SEMIGROUP AND POINCARÉ SERIES FOR DIVISORIALVALUATIONSWILLEM VEYS

1. Introduction
The semigroup of values and the Poincaré series are introduced to study properties of afinite set of divisorial valuations. We mention for example [5], [6], [7], [8] and [10]. The
two-dimensional case is the most understood and well behaved. There it is known ([6])
that a set of valuations coming from a modification of K 2 , with K algebraically closed,
has a finite generating sequence and that the semigroup of values is finitely generated.
Besides, there exists an explicit description of the generating sequence in terms of the dualgraph of the modification; this graph even determines the generators of the semigroup.
Furthermore, when K = C, there exists an explicit formula for the Poincaré series interms of the topology of the exceptional locus of the modification.
In higher dimensions, Lemahieu ([10]) looks at toric constellations on Cd . In particular,
all valuations
considered, are monomial. She shows that then the Poincaré series
Qd that areν(xi )
equals 1/ i=1 (1 − t), where x1 , . . . , xd is the coordinate system of Cd and where ν(xi )is the valuation vector of xi .
Here we present in arbitrary dimension the work of Van Langenhoven and the author([13]), generalizing the results above, assuming that the given set of valuations has afinite generating sequence.
We use the notation N = {n ∈ Z | n > 0} and N∗ = {n ∈ Z | n > 0}. The homogeneous
maximal ideal of the ring K[x1 , . . . , xd ], i.e., (x1 , . . . , xd ), is denoted by m.
2. PreliminariesDefinition 2.1. Let K be a field. In this text a modification π of K d is a composition ofblow-ups
πr−1π1π0πr
Xr −→Xr−1 −→ · · · −→
X0 −→
K d,
π = π0 ◦ π1 ◦ · · · ◦ πr , where π0 is the blow-up of K d at the origin. All the other πσ , 1 ≤σ ≤ r, are blow-ups at smooth, irreducible centers Zσ (⊆ Xσ−1 ) that are defined over K,
have codimension at least 2, and are contained in and have normal crossings with theexceptional locus of π0 ◦ · · · ◦ πσ−1 .
Denote the exceptional locus of πσ , as well as its consecutive strict transforms, by Eσ .
For a polynomial g ∈ K[x1 , . . . , xd ]\{0}, let νσ (g) be the vanishing order of g ◦ π alongEσ (and νσ (0) = ∞). The map νσ defines a divisorial valuation on K(x1 , . . . , xd ).
Now let us fix s different components Eσ1 , . . . , Eσs of the exceptional divisor E = π −1 (0),let νi = νσi for i ∈ {1, . . . , s}, and take V = {ν1 , . . . , νs }. When g ∈ K[x1 , . . . , xd ], we callν(g) = (ν1 (g), . . . , νs (g)) the valuation vector of g.
12Remark 2.2. When K is a field of characteristic 0, any finite set of divisorial valuations
centered at the origin of K d can be viewed as such a set V (using resolution of indeterminacies). So in fact our setup is very general.
Definition 2.3. The semigroup of values of V is the additive subsemigroup of Ns given by
SV = ν(g) = ν1 (g), . . . , νs (g) | g ∈ K[x1 , . . . xd ]\{0} .
Its saturation isSVsat = x ∈ Zs | kx ∈ SV for some k ∈ N∗ .
Definition 2.4. Let V = {ν1 , . . . , νs } be defined as above. Then for every v ∈ Zs we define
its valuation ideal
J(v) = {g ∈ K[x1 , . . . , xd ] | ν(g) > v} .
Following [2] and [7] we define the Poincaré series associated to V asQs
(ti − 1) X
PV (t1 , . . . , ts ) = i=1d(v)tv ,t1 · · · ts − 1 v∈Zs
where d(v) = dimK J(v)/J(v + (1, 1, . . . , 1)).
Definition 2.5. Let Λ = {qα }α∈A be a subset of m. A monomial in Λ is defined as a finite
product in the elements of Λ:
YqαMα with every Mα ∈ N∗ .
We call Λ a generating sequence for V if for every v ∈ Ns the ideal J(v) is generated (asan ideal) by all monomials in Λ that are also in J(v).
3. Results
Theorem 3.1. Let K be an infinite (or big enough) field. If Λ = {q1 , . . . , qk } is a finitegenerating sequence for V , then SV is a finitely generated semigroup.
In particular, this applies to any toric constellation, since then each V has Λ = {x1 , . . . , xd }as a generating sequence.
The proof we provide, gives us a method to compute the generators of SV . However, thismethod introduces a lot of extra variables. On the other hand, (generators of) SVsat canbe computed more easily. In our algorithms, hyperplanes defining relevant cones can becomputed by Fourier-Motzkin-elimination [12], and generators using Normaliz ([1]).
Example 3.2. We present a set of valuations V , coming from an ‘easy’ toric constellation
on C3 , but with non-saturated semigroup SV .
We start by the blow-up π0 : X0 → C3 at the origin Z0 with exceptional divisor E0 . This
blow-up can be described by three affine charts, say K1 , K2 and K3 . Next we blow up at
the origin Z1 of K1 , which gives us an extra exceptional divisor E1 and charts K1.1 , K1.2and K1.3 . Then we blow up at the origin Z2 of K2 , which gives us E2 and three extra
affine charts. After this, we blow up at the origins of K1.1 and K1.2 ; this gives rise to theexceptional divisors E11 and E12 . We can sketch this modification as follows.

SEMIGROUP AND POINCARE SERIES

E1E1Z12Z0π0←−Z1Z2π

1←−π2E0Z113E2π

11←−−E0E11π12E12E0It gives us five divisorial valuations V = {ν0 , ν1 , ν2 , ν11 , ν12 }. Since they are all monomial,Λ = {x, y, z} is a generating sequence for V . The associated valuation vectors of x, y andz are (1, 1, 2, 1, 2), (1, 2, 1, 3, 3) and (1, 2, 2, 3, 4), respectively.
Although the modification is relatively simple, the semigroup SV is more complex than
one might expect. For one, it is not saturated. Thanks to Normaliz [1] we can computethe 1820 generators of a certain rational cone used in the proof of Theorem 3.1. From
this we computed the following 22 generators of SV :(1, 1, 1, 1, 2)(1, 1, 2, 1, 2)(1, 2, 1, 2, 3)(1, 2, 1, 3, 3)(1, 2, 2, 2, 4)(1, 2, 2, 3, 4)(2, 3, 2, 3, 6)(2, 3, 2, 4, 6)(2, 4, 3, 4, 8)(2, 4, 3, 5, 8)(2, 4, 3, 6, 8)(2, 4, 4, 4, 7)(2, 4, 4, 5, 7)(3, 6, 4, 6, 12)(3, 6, 4, 7, 12)(3, 6, 4, 8, 12)(3, 6, 4, 9, 12)(3, 6, 6, 6, 10)(3, 6, 6, 7, 10)(3, 6, 6, 8, 10)(4, 8, 8, 12, 15)(5, 10, 10, 15, 18)

We see that (6, 12, 12, 18, 22) = ν(z 6 + x3 y 4 z) = (1, 2, 2, 3, 4) + (5, 10, 10, 15, 18) ∈ SV . But
it turns out that (3, 6, 6, 9, 11) cannot be an element of SV , and hence SV is not saturated.
Our method to compute SVsat yields that (n0 , n1 , n2 , n11 , n12 ) ∈ Z5 is an element of SVsat if
and only if it satisfies all the inequalitiesn2 − n0 > 0
2n0 − n1 > 0
3n2 − n12 > 0
n11 − n1 > 0
2n0 − n2 > 0
3n12 − 4n1 − n2 > 0n12 − n0 − n1 > 0 2n1 − n12 > 05n12 − 3n2 − 4n11 > 02n1 − n0 − n11 > 0It turns out that SVsat also has 22 generators, more precisely those for SV above, where(4, 8, 8, 12, 15) is replaced by (3, 6, 6, 9, 11).
Remark 3.3. In dimensional two the semigroup of values of a modification over an algebraically closed field is always finitely generated and there always exists a finite generating
sequence that can be expressed in terms of the dual graph of the modification (see [6]).
In higher dimensions, the semigroup of values of a modification is not always finitely generated. Note that in such cases there cannot be a finite generating sequence. The factthat there is different behaviour in higher dimensions, could be expected from examples
in similar settings such as [3, Remark 1.24] and [4].
Our counterexample to confirm that statement is constructed as follows. We start withthe blow-up at the origin of C3 and then we blow up at nine very general points on thefirst exceptional divisor. This gives us a set V of ten valuations, such that SV is not
finitely generated. Our proof uses the fact that the cone of curves of the blow-up of P2
at nine very general points is not finitely generated [11] [9, II Lemma 4.12]. In fact, weconstruct more precisely an explicit list of generators of SVsat .

4Theorem 3.4. If Λ = {q1 , . . . , qk } is a finite generating sequence for V , then the Poincaré
series associated to V is a rational function whose denominator is equal tokY
(1 − tν(qi ) ).
i=1

Our long and technical proof uses the following strategy. We endow the polynomial
ring K[Q1 , . . . , Qk ] with r + 1 new monomial valuations v̂i , defined by v̂i (Qj ) = νi (qj ),to which we associate a certain ‘mixed’ semigroup ΣV . We also consider a morphism
φ : K[Q1 , . . . , Qk ] → K[x1 , . . . , xd ], sending Qj to the lowest degree homogeneous component of qj . Then ker φ determines a certain ideal of the semigroup ΣV , whose (finitelymany) generators describe in a combinatorial way the dimensions in the definition of thePoincaré series. (Remember that every ν(qi ) > 1, because Λ ⊆ m.)
Remark 3.5. If ker φ = hBi is generated by one element, then the Poincaré series has thesimple expression
1 − tv̂(B).
PV (t) = Qk
ν(qi ))
i=1 (1 − tExample 3.6. Let π0 : X0 → C3 be the blow-up at the origin with exceptional divisor E0 .
We blow up further at four points in this E0 , say (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and(1 : 1 : 1). This gives us five divisorial valuations, say ν0 , ν1 , ν2 , ν3 and ν4 .

Z0π0←−ZZ4Z32E0E3 E2E1Z1π ◦π ◦12←−−−−π3 ◦π4E4E0A somewhat lengthy computation yields that Λ = {x, y, z, x−z, y−z, x−y} is a generating
sequence for the set of valuations V = {ν0 , ν1 , ν2 , ν3 , ν4 }. The corresponding valuation
vectors are ν(x) = (1, 1, 2, 2, 1), ν(y) = (1, 2, 1, 2, 1), ν(z) = (1, 2, 2, 1, 1), ν(x − z) =(1, 1, 2, 1, 2), ν(y − z) = (1, 2, 1, 1, 2) and ν(x − y) = (1, 1, 1, 2, 2).
Firstly, we construct the surjective morphism φ : C[Q1 , . . . , Q6 ] → C[x, y, z] : Q1 7→x, Q2 7→ y, Q3 7→ z, Q4 7→ x − z, Q5 7→ y − z and Q6 7→ x − y. Secondly we define theinduced monomial valuations on C[Q] where the valuations of the Qi areQ1 Q2 Q3 Q4 Q5 Q6v̂0 111111v̂1 122121v̂2 212211v̂3 221112v̂4 111222In this case ker φ = (Q1 − Q2 − Q6 , Q2 − Q3 − Q5 , Q4 − Q5 − Q6 ). It is pretty complicatedto find a basis of the relevant ideal of ΣV . Knowing such a basis, we can compute thatthe Poincaré series PV (t) is equal toSEMIGROUP AND POINCARE SERIES

5

1 + t(1,1,1,1,1) − t(1,2,1,1,1) − t(1,1,2,1,1) − t(1,1,1,2,1) − t(1,1,1,1,2)+ t(2,3,3,3,2) + t(2,3,3,2,3) + t(2,3,2,3,3) + t(2,2,3,3,3) − t(2,3,3,3,3) − t(3,4,4,4,4)
PV (t) =.
(1,2,2,1,1)(1,2,1,2,1)(1,2,1,1,2)(1 − t)(1 − t)(1 − t)
(1,1,2,2,1)(1,1,2,1,2)(1 − t)(1 − t)(1 − t(1,1,1,2,2) )Note that this numerator cannot be factored. This is different from the case where π is a
modification of C2 (see for example [6] or [7]).
Question. For a given set of divisorial valuations V , how can one ‘guess’ a generating
sequence, and prove efficiently that it is really a generating sequence?
Question. Is the Poincaré series PV (t) for the example in Remark 3.3 a rational function
or not?
References[1] W. Bruns and B. Ichim. Normaliz: algorithms for affine monoids and rational cones. J. Algebra,324(5):1098–1113, 2010.
[2] A. Campillo, F. Delgado, and S. M. Gusein-Zade. The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J., 117(1):125–156, 2003.
[3] A. Campillo, G. Gonzalez-Sprinberg, and M. Lejeune-Jalabert. Clusters of infinitely near points.
Math. Ann., 306(1):169–194, 1996.
[4] V. Cossart, C. Galindo, and O. Piltant. Un exemple effectif de gradué non noethérien associé à une
valuation divisorielle. Ann. Inst. Fourier (Grenoble), 50(1):105–112, 2000.
[5] S. D. Cutkosky, J. Herzog, and A. Reguera. Poincaré series of resolutions of surface singularities.
Trans. Amer. Math. Soc., 356(5):1833–1874 (electronic), 2004.
[6] F. Delgado, C. Galindo, and A. Núñez. Generating sequences and Poincaré series for a finite set ofplane divisorial valuations. Adv. Math., 219(5):1632–1655, 2008.
[7] F. Delgado and S. M. Gusein-Zade. Poincaré series for several plane divisorial valuations. Proc.
Edinb. Math. Soc. (2), 46(2):501–509, 2003.
[8] C. Galindo and F. Monserrat. Finite families of plane valuations: value semigroup, graded algebra
and Poincaré series. In Zeta functions in algebra and geometry, volume 566 of Contemp. Math., pages
189–212. Amer. Math. Soc., Providence, RI, 2012.
[9] J. Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics andRelated Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin,
1996.
[10] A. Lemahieu. Poincaré series of a toric variety. J. Algebra, 315(2):683–697, 2007.
[11] M. Nagata. On rational surfaces. II. Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 33:271–293,
1960/1961.
[12] A. Schrijver. Theory of linear and integer programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester, 1986. A Wiley-Interscience Publication.
[13] L. Van Langenhoven and W. Veys. Semigroup and Poincaré series for a finite set of divisorial valuations. Rev. Mat. Complutense, 28(1):191–225, 2015.
KU Leuven, Departement of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium
E-mail address: wim.veys@kuleuven.be