y ( − Z )) → H ( O C n ( − Z )) andTheorem 3.1, we obtain that h ( O Y ( − Z )) = p g ( A ) and O Y ( − Z ) is generated.If Z K X is a cycle and Z K X >
0, then we can take C = Z K X . In this case weobtain that C n = Z K Y . Since O C n ( − Z ) ∼ = O C n , we obtain that K Y Z = 0. (cid:3) Let us explain the procedure of Proposition 4.5 by an example.
Example 4.6.
Let us consider a cone singularity and check the key roles in theproof of Proposition 4.5. Let C be a nonsingular curve of genus g ≥ A = M n ≥ H ( O C ( nK C )) . By Pinkham’s formula [17], we have p g ( A ) = P n ≥ h ( O C ( nK C )) = g + 1. Let f : X → Spec A be the minimal resolution. Then E ∼ = C , O E ( − E ) ∼ = O E ( K E ), and K X = − E . It follows that H ( O X ( − E )) = 0 by Theorem 2.7. From the exactsequence 0 → O X ( − E ) → O X ( − E ) → O E ( − E ) → , we see that O X ( − E ) is generated. Thus we can take W = kE ( k ≥
1) and C = Z K X = 2 E . Then B and B consists of − W E = k (2 g −
2) points and B = ∅ .Hence we have Y = Y . In this case, − Z = H n Z = k ( k + 2)(2 g − g = 2 can be realized by A = k [ x, y, z ] / ( x + y + z ). This is graded bydeg x = 3, deg y = deg z = 1. If W = E ( k = 1) and h = y ∈ H ( O X ( − W )), then I Z = ( x, y, z ) and I Z is a good ideal with multiplicity 6; this is also an Ulrich ideal(see Section 7). Let W = 2 E ( k = 2) and h = yz ∈ H ( O X ( − W )). Then I := I Z isa homogeneous ideal and I = ( yz, y , z , xy, xz ) is a good ideal with multiplicity 16.5. Good ideals for non-Gorenstein rational singularities
In this section, we characterize good ideals for rational singularities, which givesa generalization of [5, Section 7].
Theorem 5.1.
Assume that ( A, m ) is a two-dimensional rational singularity. Let I be an m -primary ideal of A . Then the following conditions are equivalent: I is a good ideal, that is, I = QI and I = Q : I for some minimal reduction Q of I . (2) I is an integrally closed ideal that is represented on the minimal resolution. Now let I be a good ideal in a rational singularity ( A, m ). Then Lemma 2.4implies that I is integrally closed and thus I is represented by some anti-nef cycle Z on some resolution of singularities X → Spec A . Then Z is a p g -cycle on X ; seeExample 3.3. Before proving that X is minimal, we study properties of p g -cycle forany normal local ring.We first show that any integrally closed ideal that is represented on non-minimalresolution is not good. We need the following lemma. Lemma 5.2.
Let E ≤ E be a ( − -curve. Then h ( O X ( E )) = p g ( A ) .Proof. Consider the exact sequence0 → O X → O X ( E ) → O E ( E ) → . Since H i ( O E ( E )) = H i ( O P ( − i = 0 ,
1, we have h ( O X ) = h ( O X ( E )). (cid:3) The implication (1) = ⇒ (2) in Theorem 5.1 follows from the following propositionand Lemma 2.4. Proposition 5.3.
Assume that Z is a p g -cycle on X and there exists a ( − -curve E such that ZE = − a < . Let φ : X → X ′ be the blowing-down of E and Z ′ = Z − aE . Consider the following conditions :(1) Z ′ is a p g -cycle on X ; (2) φ ∗ Z is a p g -cycle on X ′ ; (3) I := I Z $ I Z ′ .We have the implication: (1) ⇔ (2) ⇒ (3) ; if a = 1 , (3) implies (1) . If the condition (1) is satisfied, then I Z ′ ⊂ Q : I ; in particular, I is not good. If ( A, m ) is rational,then all conditions are satisfied.Proof. Note that Z ′ = φ ∗ φ ∗ Z and φ ∗ O X ( − Z ′ ) = O X ′ ( − φ ∗ Z ). Thus O X ( − Z ′ ) isgenerated if and only if so is O X ′ ( − φ ∗ Z ). From the spectral sequence, we have that h ( O X ′ ( − φ ∗ Z )) = h ( O X ( − Z ′ )) . Therefore the conditions (1) and (2) are equivalent. Consider the exact sequence0 → O X ( − Z ) → O X ( − Z ′ ) → O aE ( − Z ′ ) = O aE → . If (1) is satisfied, then the natural homomorphism α : H ( O X ( − Z ′ )) → H ( O aE ) issurjective, and (3) holds. If a = 1, the following three conditions are equivalent: • I $ I Z ′ ; • α is surjective; • h ( O X ( − Z )) = h ( O X ( − Z ′ )).Since the non-triviality of α implies that O X ( − Z ′ ) has no fixed components in E ,(3) implies (1). ssume that Z ′ is a p g -cycle on X . Then Z + Z ′ is also a p g -cycle on X byTheorem 3.5. The following sequence is obtained from (2.3).0 → O X ( E ) → O X ( − Z ′ ) → O X ( − Z − Z ′ ) → . By Lemma 5.2, H ( O X ( − Z ′ ) ) → H ( O X ( − Z − Z ′ )) is surjective. Thus II Z ′ = QI Z ′ . This implies I Z ′ ⊂ Q : I .If ( A, m ) is rational, then Z ′ is a p g -cycle from Example 3.3. (cid:3) Example 5.4.
Assume that A = k [[ x, y, z ]] / ( x + y + z ) and X ′ is the minimalgood resolution. Then the exceptional set E ′ = E ′ + · · · + E ′ is star-shaped and all E i are rational curves. Suppose that E ′ is the central curve and ( − E ′ , . . . , − E ′ ) =(1 , , , X ′ is Z K X ′ = E ′ + E ′ . Let Z be thefundamental cycle on X ′ . Then O X ′ ( − Z ) has no fixed components in E ′ , and hasa base point p ∈ E ′ \ E ′ . Let φ : X → X ′ be the blowing-up of the point p and E the exceptional set, and let Z ′ = φ ∗ Z and Z = Z ′ + E . Then Z K X = φ − ∗ Z K X ′ and O Z KX ( − Z ) ∼ = O Z KX . Since O X ( − Z ) is generated and h ( O X ( − Z ) ≥ h ( O Z X ) = p g , Z is a p g -cycle and I Z is good by Proposition 4.2. However, I Z = I Z ′ .In what follows, we prove (2) = ⇒ (1) in the theorem. Lemma 5.5.
Assume that ( A, m ) is rational and that f : X → Spec( A ) is minimal.Let D and D be effective cycles. Suppose that they have no common irreduciblecomponents and D = 0 . Then H ( O X ( D − D )) = 0 .Proof. Consider the exact sequence0 → O X ( D − D ) → O X ( D ) → O D ( D ) → . Since O D ( D ) is nef on its support, H ( O D ( D )) = 0. Therefore, it suffices toshow that H ( O X ( D )) = 0. Since H ( O D ( D )) = 0 (Wahl [20, (2.2)]), we havethat h ( O D ( D )) = − χ ( O D ( D )) = ( K X D − D ) / . Since X is minimal and D = 0, we have h ( O D ( D )) = 0. Therefore h ( O X ( D )) ≥ h ( O D ( D )) ≥ (cid:3) Assume that A is a rational singularity. Let Z > I = H ( O X ( − Z )). Let Q be a minimal reduction of I . From the Koszul complexassociated with generators of Q (cf.(2.3)), we obtain the exact sequence(5.1) 0 → O X ( Z ) → O X → O X ( − Z ) → . This implies the exact sequence(5.2) 0 → Q → I → H ( O X ( Z )) → . The next lemma holds without rationality of the singularity.
Lemma 5.6.
Assume that I = QI . Then core( I ) = Q : I = ( Q : I ) I = ( Q : I ) Q. roof. By Goto–Shimoda [9], the Rees algebra R ( I ) = ⊕ n ≥ I n t n is Cohen-Macaulay.Therefore it follows from Corollary 5.1.1 and Remark 5.1.2 of Hyry–Smith [10] thatcore( I ) = Q : I . Let x = P mi =1 a i h i ∈ ( Q : I ) I , where a i ∈ Q : I and h i ∈ I . Then xI ⊂ P a i I = P a i IQ ⊂ Q . Thus ( Q : I ) I ⊂ Q : I . Conversely assume that x ∈ Q : I . Suppose that Q is generated by f and g . Since core( I ) ⊂ Q , thereexist a, b ∈ A such that x = af + bg . For any h ∈ I , there exist c , c , c ∈ A suchthat ( af + bg ) h = c f + c f g + c g . Since f, g form a regular sequence, we have ah − c f ∈ ( g ) and bh − c g ∈ ( f ). Thus ah, bh ∈ ( f, g ) = Q . This shows that a, b ∈ Q : I . Hence x = af + bg ∈ ( Q : I ) Q . (cid:3) Proposition 5.7.
Assume that ( A, m ) is rational and that f : X → Spec( A ) isminimal. Let Z > be an anti-nef cycle on X and I = I Z . Then I = QI and Q : I = I .Proof. By Proposition 3.6, it suffices to show that Q : I ⊂ I . Let h ∈ Q : I and Z h = div( f ∗ h ) − f − ∗ (div( h )) (the exceptional part of the divisor of h on X ). Wehave to show that Z h ≥ Z . To this end, we may assume that h is a general elementof Q : I so that Q : I ⊂ I Z h . From the exact sequence0 → O X ( Z ) × h −→ O X ( Z − Z h ) → C → h : H ( O X ( Z )) × h −→ H ( O X ( Z − Z h )) . We shall show that h is trivial. Since H ( O X ( − Z h )) = 0, tensoring the exactsequence (5.1) with O X ( − Z h ) we obtain the exact sequence0 → QI Z h → I Z + Z h → H ( O X ( Z − Z h )) → . Therefore we may regard h as a homomorphism h : I/Q × h −→ I Z + Z h /QI Z h . However it follows from Lemma 5.6 that hI ⊂ ( Q : I ) Q ⊂ I Z h Q . Hence the map h should be trivial. By Lemma 5.5, we obtain that Z − Z h ≤ (cid:3) Number of minimal generators of integrally closed ideals.
The aim of this section is to study the number of minimal set of generators forintegrally closed ideals in A . In what follows, let M denote the maximal ideal cycleof a given resolution of singularities f : X → Spec A ; see [24, Definition 2.11].Furthermore, we always assume that an integrally closed m -primary ideal I = I Z is represented by Z . Theorem 6.1.
Let ( A, m ) and f : X → Spec A be as above. Let I = I Z be anintegrally closed m -primary ideal and assume that m O X = O X ( − M ) . Then we havean inequality − M Z + 1 ≥ µ A ( I ) ≥ − M Z + 1 − p g ( A ) . More precisely, we have ℓ A ( I/I m ) = − M Z + 1 − ε ( Z, M ) nd we have equality µ A ( I ) = − M Z + 1 if I m = f I + g m for general elements f ∈ m and g ∈ I .Proof. In the proof, we write h ( − Z ) instead of h ( O X ( − Z )) for any anti-nef cycle Z on X . Note that µ A ( I ) = ℓ A ( I/I m ) ≥ ℓ A ( I/I m ) and I m = I M + Z . By Riemann-Roch formula 2.8, we have ℓ A ( A/I m ) = p g ( A ) − h ( − M − Z ) − ( M + Z ) + K X ( M + Z )2 ,ℓ A ( A/I ) = p g ( A ) − h ( − Z ) − Z + K X Z , ℓ A ( A/ m ) = p g ( A ) − h ( − M ) − M + K X M . It follows that ℓ A ( I/I m ) = ℓ A ( A/I m ) − ℓ A ( A/I ) − ℓ A ( A/ m ) + 1= − M Z + 1 − (cid:8) p g ( A ) − h ( − M ) − h ( − Z ) + h ( − Z − M ) (cid:9) = − M Z + 1 − ε ( Z, M ) . Hence the theorem follows from Proposition 2.6. (cid:3)
Example 6.2. If I Z or m is a p g -ideal, then µ A ( I Z ) = − M Z + 1. Actually, we getthe equalities I m = I m = f I Z + g m by Theorem 3.5. Corollary 6.3. If A is a rational singularity and I is an integrally closed m -primaryideal, then µ A ( I ) = − M Z + 1 . Example 6.4.
Assume that the exceptional set of the minimal resolution of Spec A consists of one curve E ∼ = P with E = − r . Then µ A ( I ) of integrally closed ideal I is of the form µ A ( I ) = nr + 1 for some positive integer n . If A is a simpleelliptic singularity of e ( m ) = r ≥
3, then µ A ( I ) = nr + 1 or nr for every integrallyclosed ideal I . Actually, if I = I Z for some anti-nef cycle Z on X and if we denote f : X → X , where X is the minimal resolution, then − M Z = − f ∗ ( Z ) E = nr ,where E is the unique elliptic curve on X . Remark . In any two-dimensional normal local ring A , if I ′ ⊂ I are integrallyclosed ideals in A , then µ A ( I ′ ) ≥ µ A ( I ) holds true by [22, Theorems 3,5].7. Ulrich ideals of minimally elliptic and simple ellipticsingularities.
Let ( A, m ) be a Cohen-Macaulay local ring with infinite residue field, and let I be an m -primary ideal of A and Q a minimal reduction of I . Then I is called an Ulrich ideal if I = QI and I/I is A/I -free. When A is Gorenstein, I is an Ulrichideal if and only if it is a good ideal and µ A ( I ) = dim A + 1; see also subsection2.5. Thus in the Gorenstein case, we can regard Ulrich ideals as typical example ofgood ideals. In [6, 7, 8], the last two authors classified all Ulrich ideals for simplesingularities and two-dimensional rational singularities. So the following problem isnatural. roblem 7.1. Let A be a two-dimensional normal local ring. Classify all Ulrichideals of A .In this section, we will prove non-existence theorem for minimally elliptic singu-larities (namely, Gorenstein rings with p g ( A ) = 1) with high multiplicity and we willcomplete solution for Problem 7.1 in the case of simple elliptic singularities. Notethat in our case, Ulrich ideal I is a good ideal with µ A ( I ) = 3.First, by Lemma 2.4 and Proposition 3.6, we have the following. Proposition 7.2.
Let I be a good ideal and assume I O X = O X ( − Z ) is invertiblewith h ( O X ( − Z )) = p g ( A ) . Then I and I are integrally closed.Remark . If p g ( A ) = 1 and h ( O X ( − Z )) = 0, then one of the following 2 casesoccur; see (2.3):(1) I = QI .(2) ℓ A ( I /QI ) = 1 and I is integrally closed.Let us recall some fundamental facts for minimally elliptic singularities. Let Z f be the fundamental cycle and e = − Z f . Then Z f = M = Z K X on the minimalresolution, O X ( − M ) has no fixed component, O X ( − M ) is generated (i.e., m O X = O X ( − M )) if e ≥
2. Moreover, m n are integrally closed for all n ≥ e ≥
3; seeLaufer [13].
Lemma 7.4.
Let ( A, m ) be a minimally elliptic singularity of degree e ≥ . Then M + K X M = 0 and h ( O X ( − M )) = 0 .Proof. Let Q be a minimal reduction of m . As A is Gorenstein with e ( m ) ≥
3, wemust have m = Q m . In particular, m is not a p g -ideal and h ( O X ( − M )) = 0. Alsoby Riemann-Roch Theorem 2.8 for 1 = ℓ A ( A/ m ), we have M + K X M = 0. (cid:3) The following lemma plays an essential role.
Lemma 7.5.
Let ( A, m ) be a minimally elliptic singularity and let I be an m -primaryideal such that ¯ I is represented by some cycle Z on a resolution X of Spec A andassume that I = QI . Then I is a good ideal if and only if one of the following casesoccurs: (1) h ( O X ( − Z )) = 1 , K X Z = 0 and I is integrally closed. (2) h ( O X ( − Z )) = 0 and K X Z = 2(1 + ℓ A ( ¯ I/I )) .Proof. Since A is Gorenstein, I is good if and only if I = QI and e ( I ) = 2 · ℓ A ( A/I ).The result follows from Lemma 2.4 and Riemann-Roch Theorem 2.8. (cid:3)
Next, let us discuss how far is an Ulrich ideal from integrally closed.
Lemma 7.6.
Let A be a minimally elliptic singularity of degree e ≥ . If I is anUlrich ideal of A , then ℓ A ( ¯ I/I ) ≤ .Proof. We saw a good ideal I is integrally closed if ¯ I is a p g -ideal in Proposition 7.2.Hence we may assume that ¯ I = I Z with h ( O X ( − Z )) = 0 on some resolution X .Now, by Theorem 6.1, µ ( I Z ) ≥ − M Z , where M is the maximal ideal cycle on X .Now let X be the minimal resolution of Spec A and f : X → X be the contraction. hen we know that M = f ∗ ( M ) and M = − K X , where M is the maximal idealcycle on X . Now, write Z = f ∗ ( f ∗ ( Z )) + Y and K X = f ∗ ( K X ) + L. Then ZK X = f ∗ ( Z ) K X + Y L and
Y L = ZL ≤ L ≥ Z is anti-nef.Moreover, M Z = M f ∗ ( Z ) = − K X f ∗ ( Z ) ≤ − K X Z . Thus we have3 = µ A ( I ) ≥ µ A ( ¯ I ) − ℓ A ( ¯ I/I ) ≥ − M Z − ℓ A ( ¯ I/I ) ≥ K X Z − ℓ A ( ¯ I/I ) = ℓ A ( ¯ I/I ) + 2and we get the desired inequality. (cid:3)
Theorem 7.7.
Let A be a minimally elliptic singularity of degree e ≥ . (1) If e ≥ , then A has no Ulrich ideals. (2) If e = 4 and I is an Ulrich ideal with ¯ I = I Z , then ℓ A ( ¯ I/I ) = 1 and I isrepresented on the minimal resolution X and − M Z = 4 , where M is themaximal ideal cycle on X . (3) If A has an Ulrich ideal which is a p g -ideal, then e ≤ .Proof. Note that e = − M ≤ − M Z for any anti-nef cycle Z . If I is Ulrich, then µ A ( I ) = 3 and putting ¯ I = I Z , µ A ( I Z ) ≤
4. On the other hand, µ A ( I Z ) ≥ − M Z ≥ e and we have e ≤
4. If I Z is an Ulrich p g -ideal, then by Example 6.2, we have3 = µ A ( I ) = − M Z + 1 ≥ e + 1.If e = 4 and I is an Ulrich ideal with ¯ I = I Z , then ℓ A ( ¯ I/I ) = 1 and − M Z = 4since µ A ( I Z ) ≥ − M Z ≥ e . Moreover, by Lemma 7.5, we have K X Z = 4. Using thenotation of Lemma 7.6, we have ZL = 0. Therefore ¯ I can be represented on theminimal resolution. (cid:3) Remark . Let A be a minimally elliptic singularity of degree e = 4. Then itis known that A is a complete intersection of codimension 2. Now, let I be an m primary ideal generated by 3 elements among the minimal generating systemof m containing some minimal reduction Q of m . Then we can show that I is anUlrich ideal. Let E = f − ( m ), where f : X → Spec A is the blowing up of themaximal ideal. Now, consider the family of ideals I generated by 3 elements amongthe minimal generating system of m . Then the condition I contains a minimalreduction of m is equivalent to say that the generators of I have no common zeroon E as sections of a line bundle m O X . Hence the family of such ideals forms anopen subset of P = P ( m / m ). If I is an Ulrich ideal, then ¯ I = m by Theorem 7.7. Example 7.9.
Let A = k [[ x, y, z, w ]] / ( y − xz, w − y ( z − x )) be a complete inter-section, which is a minimally elliptic singularity of degree e = 4. Then the minimalresolution of Spec A is a star-shaped graph with central curve E ∼ = P with E = − E i ( i = 1 , , ,
4) with E i = −
3. Here, we have M = 2 E + P i =1 E i .Let Z = 3 E + P i =1 E i . Then I Z = ( x, y, z, w ) and I = ( x, y, z ) is an Ulrich idealwith ¯ I = I Z .In the following, let ( A, m ) be a simple elliptic singularity of degree e (see Section2). Let us classify all of the Ulrich ideals of those rings. Also, let I be an m -primaryideal and Q its minimal reduction. We assume I O X = O X ( − Z ) for some resolution X so that ¯ I = I Z = H ( X, O X ( − Z )) is the integral closure of I .Now we state our classification. heorem 7.10. Let ( A, m ) be a simple elliptic singularity of degree e . Then :(1) If e ≥ , then A has no Ulrich ideals. (2) If e = 4 and I is an Ulrich ideal, then ℓ A ( A/I ) = 2 and ¯ I = m . (3) If e = 3 and I is an Ulrich ideal, then ℓ A ( A/I ) = 2 and I is integrally closed.Such ideals consist a family parametrized by the elliptic curve E . (4) If e = 2 , then an Ulrich ideal I is one of the followings ;(a) m . (b) I = ¯ I and ℓ A ( A/I ) = 2 . (c) I = ¯ I and ℓ A ( A/I ) = 3 . (d) ¯ I = m and ℓ A ( A/I ) = 4 .There are ideals of type ( c ) , and the ideals of type ( b ) is parametrized by P \ { points } . (5) If e = 1 , then A has integrally closed Ulrich ideals with ℓ A ( A/I ) = 1 , , , .The ones with ℓ A ( A/I ) = 2 (resp. ℓ A ( A/I ) = 3 ) are parametrized by E ( resp. P \{ points } ) and there are exactly Ulrich ideals with ℓ A ( A/I ) = 4 . Incidentally, A is not a hypersurface or complete intersection if and only if e ≥ Question 7.11.
Let ( A, m ) be Gorenstein normal local ring of dimension ≥
2. Upto now, if A has an Ulrich ideal, then A is a complete intersection. Is A a completeintersection if it has an Ulrich ideal? Proof of Theorem 7.10.
The cases with e ≥ e ≤
3. In the following, let I be an Ulrich ideal of A with¯ I = I Z for some anti-nef cycle Z on some resolution X → Spec A . Let X be theminimal resolution with exceptional set E and let f : X → X be the contraction.Suppose that f ∗ ( Z ) = nE .(3) If e = 3, and I = I Z is a p g -ideal, then by Example 6.2, µ A ( I ) ≥
4. Hence,we may assume I Z is not a p g -ideal. If n ≥
2, then µ A ( I Z ) ≥ µ A ( I ) ≥
5. Hence n = 1 and K X Z ≤ K X E = 3. By Lemma7.5, we have K X Z = 2 and I is integrally closed. Since K X Z = 2, we need exactlyone blowing-up from X and hence we may assume that f is the blowing up of apoint P on E and Z = f ∗ ( E ) + E , where E = f − ( P ). In this case, ℓ A ( A/I Z ) = 2.Such I is determined by P ∈ E .Now let us show that such I = I Z is actually an Ulrich ideal. Since ℓ A ( A/I ) = 2and e ( I ) = − Z = 4, we have only to show that I = QI for a reduction Q of I .For that purpose, we need to show that I is not integrally closed; see Remark 7.3.Now, by Riemann-Roch Theorem 2.8, we get ℓ A ( A/I ) = 7. On the other hand,since I is generated by m and 2 linear forms, ℓ A ( A/ ( I + m )) = 7. Hence if I = I ,then I + m = I , or I ⊃ m . This contradicts the fact 3 f ∗ ( E ) Z .(4) Next, assume e = 2. If ¯ I = I Z is a p g -ideal, then by Lemma 7.5, we have I Z = ¯ I = I and K X Z = 0. Also µ A ( I ) ≥ n + 1 and hence n = 1. Then the cases(b), (c) of the theorem occur. Actually, we know that ˆ A ∼ = k [[ x, y, z ]] / ( x − φ ( y, z )),where φ ( y, z ) is a homogeneous polynomial of degree 4 in ( y, z ). Take any linearform l ∈ I . We may assume that l is a form of y and z . If l is not a factor of φ ,then the line l = 0 intersects with E in 2 points P , P . Let f : X → X be the lowing up of these 2 points, let E i = f − ( P i ). and Z = f ∗ ( E ) + E + E , thenwe get the case (b). If l is a factor of φ , then l = 0 intersects E at a point P with multiplicity 2. Let f : X → X be the blowing up of P , let E i = f − ( P ) and Z = f ∗ ( E ) + 2 E , then we get the case (c). The ideal is I = ( l, x, ( y, z ) ) in case (b)and I = ( l, x, ( y, z ) ) in case (c).Next assume that ¯ I = I Z is not a p g -ideal. By Lemma 7.5, we have K X Z =2(1 + ℓ A ( ¯ I/I )). Also, Theorem 6.1 and Lemma 7.6 imply that K X Z ≤ n = − M Z ≤ µ A ( I Z ) ≤ µ A ( I ) + ℓ A ( ¯ I/I ) ≤ . Hence n ≤ K X Z ≤ n . It turns out that we have K X Z = 2 n in case n = 1 , A = k [[ x, y, z ]] / ( x + y + z ), the Ulrich ideals are calculated inExample 7.14 using the theory of simple singularities.(5) Finally let us treat the case e = 1. In this case, m O X has a base point andlet X be the blowing up of the base point. We choose X as starting point. Theexceptional set of X is E ∪ E , where E is an elliptic curve with E = − , E ∼ = P with E = − E E = 1. Here, m is defined by M = E + 2 E . Note that K X M = 0 and m is a p g -ideal.Let I be an Ulrich ideal of A with ¯ I = I Z , f : X → X be contraction and weput f ∗ ( Z ) = aE + bE . Hence − M f ∗ ( Z ) = b and K X f ∗ ( Z ) = 2 a − b . Since f ∗ ( Z )is anti-nef, a ≤ b .First assume that I Z is a p g -ideal. Then from µ ( I ) = 3, we get b = 2. Hereif a = 1, K X f ∗ ( Z ) = 0 and Z = M, I = m . If a = 2, we get the ones with ℓ A ( A/I ) = 3 , A ∼ = k [[ x, y, z ]] / ( x − φ ( y, z )), where φ is a homogeneouspolynomial of degree 3 with no multiple roots. Since a ≥ I is contained in( x, y, z ). Take any linear form l ∈ I . We may assume that l is a form of y and z .If l is not a factor of φ , then l = 0 defines 2 points P , P on E . Let f : X → X be the blowing up of these 2 points and put F i = f − ( P i ) ( i = 1 , Z = f ∗ (2 E + 2 E ) + F + F , we get K X Z = 0 and I Z is an Ulrich ideal of ℓ A ( A/I Z ) = 3. If l is one of 3 factors of φ , then l = 0 intersects E at a point P with multiplicity 2. Let f : X → X be the blowing up of P and put F = f − ( P ) .Then putting Z = f ∗ (2 E + 2 E ) + 2 F , we get K X Z = 0 and I Z is an Ulrich idealof ℓ A ( A/I Z ) = 4.Next assume that h ( − Z ) = 0 with f ∗ ( Z ) = aE + bE . Then by Example 6.2, ℓ A ( ¯ I/ m I ) = − M Z + 1. Hence3 = µ A ( I ) ≥ − M Z + 1 − ℓ A ( ¯ I/I )= − M X f ∗ ( Z ) + 1 − ℓ A ( ¯ I/I )= ( K X − E ) f ∗ ( Z ) + 1 − ℓ A ( ¯ I/I ) ≥ K X Z + 1 − ℓ A ( ¯ I/I )= 2(1 + ℓ A ( ¯ I/I )) + 1 − ℓ A ( ¯ I/I )= 3 + ℓ A ( ¯ I/I ) . hus we have ¯ I = I . Then by Lemma 7.5, we must have K X Z = 2. On the otherhand, 2 = K X Z ≤ K X f ∗ ( Z ) = 2 a − b and 3 = µ A ( I ) ≥ − M Z + 1 = b + 1. Hencewe must have a = b = 2, I = ( x, y, z ).In the case A = k [[ x, y, z ]] / ( x + y + z ), the Ulrich ideals are calculated inExample 7.13 using the theory of simple singularities. (cid:3) Example 7.12.
Let ( A, m ) be the local ring of the vertex of the cone over smoothcubic curve E ⊂ P k . Then A is a simple elliptic singularity of degree e = 3.The minimal resolution X of A is obtained by blowing-up of the maximal idealand the exceptional set is E with E = −
3. Take a line l = 0 in P k intersectingwith E at 3 distinct points P , P , P . Let π : X → X be obtained by blowing-up these 3 points. We denote E i = π − ( P i ) the corresponding exceptional curve( i = 1 , ,
3) and we denote E the elliptic curve. Put Z = E + 2( E + E + E ).Then O X ( − Z ) ⊗ O E ∼ = O E and actually, we get h ( O X ( − Z )) = 1. If we put I = H ( X, O X ( − Z )), then I is generated by m and the linear form l and I is agood ideal with e ( I ) = 6 and ℓ A ( A/I ) = 3. Since µ A ( I ) = 4, I is not an Ulrichideal. Example 7.13.
Let A = k [[ x, y, z ]] / ( x + y + z ) be a hypersurface, which is asimple elliptic singularity of degree e = 1. Then(1) m = ( x, y, z ) is an Ulrich ideal of colength 1 with minimal reduction Q =( y, z ).(2) I = ( x, y, z ) is an Ulrich ideal of colength 2 with minimal reduction Q =( y, z ).(3) For any ε ∈ C , ( x, y + εz , z ) is an Ulrich ideal of colength 3 with minimalreduction Q = ( y + εz , z ).(4) For an ε ∈ C × , ( x, y + εz , z ) is an Ulrich ideal of colength 4 if and only if ε = 1. Then Q = ( y + εz , z ) gives a minimal reduction. Example 7.14.
Let A = k [[ x, y, z ]] / ( x + y + z ) be a hypersurface, which is asimple elliptic singularity of degree e = 2.(1) m = ( x, y, z ) is an Ulrich ideal of colength 1 with minimal reduction Q =( y, z ).(2) If we put I = ( x, y + εz, z ) for some ε ∈ C , then I is an Ulrich ideal ofcolength 2 with minimal reduction Q = ( y + εz, z ).(3) If we put I = ( x, y + εz, z ) for some ε ∈ C with ε = −
1, then I is an Ulrichideal of colength 3 with minimal reduction Q = ( y + εz, z ).(4) I = ( x, y , z ), ( x, y ± z , yz ) are Ulrich ideals of ℓ A ( A/I ) = 4 and I = m =( x, y , yz, z ).We know that any diagonal hypersurface admits an Ulrich ideal if some exponentis an even number. How about the case that all exponents are odd numbers? Example 7.15.
Let A = k [[ x, y, z ]] / ( x n − + y n − + z n − ) be a hypersurface with p g ( A ) = (cid:0) n − (cid:1) , where n ≥ A has an Ulrich ideal I = ( x + y + y + · · · + y n − , y n , z ). Actually, if we put Q = ( x + y + y + · · · + y n − + y n , z ), thenit is a minimal reduction of I such that I = QI and ℓ A ( A/Q ) = 2 · ℓ A ( A/I ) = 2 n .Note that if n = 2, then A is a simple elliptic singularity of degree e = 3. cknowledgement. The authors thank Shiro Goto for valuable discussions ongood ideals and the number of generators of integrally closed ideals.
References [1] M. Artin,
On isolated rational singularities of surfaces , Amer. J. Math. (1966) 129–136.[2] A. Corso, C. Polini and B. Ulrich, Core and residual intersection of ideals , Trans.Amer.Math.Soc. , (2002), 2579–2594.[3] S.D. Cutkosky, A new characterization of rational surface singularities Inventiones math. (1990), 157-177.[4] J. Giraud,
Improvement of Grauert-Riemenschneider’s Theorem for a normal surface , Ann.Inst. Fourier, Grenoble (1982),13–23.[5] S. Goto, S. Iai, and K.-i. Watanabe, Good ideals in Gorenstein local rings , Trans. Amer. Math.Soc., (2000), 2309–2346.[6] S.Goto, K.Ozeki, R.Takahashi, K.-i.Watanabe, K.Yoshida,
Ulrich ideals and modules , Math.Proc. Camb. Phil. Soc. (2014), 137–166.[7] S.Goto, K.Ozeki, R.Takahashi, K.-i.Watanabe, K.Yoshida,
Ulrich ideals and modules over two-dimensional rational singularities , submitted.[8] S.Goto, K.Ozeki, R.Takahashi, K.-i.Watanabe, K.Yoshida,
Ulrich ideals and modules for simplesingularities , in preparation.[9] S. Goto and Y. Shimoda,
On the Rees algebras of Cohen-Macaulay local rings , Commutativealgebra (Fairfax, Va., 1979), Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, NewYork, 1982, pp. 201–231. MR 655805 (84a:13021)[10] E. Hyry and K. E. Smith,
On a non-vanishing conjecture of Kawamata and the core of anideal , Amer. J. Math. (2003), no. 6, 1349–1410.[11] M. Kato,
Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension 2 , Math.Ann. , (1976), 243–250.[12] H.Laufer,
On rational singularities , Amer. J. Math. , (1972), 31–62.[13] H.Laufer, On minimally elliptic singularities , Amer. J. Math. , (1975), 1257–1295.[14] J. Lipman, Rational singularities with applications to algebraic surfaces and unique factoriza-tion , Inst. Hautes ´Etudes Sci. Publ. Math. (1969), 195–279.[15] M. Morales, Calcul des quelques invariants des singularit´es de surface normale , Enseign. Math. , (1983), 191–203.[16] T. Okuma, Numerical Gorenstein elliptic singularities , Math. Z., , (2005), 31–62.[17] H. Pinkham,
Normal surface singularities with C ∗ action , Math. Ann. (1977), 183–193.[18] M. Reid, Chapters on algebraic surfaces , Complex algebraic geometry, IAS/Park City Math.Ser., vol. , Amer. Math. Soc. Providence, RI, 1997, pp. 3–159.[19] A. R¨ohr, A vanishing theorem for fiber bundles on resolution of surface singularities , Abh.Math. Sem. Univ. Hamburg, , (1995), 215–223.[20] J. Wahl, Vanishing Theorems for Resolutions of Surface Singularities , lnventiones math. (1975), 17–41.[21] J. Wahl, A characteristic number for links of surface singularities , J. Amer. Math. Soc. ,(1990), 625–637.[22] J. Watanabe, m-full ideals , Nagoya Math. J. (1987), 101–111.[23] K.-i.Watanabe and K.Yoshida, Hilbert-Kunz multiplicity, McKay correspondence and goodideals in two-dimensional rational singularities , manuscripta math. (2001), 275–294.[24] S.S.-T. Yau,
On maximally elliptic singularities , Trans. Amer. Math. Soc. (1980), 269–329. Tomohiro Okuma)
Department of Mathematical Sciences, Faculty of Science, Ya-magata University, Yamagata, 990-8560, Japan.
E-mail address : [email protected] (Kei-ichi Watanabe) Department of Mathematics, College of Humanities and Sci-ences, Nihon University, Setagaya-ku, Tokyo, 156-8550, Japan
E-mail address : [email protected] (Ken-ichi Yoshida) Department of Mathematics, College of Humanities and Sci-ences, Nihon University, Setagaya-ku, Tokyo, 156-8550, Japan
E-mail address : [email protected]@math.chs.nihon-u.ac.jp