Toric varieties

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ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927872.2011.552084

A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER IDEALS ON TORIC VARIETIES Jen-Chieh Hsiao

Department of Mathematics, Purdue University, West Lafayette, Indiana, USA

We construct a 3-dimensional 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 2-dimensional normal toric varieties is also provided.

Key Words: Multiplier ideals; Subadditivity formula; Toric varieties.

2010 Mathematics Subject Classification: 14F18; 14M25.

1. INTRODUCTION

Demailly 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, 2010. Communicated by I. Swanson.

Address correspondence to Jen-Chieh Hsiao, Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, USA; E-mail: jhsiao@math.purdue.edu

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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 2-dimensional 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 KX

of X is Cartier, so there exists a u₀ ∈ M ∩ %^∨ such that !u₀' n_i"= 1 where the n^′is 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 + u0 ∈ 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 3-dimensional normal semigroup ring R= %$x²y' xy' xy²' z&, % a field. Notice that R is a complete intersection, and hence Gorenstein. Note also that

u₀ = !1' 1' 1"#

Consider the following two ideals of R:

"= &x²y⁴' x¹⁰y⁶z²('

#= &x¹²y⁷' x¹⁰y⁶z²(#

Then "#= !x¹⁴y¹¹' x¹²y¹⁰z²' x²²y¹³z²' x²⁰y¹²z⁴"#Denote w₁ = !14' 11' 0"' w₂ = !12' 10' 2"' w₃ = !22' 13' 2"' w₄= !20' 12' 4"#

Observe that the lattice point

v= !18' 12' 2" ∈ relint Newt!"#"#

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To see this, consider the four points

v₁ = w1 = !14' 11' 0"' v₂= w1+ !4' 2' 0" = !18' 13' 0"' v₃= w1+ !2' 1' 4" = !16' 12' 4"' v₄= 1

2!w₃+ w4"=

! 21'25

2 '3

"

#

They are in Newt!"#" and do not lie on a plane, namely, they are affinely independent. Since

v= 5

16v₁+ 1

16v₂+1 8v₃+ 1

2v₄' it is in relint Newt!"#".

Now, since −u0+ v = !17' 11' 1", by (2.1) x¹⁷y¹¹z∈ !!"#"#

We claim that

x¹⁷y¹¹z* !!""!!#"#

An element in !!""!!#" is a finite sum of monomials of the form c· x⁽x⁾ where c∈ k, (' )∈ M ∩ %^∨, (+ u0∈ relint Newt!"", and ) + u0∈ relint Newt!#". If x¹⁷y¹¹z∈

!!""!!#", then

−u0+ v = ( + )

for some (, ) as above. This means v= !18' 12' 2" can be written as a sum of a lattice point !(+ u0"in relint Newt!"" and a lattice point ) in−u0+ relint Newt!#".

We check that this is not possible.

Suppose ( and ) are lattice points satisfying (+ u0+ ) = v = !18' 12' 2".

Write (^′ = ( + u0= !a1' a₂' a₃" and )= !b1' b₂' b₃", so

!a₁+ b1' a₂+ b2' a₃+ b3"= v = !18' 12' 2"#

We will show that in each case either (^′ * relint Newt!#" or ) + u0 * 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)

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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'

4x− 2y + 3z > 34' z > 0+# (3.2) We consider the following cases:

Case I: If a₁≥ 7, then b2≤ 5 and ) + u0* relint Newt!#". To see this, suppose )+ u0= !b1+ 1' b2+ 1' b3+ 1" ∈ relint Newt!#". By (3.2), 2!b1+ 1" −

!b₂+ 1" > 14 and −!b1+ 1" + 2!b2+ 1" > 2. So 4!b2+ 1" − 4 > 2!b1+ 1" > 14 +

!b₂+ 1" and hence b2>5, which is a contradiction.

Case II. If a₂ ≤ 4, then (^′ * relint Newt!"". Indeed, suppose (^′ = !a1' a₂' a₃∈ relint Newt!"". By (3.1), 2a1− a2 >0 and −a1+ 4a2>14. So 8a₂− 28 > 2a1 > a₂ and hence a₂>4.

Case III. Suppose a₂= 5 and b2= 7.

a) If a₁≥ 6, then (^′ * relint Newt!"". Indeed, suppose (^′ = !a1' a₂' a₃"∈ relint Newt!"". By (3.1), −a1+ 4a2>14 and hence a₁ <4a₂− 14 = 6.

b) If a₁ ≤ 5, then b1≥ 13. This implies ) + u0* relint Newt!#". Indeed, suppose ) + u₀= !b1+ 1' b2+ 1' b3+ 1" ∈ relint Newt!#". By (3.2), −!b1+ 1" + 2!b2+ 1" > 2 and hence b₁ <2!b₁+ 1" − 3 = 13.

Case IV: Suppose a₂ = b2− 6.

a) If b₁ ̸= 10, then ) + u0* relint Newt!#". To see this, suppose ) + u0= !b1+ 1' b₂+ 1' b3+ 1" ∈ relint Newt!#". By (3.2) again, 2!b1+ 1" − !b2+ 1" > 14 and

−!b1+ 1" + 2!b2+ 1" > 2. This forces b1 = 10.

b) If b₁= 10, then (^′ = !a1' a₂' a₃"= !8#6' a3"and ) = !b1' b₂' b₃"= !10' 6' b3".

i) If a₃ ≤ 0, then (^′ * relint Newt!"" by (3.1).

ii) If a₃ >2, then b₃ <0. In this case, )+ u0* relint Newt!#" by (3.2).

iii) If (^′ = !a1' a₂' a₃"= !8' 6' 1", then −a1+ 2a2+ 2a3 = 6. So (^′ * relint Newt!"" by (3.1).

iv) If (^′ = !a1' a₂' a₃"= !8' 6' 2", then ) = !b1' b₂' b₃"= !10' 6' 0". So 4!b1+ 1"− 2!b2+ 1" + 3!b3+ 1" = 33 < 34. Hence ) + u0* 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:

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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^′ = %$x²y' xy' xy²&' r = x⁸y⁶' I^′ = &x²y⁴(' J^′ = &x¹²y⁷('

Z= x¹⁰y⁶z²#

Then rZ= x¹⁸y¹²z² is exactly the crucial point we considered in the example.

4. TWO-DIMENSIONAL CASE

Let R = %$M ∩ %^∨&, % a field, be a 2-dimensional normal toric ring and denote X= Spec!R". Then there exists a primitive lattice point w0∈ M ∩ %^∨ such that !w₀' 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 u₀= w0/r. By Theorem 4.8 in [7], for any monomial ideal " in R

!!""= &x^w ∈ R ' w + u0∈ relint Newt!""(# (4.1) The following theorem establishes the subadditivity formula on two-dimensional normal toric rings.

Theorem 4.1. For any pair of monomial ideal ", # in R,

!!"#"⊆ !!""!!#"#

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Proof. Write " = &xâ ' a ∈ A( and # = &x^b ' b ∈ B( for some finite sets A and B in M∩ %^∨. We assume that *xâ' a ∈ A+ and *x^b' b ∈ B+ are the sets of monomial minimal generators of " and #, respectively. Then "# = &xâ^+b' a ∈ A and b ∈ B(.

Let (₁' # # # ' (_k be the vertices of the Newton polyhedron Newt!"#" such that (₁+ ,1'conv*(₁' (₂+' # # # 'conv*(_k₋₁' (_k+' and (_k+ ,2

form the boundary of Newt!"#", where ,1' ,₂ are the two rays of %^∨. Then

Newt!"#"=

k#−1 i=1

!conv*(_i' (_i₊₁++ %^∨"#

Note also that the (^′_is are of the form a_i+ bi for some a_i ∈ A and bi ∈ B. Suppose that for some i∈ *1' # # # ' k − 1+, we have ai ̸= ai+1 and b_i ̸= bi+1. Then a_i + bi+1= a_i₊₁+ bi, 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 + bi+1 to the sequence (₁' # # # ' (_k. So we obtain a sequence, say )₁ = a^′₁+ b^′1' # # # ' )_s = a^′s+ bs^′, such that, for each i∈ *1' # # # ' s − 1+, either a^′i = a^′i+1 or b^′_i = b^′i+1, and that

Newt!"#"=

s#−1 i=1

!conv*)_i' )_i₊₁++ %^∨"#

Now, observe that

relint Newt!"#"⊆

s#−1 i=1

!relint -_i"'

where -_i = conv*)i' )_i₊₁++ %^∨. If x^p ∈ !!"#", then by (4.1) p + u0∈ relint Newt!"#"

and hence in relint -_i₀ for some i₀. Without loss of generality, we may assume a^′_i₀ = a^′_i₀₊₁. So

p+ u0∈ relint -i₀ = a^′i₀ + $relint !conv*b^′i₀' b^′_i

0+1++ %^∨"&⊆ a^′i₀ + relint Newt!#"#

Therefore, p∈ a^′i₀ + $−u0+ relint Newt!#"&# Since a^′i₀ + u0 ∈ relint Newt!"", by (4.1) we conclude that x^p ∈ !!""!!#", as desired. "

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+ u0∈ arelint Newt!#". This cannot be extended to the higher dimensional case. From the example in Section 3, x¹⁷y¹¹z∈ !!"#" and u0 = !1' 1' 1". Newt!"" is minimally generated by x²y⁴ and x¹⁰y⁶z². But !16' 8' 2" = !18' 12' 2" − !2' 4' 0" and !8' 6' 0" = !18' 12' 2" − !10' 6' 2"

are not in relint Newt!#" by (3.2). Similarly, Newt!#" is minimally generated by x¹2y⁷ and x¹⁰y⁶z². But !6' 5' 2"= !18' 12' 2" − !12' 7' 0" and !8' 6' 0" = !18' 12' 2" −

!10' 6' 2" are not in relint Newt!"" by (3.1).

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ACKNOWLEDGMENTS

The author was partially supported by NSF under grant DMS 0555319 and DMS 0901123. 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.

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