A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER IDEALS ON TORIC VARIETIES


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1 Communications in Algebra, 40: , 2012 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER IDEALS ON TORIC VARIETIES JenChieh Hsiao Department of Mathematics, Purdue University, West Lafayette, Indiana, USA Key Words: We construct a 3dimensional complete intersection toric variety on which the subadditivity formula doesn t hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2dimensional normal toric varieties is also provided. Multiplier ideals; Subadditivity formula; Toric varieties Mathematics Subject Classification: 14F18; 14M INTRODUCTION D ly et al. [2] proved the subadditivity theorem for multiplier ideals on smooth complex varieties, which states This theorem is responsible for several applications of multiplier ideals in commutative algebra, in particular to symbolic powers [3] and Abhyankar valuations [4]. In a later article, Takagi and Watanabe [9] investigated the extent to which the subadditivity theorem remains true on singular varieties. They showed that on Gorenstein normal surfaces, the subadditivity formula holds if and only if the variety is log terminal [9, Theorem 2.2]. Furthermore, they gave an example of a Gorenstein normal toric threefold on which the formula is not satisfied [9, Example 3.2]. This led Takagi and Watanabe to ask the following question. Question 1.1. Let R be a Gorenstein toric ring and, be monomial ideals of R. Is it true that? Received August 17, 2010; Revised December 27, Communicated by I. Swanson. Address correspondence to JenChieh Hsiao, Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, USA;
2 A COUNTEREXAMPLE FOR SUBADDITIVITY FORMULA 1619 The purpose of this article is to provide a counterexample to Question 1.1. We will also give, in Section 4, a combinatorial proof of the subadditivity formula on any 2dimensional normal toric rings. The standard notation and facts in [5] will be used freely in the presentation. 2. MULTIPLIER IDEALS ON TORIC VARIETIES Let be a field and R = M be the coordinate ring of an affine normal Gorenstein toric variety. Denote X = Spec R. In this case, the canonical divisor K X of X is Cartier, so there exists a u 0 M such that u 0 n i = 1 where the n i s are the primitive generators of. For any monomial ideal of R, denote Newt the Newton polyhedron of and relint Newt the relative interior of Newt. The multiplier ideal of in R admits a combinatorial description. Proposition 2.1. = x w R w + u 0 relint Newt (2.1) This is a result by Hara and Yoshida [7, Theorem 4.8] which is generalized by Blickle [1] to arbitrary normal toric varieties. 3. THE EXAMPLE Consider the 3dimensional normal semigroup ring R = x 2 y xy xy 2 z, a field. Notice that R is a complete intersection, and hence Gorenstein. Note also that Consider the following two ideals of R: u 0 = = x 2 y 4 x 10 y 6 z 2 = x 12 y 7 x 10 y 6 z 2 Then = x 14 y 11 x 12 y 10 z 2 x 22 y 13 z 2 x 20 y 12 z 4 Denote w 1 = w 2 = w 3 = w 4 = Observe that the lattice point v = relint Newt
3 1620 HSIAO To see this, consider the four points v 1 = w 1 = v 2 = w = v 3 = w = v 4 = 1 ( 2 w 3 + w 4 = ) 2 3 They are in Newt and do not lie on a plane, namely, they are affinely independent. Since v = 5 16 v v v v 4 it is in relint Newt. Now, since u 0 + v = , by (2.1) We claim that x 17 y 11 z x 17 y 11 z An element in is a finite sum of monomials of the form c x x where c k, M, + u 0 relint Newt, and + u 0 relint Newt. If x 17 y 11 z, then u 0 + v = + for some, as above. This means v = can be written as a sum of a lattice point + u 0 in relint Newt and a lattice point in u 0 + relint Newt. We check that this is not possible. Suppose and are lattice points satisfying + u 0 + = v = Write = + u 0 = a 1 a 2 a 3 and = b 1 b 2 b 3, so a 1 + b 1 a 2 + b 2 a 3 + b 3 = v = We will show that in each case either relint Newt or + u 0 relint Newt. First, note that the Newton polyhedron Newt is the intersection of halfspaces determined by the following five hyperplanes: 2x y = 0 x + 4y = 14 x + 2y = 2 x + 2y + 2z = 6 z= 0. So we have relint Newt = x y z M 2x y>0 x + 4y >14 x + 2y >2 x + 2y + 2z >6 z>0 (3.1)
4 A COUNTEREXAMPLE FOR SUBADDITIVITY FORMULA 1621 Also, Newt is the intersection of the halfspace determined by the following four hyperplanes: 2x y = 14 x + 2y = 2 4x 2y + 3z = 34 z= 0. We have relint Newt = x y z M 2x y>14 x + 2y >2 We consider the following cases: 4x 2y + 3z >34 z>0 (3.2) Case I: If a 1 7, then b 2 5 and + u 0 relint Newt. To see this, suppose + u 0 = b b b relint Newt. By (3.2), 2 b b >14 and b b >2. So 4 b > 2 b >14 + b and hence b 2 > 5, which is a contradiction. Case II. If a 2 4, then relint Newt. Indeed, suppose = a 1 a 2 a 3 relint Newt. By (3.1), 2a 1 a 2 > 0 and a 1 + 4a 2 > 14. So 8a 2 28 > 2a 1 >a 2 and hence a 2 > 4. Case III. Suppose a 2 = 5 and b 2 = 7. a) If a 1 6, then relint Newt. Indeed, suppose = a 1 a 2 a 3 relint Newt. By (3.1), a 1 + 4a 2 > 14 and hence a 1 < 4a 2 14 = 6. b) If a 1 5, then b This implies + u 0 relint Newt. Indeed, suppose + u 0 = b b b relint Newt. By (3.2), b b >2 and hence b 1 < 2 b = 13. Case IV: Suppose a 2 = b 2 6. a) If b 1 10, then + u 0 relint Newt. To see this, suppose + u 0 = b b b relint Newt. By (3.2) again, 2 b b >14 and b b >2. This forces b 1 = 10. b) If b 1 = 10, then = a 1 a 2 a 3 = 8 6 a 3 and = b 1 b 2 b 3 = 10 6 b 3. i) If a 3 0, then relint Newt by (3.1). ii) If a 3 > 2, then b 3 < 0. In this case, + u 0 relint Newt by (3.2). iii) If = a 1 a 2 a 3 = 8 6 1, then a 1 + 2a 2 + 2a 3 = 6. So relint Newt by (3.1). iv) If = a 1 a 2 a 3 = 8 6 2, then = b 1 b 2 b 3 = So 4 b b b = 33 < 34. Hence + u 0 relint Newt by (3.2). Remark 3.1. We briefly explain the idea behind the example. Recall that the integral closure I of a monomial ideal I in a normal toric ring R is determined by Newt I (see, for example, [8]): I = x w R w Newt I So Question 1.1 is closely related to the containment I J IJ for monomial ideals of R. Huneke and Swanson provide a trick to construct examples where the strict containment I J IJ occur (see [6, Example 1.4.9] and the remark after it). We repeat their construction here:
5 1622 HSIAO Choose a ring R and a pair of ideal I, J in R such that I + J I + J Pick an element r I + J \ I + J Set R = R Z for some variable Z over R and set I = I R + ZR J = J R + ZR Then I and J are integrally closed and rz IJ\I J This kind of construction doesn t always guarantee a counterexample to Question 1.1. However, a suitable choice of r, Z, R, I, and J will do. In our example, take R = x 2 y xy xy 2 r = x 8 y 6 I = x 2 y 4 J = x 12 y 7 Z = x 10 y 6 z 2 Then rz = x 18 y 12 z 2 is exactly the crucial point we considered in the example. 4. TWODIMENSIONAL CASE Let R = M, a field, be a 2dimensional normal toric ring and denote X = Spec R. Then there exists a primitive lattice point w 0 M such that w 0 n i = r 0 where the n is are the primitive generators of. So the canonical divisor K X of X is Cartier and R is Gorenstein. Set u 0 = w 0 /r. By Theorem 4.8 in [7], for any monomial ideal in R = x w R w + u 0 relint Newt (4.1) The following theorem establishes the subadditivity formula on twodimensional normal toric rings. Theorem 4.1. For any pair of monomial ideal, in R,
6 A COUNTEREXAMPLE FOR SUBADDITIVITY FORMULA 1623 Proof. Write = x a a A and = x b b B for some finite sets A and B in M. We assume that x a a A and x b b B are the sets of monomial minimal generators of and, respectively. Then = x a+b a A and b B. Let 1 k be the vertices of the Newton polyhedron Newt such that conv 1 2 conv k 1 k and k + 2 form the boundary of Newt, where 1 2 are the two rays of. Then k 1 Newt = conv i i+1 + i=1 Note also that the i s are of the form a i + b i for some a i A and b i B. Suppose that for some i 1 k 1, we have a i a i+1 and b i b i+1. Then a i + b i+1 = a i+1 + b i, lie on boundary segment conv i i+1, since otherwise they lie on different sides of conv i i+1 which is a contradiction. For any such i, we insert the point a i + b i+1 to the sequence 1 k. So we obtain a sequence, say 1 = a 1 + b 1 s = a s + b s, such that, for each i 1 s 1, either a i = a i+1 or b i = b i+1, and that Now, observe that s 1 Newt = conv i i+1 + i=1 s 1 relint Newt relint i where i = conv i i+1 +. If x p, then by (4.1) p + u 0 relint Newt and hence in relint i0 for some i 0. Without loss of generality, we may assume a i 0 = a i So p + u 0 relint i0 = a i 0 + relint conv b i 0 b i a i 0 + relint Newt Therefore, p a i 0 + u 0 + relint Newt Since a i 0 + u 0 relint Newt, by (4.1) we conclude that x p, as desired. i=1 Remark 4.2. As one can see in the proof of Theorem 4.1, the choice of i s is essential. For any x p we are able to choose a = Newt such that x a is in the set of monomial minimal generators of and that p + u 0 arelint Newt. This cannot be extended to the higher dimensional case. From the example in Section 3, x 17 y 11 z and u 0 = Newt is minimally generated by x 2 y 4 and x 10 y 6 z 2. But = and = are not in relint Newt by (3.2). Similarly, Newt is minimally generated by x 1 2y 7 and x 10 y 6 z 2. But = and = are not in relint Newt by (3.1).
7 1624 HSIAO ACKNOWLEDGMENTS The author was partially supported by NSF under grant DMS and DMS The author would like to thank his advisor, Uli Walther, for his encouragement during the preparation of this work. He is also grateful to the refree for the careful reading and useful suggestions. REFERENCES [1] Blickle, M. (2004). Multiplier ideals and modules on toric varieties. Math. Z. 248(1): [2] D ly, J.P., Ein, L., Lazarsfeld, R. (2000). A subadditivity property of multiplier ideals. Michigan Math. J. 48: [3] Ein, L., Lazarsfeld, R., Smith, K. E. (2001). Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144(2): [4] Ein, L., Lazarsfeld, R., Smith, K. E. (2003). Uniform approximation of Abhyankar valuation ideals in smooth function fields. Amer. J. Math. 125(2): [5] Fulton, W. (1993). Introduction to Toric Varieties. Annals of Mathematics Studies, Vol Princeton, NJ: Princeton University Press. The William H. Roever Lectures in Geometry. [6] Huneke, C., Swanson, I. (2006). Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, Vol Cambridge: Cambridge University Press. [7] Hara, N., Yoshida, K.I. (2003). A generalization of tight closure and multiplier ideals. Trans. Amer. Math. Soc. 355(8): (electronic). [8] Teissier, B. (2004). Monomial Ideals, Binomial Ideals, Polynomial Ideals. Trends in Commutative Algebra, Math. Sci. Res. Inst. Publ., Vol. 51. Cambridge: Cambridge University Press, pp [9] Takagi, S., Watanabe, K.I. (2004). When does the subadditivity theorem for multiplier ideals hold? Trans. Amer. Math. Soc. 356(10): (electronic).
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