Comptes Rendus Mathématique

. Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special ﬁber of an equisingular family whose generic ﬁber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.


Introduction
The monodromy conjecture for ideals predicts a relation between two invariants associated with an ideal, one originating from number theory and the other from differential topology.More precisely, it states that every pole L −s 0 of the motivic Igusa zeta function of an ideal I ⊂ C[x 0 , . . ., x n ] induces an eigenvalue of monodromy e 2πi s 0 associated with I .To date, this conjecture has only been proven in full generality for ideals in two variables [13].In higher dimension, there exist various partial results for one polynomial (see for instance the introduction of [2] for a list of references), while for multiple polynomials, the most general result so far is a proof for monomial ideals [4].
The motivic Igusa zeta function for a polynomial f ∈ C[x 0 , . . ., x n ] was introduced by Denef and Loeser [3] in analogy with the p-adic Igusa zeta function.Roughly speaking, it counts the C[t ]/(t m+1 )-points on the hypersurface X = { f = 0} and is defined in terms of the jet schemes of X .In a straightforward way, this definition can be generalized to any ideal I ⊂ C[x 0 , . . ., x n ] with associated subscheme X in C n+1 .Furthermore, the motivic Igusa zeta function is a rational function with poles of the form L −s 0 , where L denotes the class of the affine line in the Grothendieck ring of complex varieties and s 0 ∈ Q.
The eigenvalues of monodromy were originally introduced for one polynomial f ∈ C[x 0 , . . ., x n ] as follows.Assume that f (0) = 0 and consider f as a function f : C n+1 → C. Take a point x ∈ X = f −1 (0) and a small ball B in C n+1 with center x.Milnor [7] showed that the restriction f | B is a smooth locally trivial fibration over a small enough pointed disc D * = D \ {0} in C with center 0. The corresponding fiber is called the (local) Milnor fiber F x of f at x.The lifting of a loop in D * going once around the origin counterclockwise induces well-defined automorphisms on the cohomologies H m (F x , C), which are called the (local) monodromy transformations of f at x.An eigenvalue of monodromy or a monodromy eigenvalue of f is an eigenvalue of such a monodromy action at some point x ∈ X .Although the notion of local Milnor fiber is not well-defined for a general ideal I ⊂ C[x 0 , . . ., x n ], there is an abstract construction of Verdier [14] in this setting, yielding Verdier monodromy eigenvalues.Furthermore, A'Campo [1] showed an expression for the monodromy eigenvalues of a polynomial f in terms of an embedded resolution of singularities of f , which can be generalized to an ideal I by using a principalization of I , see [13].It is also wellknown that all eigenvalues of monodromy are roots of unity, or thus, of the form e 2πi s 0 for some s 0 ∈ Q.
This note investigates the monodromy conjecture for the class of binomial ideals in arbitrary dimension defining so-called space monomial curves Y ⊂ C g +1 with g ≥ 1.These curves arise as the special fibers of equisingular families of curves whose generic fibers are isomorphic to some plane branch.After introducing these curves in more detail, we show the results, without proofs, obtained in [10]; by studying the jet schemes of a space monomial curve Y ⊂ C g +1 , we are able to compute the motivic Igusa zeta function and to determine its poles, see Theorem 3 and Theorem 4, respectively.Then, we explain the approach of [6], again without proofs, to reduce the problem of studying the monodromy eigenvalues associated with Y ⊂ C g +1 by considering Y as a Cartier divisor on a generic embedding surface S.This way, we can use an A'Campo formula in terms of an embedded Q-resolution of Y ⊂ S to compute the monodromy zeta function of Y , see Theorem 8. Finally, we combine all results to conclude the monodromy conjecture for a space monomial curve Y ⊂ C g +1 in Theorem 10.

Space monomial curves with a plane semigroup
We start this note with introducing the type of singularities we are interested in.They are defined by binomial equations and arise as (equisingular) deformations of irreducible plane branches.
More precisely, let C := { f = 0} ⊂ (C 2 , 0) be the germ at the origin of a complex plane curve defined by an irreducible series f ∈ C[[x 0 , x 1 ]] satisfying f (0) = 0, and let be its associated valuation.The semigroup Γ(C ) of C is the image of this valuation and can be generated by a unique minimal set of generators (β 0 , . . ., β g ) with β 0 < • • • < β g and gcd(β 0 , . . ., β g ) = 1 (gcd being the greatest common divisor), see for instance [15].In terms of these generators, we define (Y , 0) ⊂ (C g +1 , 0) as the image of the monomial map This is an irreducible (germ of a) curve which is smooth outside the origin and which has the "plane" semigroup Γ(C ) as semigroup [12].We call Y the monomial curve associated with C .
The curve Y can be seen as a deformation of C as follows.Firstly, we put e i := gcd(β 0 , . . ., β i ) for i = 0, . . ., g , and n i := e i −1 /e i for i = 1, . . ., g .These are positive integers satisfying β 0 = e 0 > e 1 > • • • > e g = 1 and n i ≥ 2. Secondly, every n i β i for i = 1, . . ., g belongs to the semigroup generated by β 0 , . . ., β i −1 , or thus, there exist non-negative integers b i j for 0 ≤ j < i such that These integers are unique if we require that b i j < n j for j = 0.For notational reasons, we put n 0 := b 10 .Thirdly, we can identify a minimal generating sequence (x 0 , . . ., x g ) of the valuation ν C , consisting of elements x i ∈ C[[x 0 , x 1 ]] for i = 0, . . ., g such that ν C (x i ) = β i and which satisfy equations of the form where x g +1 = 0, c i ∈ C \ {0}, c i ,γ ∈ C, 0 ≤ γ j < n j for 1 ≤ j ≤ i , and i j =0 γ j β j > n i β i .We refer to [9], [11] and [12] for more details.Finally, we can modify these equations a bit by involving an extra variable v ∈ C: For every v in (C, 0), these new equations define a germ of a curve in (C g +1 , 0) with semigroup Γ(C ).Even more, for v = 0, they are all isomorphic to C , while for v = 0, we get the curve defined by = 0 for i = 1, . . ., g .Hence, all these branches together define an equisingular family η : (χ, 0) ⊂ (C g +1 × C, 0) → (C, 0) with generic fiber isomorphic to C and special fiber given by the latter equations, in which the coefficients c i are needed to see that any irreducible plane branch is a (equisingular) deformation of a such a curve.By changing the coordinates x 0 , . . ., x g if necessary, we can always assume that every c i = 1; these are the binomial equations defining the monomial curve (Y , 0) ⊂ (C g +1 , 0) associated with C .
Clearly, the equations defining (Y , 0) in (C g +1 , 0) can also be considered in C g +1 ; from now on, we define a (space) monomial curve Y ⊂ C g +1 as the global curve defined by This is still an irreducible curve which is smooth outside the origin.Furthermore, it is a complete intersection curve given by the regular sequence f 1 , . . ., f g ∈ C[x 0 , . . ., x g ].

The motivic Igusa zeta function of a space monomial curve
In this section, we explain the main results of [10], in which the motivic Igusa zeta function of a space monomial curve Y ⊂ C g +1 is calculated by determining the structure of the jet schemes of Y .The motivic Igusa zeta function of Y ⊂ C g +1 can be written as where J Y (T ) is the Poincaré series Here,

Structure of the jet schemes
Because Y 0 = Y and π m induces a trivial fibration over Y \ {0} with fiber isomorphic to C m , it remains to investigate the fibers π −1 m (0) for m ≥ 1.Because these fibers for g = 1 have already been determined in [8,Corollary 4.4], we assume that g ≥ 2. For this purpose, we stratify each fiber π −1 m (0) r ed with its reduced structure as follows.Let l ∈ N be such that l n 0 n 1 < m ≤ (l +1)n 0 n 1 .Then where To explain the structure of these strata, we introduce some notation.First, we denote by [a/b] the integer part of a rational number a/b.Second, for k ≥ 1, let j (k) ∈ N be defined by where β g +1 := +∞ by convention, we define Finally, let Y i for i = 1, . . ., g be the complete intersection curve defined in C i +1 by the first i equations f 1 , . . ., f i of the defining equations (1) of Y .
Proposition 1.Consider m ≥ 1 and let l ∈ N be such that l n 0 n 1 < m ≤ (l + 1)n 0 n 1 .We have the following.
(1) Let 1 ≤ k ≤ l .If there exists some i ∈ N such that 1 ≤ i < j (k) and , then the stratum D m,k is isomorphic to In particular, C m,k := D m,k is irreducible and its codimension in is irreducible with codimension g + (l + 1)(n 0 + n 1 ) This result implies that the stratification (2) induces a decomposition . The next theorem tells us that this is a decomposition into irreducible components.

Formula for the motivic Igusa zeta function and its poles
With the structure of the jet schemes following from Proposition 1 for g ≥ 2 and from [8, Corollary 4.4] for g = 1, we can compute the motivic Igusa zeta function of Y ⊂ C g +1 .Theorem 3. Consider a space monomial curve Y ⊂ C g +1 defined by the equations (1).Let N i and ν i for i = 1, . . ., g be the positive integers defined as where lcm denotes the least common multiple.The motivic Igusa zeta function associated with Y ⊂ C g +1 is given by Here, Z i (T ) where i ,k for i = 1, . . ., g − 1 and k, p ∈ N is the interval This expression immediately yields a complete list of g + 1 candidate poles of Z mot Y (T ): Using residues and the related Igusa zeta function, we are able to show that all these candidates are actual poles.
Theorem 4. A complete list of the poles of the motivic Igusa zeta function associated with a space monomial curve Y ⊂ C g +1 is given by and all these poles have order 1.

The monodromy zeta function of a space monomial curve
This section provides an overview of the strategy in [6] to study the monodromy eigenvalues of a space monomial curve Y ⊂ C g +1 by considering Y as a Cartier divisor on a generic embedding surface and computing an embedded Q-resolution of this situation.Because the monodromy conjecture is already well-known in the case where g = 1 (a monomial curve Y ⊂ C 2 of the above type is just a cusp), we assume that g ≥ 2. The starting point is the following generalization of A'Campo's formula to express the monodromy eigenvalues associated with Y in terms of a principalization ϕ : X → C g +1 of its defining ideal I = ( f 1 , . . ., f g ).Denote by E j for j ∈ J the irreducible components of ϕ −1 (Y ) and put E • j := E j \ ∪ i = j (E i ∩ E j ) for every j ∈ J .Let N j and ν j − 1 be the multiplicity of E j in the divisor on X of ϕ * I and ϕ * (d . By the universal property of the blow-up, there exists a unique morphism ψ : X → X such that σ•ψ = ϕ.It was proven in [13] that a complex number is a monodromy eigenvalue associated with Y if and only if it is a zero or pole of the monodromy zeta function at a point e ∈ E given by where χ is the topological Euler characteristic.For a more elaborate introduction to eigenvalues of monodromy and this formula, see for instance [6,Section 2].In other words, to know the monodromy eigenvalues of Y , we need to compute these zeta functions.For example, it is not hard to show that the formula (3) for an arbitrary point e ∈ E \σ −1 (0) gives Z mon Y ,e (t ) = 1−t , yielding the trivial monodromy eigenvalue 1.It thus remains to study the monodromy zeta functions at points in σ −1 (0).
Because Sing(S) = Sing(Y ) = {0}, we can apply the original formula of A'Campo [1] to compute the monodromy zeta function Z mon Y ,0 (t ) of Y ⊂ S at the origin in terms of an embedded resolution of singularities of Y = { f 1 = 0} on S. To see the link between the latter formula and the eigenvalues associated with Y ⊂ C g +1 , we consider the strict transform S = σ −1 (S \ Y ) of S under the blowup σ.By the properties of the blow-up, we know that S is isomorphic to S with Y := S ∩ E isomorphic to Y .In particular, the intersection S ∩ σ −1 (0) consists of a single point.We denote this point by p and refer to it as the generic point associated with S. It turns out that the monodromy zeta function Z mon Y ,p (T ) of Y ⊂ C g +1 at the generic point p is equal to the monodromy zeta function Z mon Y ,0 (t ) of Y ⊂ S at the origin.As the next theorems shows, this result is true for a larger class of ideals.Hence, it could be possibly useful for studying the monodromy eigenvalues of other ideals in this class.Theorem 6.Consider a complete intersection curve Y = V (I ) ⊂ C g +1 whose ideal I = ( f 1 , . . ., f g ) is generated by a regular sequence f 1 , . . ., f g ∈ C[x 0 , . . ., x g ], and whose singular set is Sing(Y ) = {0}.Let S = S(λ 2 , . . ., λ g ) be a generic embedding surface of Y defined by the equations (4).Denote by σ : X → C g +1 the blow-up of C g +1 with center Y and by S the strict transform of S under σ.Then, the monodromy zeta function Z mon Y ,p (t ) of Y considered in C g +1 at the generic point p = S ∩ σ −1 (0) is equal to the monodromy zeta function Z mon Y ,0 (t ) of Y considered as a Cartier divisor on S at the origin.
We will see later on that the monodromy zeta function Z mon Y ,p (t ) of Y ⊂ C g +1 at the generic point p suffices to recover all non-trivial candidate monodromy eigenvalues coming from the poles of the motivic Igusa zeta function of Y .We will refer to Z mon Y ,p (t ) as the monodromy zeta function of Y .By the previous theorem, we can find this zeta function by considering Y on a generic surface S.

Monodromy via an embedded Q-resolution
To compute the monodromy zeta function at the origin of Y considered as a Cartier divisor on a generic embedding surface S, we make use of another generalization of A'Campo's formula using an embedded Q-resolution of Y ⊂ S. Roughly speaking, an embedded Q-resolution is a resolution in which we allow the final ambient space to have abelian quotient singularities and can be constructed as a sequence of weighted blow-ups.With the formula shown in [5], the monodromy zeta function of Y ⊂ S at the origin can be written in terms of an embedded Qresolution ϕ : S → S as where {E j ,t } j =1,...,r,t ∈T is a finite stratification of the exceptional varieties E 1 , . . ., E r of ϕ such that the multiplicity m j ,t of E j along each E j ,t is constant.Here, the multiplicity m j ,t is defined as m/d where d is the order of the small group µ d acting at a general point q in E j ,t and x m : is the local equation of E j at q.More information on abelian quotient singularities, embedded Q-resolutions, weighted blow-ups and this A'Campo formula can be found for instance in [6, Section 3].
In [6,Section 5], the computation of g weighted blow-ups in higher dimension yields an embedded Q-resolution of Y ⊂ S with dual graph a tree as in Figure 1.After each blow-up, one variable can be eliminated so that the new situation is very similar to the one before the blowup, but with one equation in Y and S less.Therefore, the last step coincides with the resolution of a cusp in a Hirzebruch-Jung singularity of type 1 d (1, q) by one weighted blow-up.It is worth mentioning that this process is similar to the resolution of an irreducible plane curve with g Puiseux pairs; after each weighted blow-up, the number of Puiseux pairs is lowered by one so that the last step coincides with the resolution of an irreducible plane curve with one Puiseux pair.However, our resolution is more complicated as the strict transform of Y intersects in general the singular locus of the ambient space.For instance, the computations of the numerical data of the resolution, such as the multiplicity of the exceptional divisor in each step, its number of irreducible components and its Euler characteristic, give rise to interesting arithmetic challenges.
Theorem 7. Let Y ⊂ C g +1 be a space monomial curve defined by the equations (1) with g ≥ 2 and consider Y as a Cartier divisor on a generic surface S = S(λ 2 , . . ., λ g ) ⊂ C g +1 given by (4).There exists an embedded

which is a composition of g weighted blow-ups ϕ k with exceptional divisor E k such that the pull-back of Y is given by
intersected by only one of the components of E k , and each non-empty intersection between two components E k j and E (k−1) j consists of a single point.In particular, the dual graph of the resolution is a tree as in Figure 1.

Formula for the monodromy zeta function
By stratifying the exceptional divisor of the embedded Q-resolution of Y ⊂ S from Theorem 7 and computing the Euler characteristic of these strata, we can compute the monodromy zeta function Z mon Y ,0 (t ) of Y ⊂ S at the origin, which is equal to the monodromy zeta function of Y by Theorem 6.
Theorem 8. Let Y ⊂ C g +1 be a space monomial curve defined by the equations (1) with g ≥ 2 and consider a generic embedding surface S = S(λ 2 , . . ., λ g ) ⊂ C g +1 given by (4).Denote by σ : X → C g +1 the blow-up of C g +1 with center Y and by S the strict transform of S under σ.Then, the monodromy zeta function of Y considered in C g +1 at the generic point p = S ∩ σ −1 (0) is given by , from which it is easy to check that all four poles of the motivic zeta function induce a monodromy eigenvalue.

The monodromy conjecture for a space monomial curve
By combining the results of the previous two sections, we can show the monodromy conjecture for a space monomial curve Y ⊂ C g +1 with g ≥ 2.
Theorem 10.Let Y ⊂ C g +1 be a space monomial curve defined by the equations (1) with g ≥ 2 and denote by σ : X → C g +1 the blow-up of C g +1 with center Y .Every pole L −s 0 of the motivic Igusa zeta function associated with Y induces a monodromy eigenvalue e 2πi s 0 of Y at a point in σ −1 (Y ).
In [6,Section 7], it is shown that the pole L g and poles L which is defined as the C-scheme such that {points of Y m with coordinates in C} = {points of Y with coordinates in C[t ]/(t m+1 )}, and [Y m ] is its class in the Grothendieck ring K 0 (Var C ) of complex varieties.By L we denote the class of the affine line in K 0 (Var C ) and by M C := K 0 (Var C )[L −1 ] the localization with respect to L. We further let π m,p : Y m → Y p for all m ≥ p be the natural morphism induced by the truncation morphism C[t ]/(t m+1 ) → C[t ]/(t p+1 ) and we put π m := π m,0 : Y m → Y 0 = Y .More details can be found for instance in [10, Section 2].

Theorem 2 .
Consider m ≥ 1 and let l ∈ N be such that l n 0 n 1 < m ≤ (l + 1)n 0 n 1 .The irreducible components of π −1 m (0) r ed are C m,k = D m,k for k = 1, . . ., l such that m < kn j (k) βj (k) e 1 and B m .Furthermore, B m is a component of maximal dimension.

1 N 1
T 26 ) where P 1 (T ) is a concrete polynomial in T of degree 31 with coefficients in Z[L], see [6, Example 4.1].We see that all three candidate poles, L g = L 2 , L ν

1 .
Figure 1.Dual graph of the resolution of Y ⊂ S. Graphe dual de la résolution de Y ⊂ S.

Remark 11 .
k ∈ {1, . . ., g } with ν k /N k ∈ N induce the trivial monodromy eigenvalue 1, while every other candidate monodromy eigenvalue e −2πi ν k /N k is a pole of the monodromy zeta function Z mon Y ,p (t ) of Y .The structure of the jet schemes can also be used to compute the local motivic Igusa zeta function of a space monomial curve Y ⊂ C g +1 , see[10, Section 4].In particular, this local version has the same poles as the global one.Hence, the monodromy conjecture also holds for the local motivic zeta function: every pole L −s 0 of the local motivic Igusa zeta function associated with a space monomial curve Y ⊂ C g +1 induces a monodromy eigenvalue e 2πi s 0 of Y at a point in σ −1 (B ∩ Y ) for B a small ball around 0. Likewise, one can state and conclude the monodromy conjecture for the related global/local topological and p-adic Igusa zeta function, which are specializations of the global/local motivic Igusa zeta function, see[6, Section 7].