Resolving singularities of plane analytic branches with one toric morphism

Rebecca Goldin and Bernard Teissier

To Heisuke Hironaka

Abstract Let (C, 0) be an irreducible germ of complex plane curve. Let Γ ⊂ N be the semigroup associated to it and C^{Γ} ⊂ C^{g+1} the corresponding monomial curve, where g is the number of Puiseux exponents of (C, 0). We show, using the specialization of (C, 0) to (C^{Γ}, 0), that the same toric morphisms Z(Σ) → C^{g+1} which induce an embedded resolution of singularities of (C^{Γ}, 0) also resolve the singularities of (C, 0) ⊂ (C^{g+1}, 0), the embedding being defined by elements of the analytic algebra O_{C,0} whose valuations generate the semigroup Γ.

1 Introduction

In the last few years Mark Spivakovsky has proposed a program to prove the resolution of singularities of excellent schemes. A part of this program is a new look at Zariski’s local uniformisation theorem for arbitrary valuations. A fundamental object of study in this approach is the graded ring associated to the filtration of a local domain R naturally provided by a valuation of R.

In the special case where R is the local analytic algebra O of a plane branch C, with its unique valuation, this graded ring was studied in detail by Monique Lejeune-Jalabert and the second author (see [L-T], [T1]), as a special case of the \overline{gr}_{I}R appearing in their study of the ”\overline{ν}_{I}-filtration” of a ring (roughly, by the integral closures of the fractional powers of an ideal I of R). Recall that one calls branch a germ of analytically irreducible excellent curve. We will in this paper deal only with complex analytic branches.

The following facts which appeared at that time in that special case may be relevant to the understanding of the role of the graded ring in the general case:

– The graded ring is the ring of the monomial curve C^{Γ} with the same semigroup Γ as the given plane branch (see definitions below).

– The generators of the semigroup are the intersection numbers with the given plane branch C of a transversal non-singular germ x = 0 and of plane branches f_{j}(x, y) = 0 with a smaller number j , 0 ≤ j ≤ g − 1, of Puiseux exponents and having with C maximal contact in the sense defined by M. Lejeune-Jalabert (see [Z], pp.16-17, [L-J]). In particular the initial forms in the graded ring of the images in the algebra O of the equations of these branches generate it as a C-algebra.

– There exists a one parameter deformation of C^{Γ} having all its fibers except the special one isomorphic to C, and this deformation is equisingular in the sense that it has a simulateous resolution of singularities by normalization; the normalization of the total space of the family is non-singular and induces normalization for each fiber (see [T1]).

We may therefore hope that a process of resolution for C^{Γ} will induce resolution for C. In this paper we show that the curve C^{Γ} can be resolved by a single toric modification of its ambient space C^{g+1} (where g is the number of characteristic Puiseux exponents of the plane branch C) and that this toric modification also resolves the curve C if we view it as embedded in C^{g+1} by g + 1 elements of its maximal ideal whose valuations generate the semigroup Γ. This is shown by a generalization of the usual non degeneracy argument, which can be found also in [O2], [O3]. We need more details than in these papers because we want to show that the toric modification resolves not only C^{Γ} but also C.

¿From this follows the rather interesting fact that any plane branch with g Puiseux exponents can be embedded in C^{g+1} in such a way as to be resolved by one single toric modification, i.e., it is in some sense non-degenerate with respect to its Newton polyhedron. Moreover, if we denote by π: Z → C^{g+1} a toric map resolving C, and by π|S′: S′ → S the strict transform by this toric map of a non-singular surface S containing our plane branch C, we can identify π|S′ to the composition of g toric modifications of non-singular surfaces, thus recovering the known fact that a plane branch can be resolved by a composition of g toric morphisms. The first non trivial example, the simplest branch with two characteristic exponents, is computed in an Appendix.

The description of this composition of toric maps of surfaces obtained in this way has the advantage over the “static” one of [O1] of showing explicitly its analytic dependence on the coefficients of the equation or the parametrization of the curve, in view of the results of [T1], and also of giving a geometric vision of the relationship with the resolution process of the “singular curves with maximal contact” of M. Lejeune-Jalabert, and probably also of the approximate roots à la Abhyankar. Indeed, the embedding of the plane branch C in C^{g+1} is obtained by adding to the coordinates x, y the images in O of the equations of plane branches with < g characteristic exponents having maximal contact with C. The disadvantage is that the construction given here is for the time being restricted to irreducible germs.

It is tempting to ask whether in general, given any germ of an algebraic or analytic space and a valuation of its local algebra, the germ can be embedded in an affine space in such a way as to be non-degenerate with respect to its Newton polyhedron and to the given valuation, in the sense that its strict transform is non-singular at the point specified by the valuation in a toric modification of the ambient space subordinate to the Newton polyhedron.

This would be an effective local uniformization theorem and the results of this paper indicate that perhaps the specialization to the graded algebra is the key to such a result.

In the general case the graded algebra associated with a valuation is not even Nœtherian, and as the reader will see, the proofs in this paper rely on the extensive knowledge we have of the structure of the semigroup of a plane branch, a knowledge essentially lacking in dimension ≥ 3 (the case of dimension 2 is currently under study). In the paper “Valuations, Deformations, and toric Geometry ” following this one the second author begins to prepare the way for a proof of the local uniformization theorem along these lines. It is worth noting that one of the results of [L-T] is that the graded algebra associated with the \overline{ν}_{I} filtration, i.e., \overline{gr}_{I}R, is an R-algebra of finite type. Some very interesting finiteness results on the graded algebra associated with a valuation and its relation with Abhyankar’s inequality have recently been obtained by O. Piltant ([P]).

This text is based on a lecture given at the singularities Seminar by the second author in January 1994, where examples largely stood in for proofs. The ideas are presented in the framework of complex geometry for simplicity, but the interested reader will see how to transform this into a characteristic-blind resolution for onedimensional excellent henselian equicharacteristic local integral domains with an algebraically closed residue field.

We are grateful to Tadao Oda and Michel Vaquié for pointing out some errors and imprecisions in preliminary versions of this text.

2 Puiseux expansion and the semigroup of a curve

Definition. Let (C, 0) ⊂ (C^{2}, 0) be a germ of an irreducible analytic curve (a plane branch) defined by the equation f(X, Y) = 0, f ∈ C{X, Y }. We will call O = C{X, Y }/(f) the local algebra of the curve C, and denote by m_{O} its maximal ideal. Note that O = C{x, y} where x and y are the residue classes in C{X, Y }/(f) of X and Y respectively.

According to Newton and Puiseux, there is a parametric representation of C, called the Newton-Puiseux expansion, i.e an injection of the algebra O into C{t} described in suitable coordinates X, Y, t by:

\begin{aligned}X & = & x(t) = t^{n} (*) \\ Y & = & y(t) = ∑_{i≥m} a_{i}t^{i} with m > n. \end{aligned}

The Puiseux expansion of x and y; since the two series converge for small enough ∥t∥, say ∥t∥ < ϵ. Denoting by D(0, ϵ) the disk with center 0 and radius ϵ in C, the image of the map D(0, ϵ) → C^{2} defined by x(t), y(t) is a representative of the germ (C, 0) ⊂ (C^{2}, 0). From the integer n and the exponents appearing in the expansion of y(t) one can extract the Puiseux characteristic exponents, which characterize the topological type of a small representative of our branch. (see [Z]).

In particular, we may choose the coordinates X, Y in such a way that the exponent m appearing in the expansion of y(t) is not divisible by the multiplicity n. It is then generally denoted by β_{1}. We shall do so from now on. More generally, any analytically irreducible germ of a curve has a normalization isomorphic to C{t}, and its algebra is an analytic subalgebra of C{t}.

Given this parametrization, certain properties of an analytic branch C, plane or not, can be described using the valuation on the algebra O induced via the injection O ↪ C{t} by the t-adic valuation of C{t}. Definition. Given a branch (C, 0) and its algebra O ↪ C{t}, let ν be the t-adic valuation on C{t}. We define the semigroup Γ of O to be

Γ = {ν(ξ) : ξ ∈ O}.

We follow here the common usage, which is to denote by the same letter Γ the semigroup deprived of its zero element, i.e., {ν(ξ) : ξ ∈ m_{O}} We say that the branch (C, 0) has the semigroup Γ.

The semigroup Γ is finitely generated and has finite complement in N (see [Zariski, II.1] for the case of plane branches), and if (C, 0) is a plane branch, its minimal set of generators \{\overline{β}_{0}, …, \overline{β}_{g}\} can be uniquely determined by:

(a) \overline{β}_{0} = n

(b) \overline{β}_{i} = min\{z ∈ Γ | z ∉< \overline{β}_{0}, …, \overline{β}_{i−1} >\}

where < \overline{β}_{0}, …, \overline{β}_{i−1} >⊂ N is the semigroup generated by \{\overline{β}_{0}, …, \overline{β}_{i−1}\} (in particular, \overline{β}_{1} = β_{1}). We then have that

1. \{\overline{β}_{0}, …, \overline{β}_{g}\} generates Γ,

2. \overline{β}_{0} < ⋯ < \overline{β}_{g}, and

3. gcd(\overline{β}_{0}, …, \overline{β}_{g}) = 1

We call the vector (\overline{β}_{0}, …, \overline{β}_{g}) ⊂ Z^{g+1} the weight vector. Of course the datum of its semigroup Γ does not determine a branch up to analytic isomorphism. In fact, the datum of the semigroup Γ of a plane branch is equivalent to the datum of its Puiseux characteristic exponents (see [Z]). Among all the branches, plane or not, having a given semigroup, one may single out the affine curve C^{Γ} ⊂ C^{g+1} defined by the parametrization:

C^{Γ} : u_{i} = t^{\overline{β}^{i}} 0 ≤ i ≤ g.  (1)

Consider (C^{Γ}, 0) as the germ of a curve. Its algebra

O_{C^{Γ},0} = C\{t^{\overline{β}^{0}}, …, t^{\overline{β}^{g}}\} ⊂ C\{t\}

clearly has semigroup Γ. It is shown in [T1] that all branches having a given semigroup Γ are complex analytic deformations of (C^{Γ}, 0) and these deformations are in some sense equisingular.

Definition. Given a branch (C, 0) and its semigroup Γ, we call the curve C^{Γ} described above by the parametrization (1) the monomial curve associated to C. If (C, 0) is a plane germ, the monomial curve C^{Γ} is a complete intersection in C^{g+1}, where g is the number of Puiseux characteristic exponents of (C, 0).

3 Deforming Curves

The curve C^{Γ} mentioned above, as we shall see, is the center of a finite-dimensional flat family of branches which contains (up to complex analytic isomorphism of germs) every germ of analytically irreducible curve (or branch) with semigroup Γ.

We begin by showing (see [T1]) that given any branch (C, 0), one can construct explicitly a one-parameter analytic deformation of (C^{Γ}, 0), whose general fibres are isomorphic to the curve (C, 0). Let us take for simplicity the case of a plane branch. Consider the parametric representation of the curve C as in (*). After a change of coordinates we may assume that it has the form :

\begin{aligned}x(t) & = & t^{\overline{β}^{0}} \\ y(t) & = & t^{\overline{β}^{1}} + ∑_{j>\overline{β}_{1}} c_{j}^{(1)}t^{j}. \end{aligned}

Note that \overline{β}_{0} is the multiplicity n of C. By the definition of the \overline{β}′_{i}s, there exist elements ξ_{i}(t) ∈ O = C{x, y} , 2 ≤ i ≤ g such that

ξ_{i}(t) = t^{\overline{β}^{i}} + ∑_{j>\overline{β}_{i}} c_{j}^{(i)}t^{j}.

where ξ_{0} = x(t). Now consider the family, parametrized by v, of curves parametrized by t, in C^{g+1} × C:

− χ(v) \{ \begin{matrix}x & = & t^{\overline{β}^{0}} \\ y & = & t^{\overline{β}^{1}} + ∑_{j>\overline{β}_{1}} c_{j}^{(1)}v^{j−\overline{β}_{1}}t^{j} \\ u_{2} & = & t^{\overline{β}^{2}} + ∑_{j>\overline{β}_{2}} c_{j}^{(2)}v^{j−\overline{β}_{2}}t^{j} \\ · & = & · \\ · & = & · \\ · & = & · \\ u_{g} & = & t^{\overline{β}^{g}} + ∑_{j>\overline{β}_{g}} c_{j}^{(g)}v^{j−\overline{β}_{g}}t^{j} . \end{matrix}

Proposition 3.1. The branch (− χ(0), 0) is isomorphic to (C^{Γ}, 0), and for any v_{0} ≠ 0, the branch (− χ(v_{0}), 0) is isomorphic to (C, 0) by the isomorphism u_{i}(t) ↦

v_{0}^{−\overline{β}_{i}}u_{i}(v_{0}t) (where we set x = u_{0}, y = u_{1}).

Proof. For v = 0 it is obvious, and for v ≠ 0, since O is generated as an analytic subalgebra of C{t} by x(t), y(t), it follows from the fact that the ξ_{i}(t) are in O (see [T1]). ⟲ The semigroups of plane branches are characterized by the following arithmetical properties (see [T1]); let us set e_{i} = gcd(\overline{β}_{0}, \overline{β}_{1}, …, \overline{β}_{i}), e_{i−1} = n_{i}e_{i}. Then

\begin{matrix}n_{i}\overline{β}_{i} ∈ & < \overline{β}_{0}, …, \overline{β}_{i−1} > \\ n_{i}\overline{β}_{i} < & \overline{β}_{i+1} \end{matrix}

The first relation implies the existence of integers ℓ_{j}^{(i)} , 1 ≤ i ≤ g , 0 ≤ j ≤ i − 1, such that

n_{i}\overline{β}_{i} = ℓ_{0}^{(i)}\overline{β}_{0} + ⋯ + ℓ_{i−1}^{(i)}\overline{β}_{i−1}

So that on the curve C^{Γ} as parametrized, we have equations:

f_{1} = u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}}  = 0
f_{2} = u_{2}^{n_{2}} − u_{0}^{ℓ_{0}^{(2)}}u_{1}^{ℓ_{1}^{(2)}}  = 0
·  ·  ·  (**)
·  ·  ·
·  ·  ·
f_{g} = u_{g}^{n_{g}} − u_{0}^{ℓ_{0}^{(g)}} ⋯ u_{g−1}^{ℓ_{g−1}^{(g)}} = 0.

It is shown in [T1] that these equations define the curve C^{Γ}, which is therefore a complete intersection in C^{g+1}. Now using the theory of miniversal deformations (see [T3]), we have:

Theorem 3.2. [Teissier] There exists a germ of a flat morphism

p: (− χ_{u}, 0) → (C^{τ_{−}}, 0)

endowed with a section σ such that (p^{−1}(0), 0) is analytically isomorphic to (C^{Γ}, 0), and for any representative of the germ of the morphism p and any branch (C, 0) with semigroup Γ, there exists v_{C} ∈ C^{τ_{−}} such that (p^{−1}(v_{C}), σ(v_{C})) is analytically isomorphic to (C, 0). Moreover, (− χ_{u}, 0) is embedded in (C^{g+1} × C^{τ_{−}}, 0) in such a way that p is induced by the second projection.

Proof. See [T1]; this is the miniversal constant semigroup deformation of C^{Γ}, and it corresponds to the part of the base of the miniversal (equivariant) deformation of C^{Γ} which is spanned by the coordinates with negative weight. It is shown in [T1], 4.4 that τ_{−} ≤ #(N \ Γ).

More precisely the miniversal constant semigroup deformation of C^{Γ} is a map p: −χ_{u} → C^{τ_{−}}, where − χ_{u} is embedded in C^{g+1} × C^{τ_{−}} in such a way that p is induced by the second projection, and defined by the equations

F_{1} = f_{1} + ∑_{r=1}^{τ_{−}} v_{r}ϕ_{r,1}(u_{0}, …, u_{g}) = 0
·  ·  = ·
·  ·  . ·
·  ·  . ·
F_{g} = f_{g} + ∑_{r=1}^{τ_{−}} v_{r}ϕ_{r,g}(u_{0}, …, u_{g}) = 0

where the ϕ_{r,j} are polynomials and each monomial in (u_{0}, …, u_{g}) appearing in ϕ_{r,j} is of weight > n_{j}\overline{β}_{j} when each u_{k} is given the weight \overline{β}_{k}; the equation f_{j} is then homogeneous of weight n_{j}\overline{β}_{j}. In fact one may choose the vectors ϕ_{r} to have only one nonzero coordinate which is a monomial ϕ_{r,j}. Moreover, the images vectors ϕ_{1}, …, ϕ_{τ_{−}} in C[C^{Γ}]^{g+1}/N, where N is a certain jacobian submodule of C[C^{Γ}]^{g+1}, are linearly independant over C.

The basic property of the miniversal deformation is that for any deformation d: (Y, y) → (S, 0) of the germ (C^{Γ}, 0) such that all the fibers have a singular point with semigroup Γ there exists an analytic map germ h: (S, 0) → (C^{τ_{−}}, 0) such that d is isomorphic to the deformation obtained from p by pull-back by h. Moreover, since C^{Γ} is quasi-homogeneous, it is also the case for p in the sense that there are weights on the v_{i} such that the equations F_{j} are homogeneous of degree n_{j}\overline{β}_{j}. These weights are negative, which is the reason for the notation C^{τ_{−}}; in fact the weight w_{r} of v_{r} is n_{j}\overline{β}_{j} minus the weight of ϕ_{r,j}, this difference being independent of j.

The recipe to find vectors ϕ_{r} making the deformation miniversal is due to J. Mather; it is the following (see [T1]): Consider the ring of algebraic functions C[C^{Γ}] on the monomial curve, and the submodule N of C[C^{Γ}]^{g} generated by the vectors ∂_{u_{i}}f where f = (f_{1}, …, f_{g}). The quotient C[C^{Γ}]^{g}/N is a finite dimensional vector space over C, and so we may choose vectors ψ_{i} ∈ C[u_{0}, …, u_{g}]^{g}, each having a single nonzero coordinate which is a monomial, such that the natural images of the ψ_{i} in C[C^{Γ}]^{g}/N form a basis of this vector space. The ϕ_{r} are those among the ψ_{i} = (0, …, 0, u^{q_{i,j}}, 0, …, 0) such that if u^{q_{i,j}} is at the j-th line, the weight of u^{q_{i,j}} is > n_{j}\overline{β}_{j}, so that in the corresponding deformation the only equation modified, f_{j} + v_{i}u^{q_{i,j}} is modified by a term of weight greater than that of f_{j}.

In particular, the one-parameter specialization of a branch C to C^{Γ} which we have seen above can be obtained in this way, up to isomorphism, by a map h_{1} : (C, 0) → (C^{τ_{−}}, 0). The map p is equivariant with respect to the action of the group C^{∗} on − χ_{u} (resp C^{τ_{−}}) which is described by u_{k} ↦ λ^{\overline{β}^{j}}u_{k}, v_{r} → λ^{w_{r}}v_{r} (resp. only the action on the v_{r}). If v_{C} ∈ C^{τ_{−}} is a point corresponding to a branch isomorphic to (C, 0), the image of h_{1} is contained in the orbit of v_{C} under this action of C^{∗}. The action of C^{∗} on C^{τ_{−}} ensures that any branch with semigroup Γ appears up to isomorphism as a fiber in any representative of the germ p: (− χ_{u}, 0) → (C^{τ_{−}}, 0). ⟲

4 Resolution Using Toric Morphisms

We produce toric morphisms π(Σ): Z(Σ) → C^{g+1} to resolve the singularities of C^{Γ} ⊂ C^{g+1}, and then show that the same morphisms which resolves C^{Γ} resolve any fibre of the miniversal constant semigroup deformation p: −χ_{u} → C^{τ_{−}}. In particular, since all curves with semigroup Γ are represented as fibers of this deformation, Theorem 3.2 implies that π(Σ) resolves (C, 0). The toric resolution of the curve C^{Γ} will be a consequence of a generalization of the work of Varchenko [V] (expounded also in [M]) for “convenient” or “commode” functions, whose Newton polyhedron intersects each axis and which are non-degenerate.

4.1 The Toric Morphism

For the notions and basic results of toric geometry used in this section, we refer to [C]. A toric morphism is locally described by monomial maps π(a): C^{g+1} → C^{g+1} where a = (a^{0}, …, a^{g}) with a^{j} ∈ N_{0}^{g+1}. for all j and span{a^{0}, …, a^{g}} = R^{g+1}:

u_{0} = y_{0}^{a_{0}^{0}} ⋯y_{g}^{a_{0}^{g}}
u_{1} = y_{0}^{a_{1}^{0}} ⋯y_{g}^{a_{1}^{g}}
·  ·
·  ·
·  ·
u_{g} = y_{0}^{a_{g}^{0}} ⋯y_{g}^{a_{g}^{g}}

More precisely, a fan (see [Oda], Chap. 1) Σ with support R_{+}^{g+1} is a decomposition of the positive quadrant R_{+}^{g+1} into rational simplicial cones σ_{α} with the properties that any face of such a cone is also a part of the fan and that the intersection of two of them is a face of each. A fan is non-singular if the primitive integral vectors of the 1-skeleton of each cone σ of dimension g + 1 form a basis of the integral lattice N_{0}^{g+1}. We have set N_{0} = N ∪ {0}.

To a fan Σ are associated a toric variety Z(Σ) and a toric (equivariant) map π(Σ): Z(Σ) → C^{g+1}. The variety Z(Σ) is obtained by glueing up affine varieties Z(σ) corresponding to each cone of maximum dimension, and if the fan is nonsingular, so is Z(Σ); in this case, for each cone σ of maximal dimension Z(Σ) is isomorphic to C^{g+1} and the map π(σ) induced by π(Σ) is equal to π(a^{0}, …, a^{g}) where the a^{j} are the primitive integral vectors of the 1-skeleton of the cone σ. An upper convex map (see [C], [Oda] A.3 p. 182-185) m: R_{+}^{g+1} → R_{+} taking integral values on the integral points and linear in each cone σ ∈ Σ determines a divisor on Z(Σ), which in the chart π(a^{0}, …, a^{g}): Z(Σ) → C^{g+1} has the equation

y_{0}^{m(a^{0})} … y_{g}^{m(a^{g})} = 0.

We will call it the exceptional divisor of the toric map π(Σ) corresponding to the upper convex function m. Just like in the absolute case, the datum of such a map is equivalent, in the case where the function m is strictly convex, to the datum of an embedding Z(Σ) ⊂ C^{g+1} × P^{N} such that the exceptional divisor is the pull back of a hyperplane section of P^{N}. Such an upper convex map is called a support function .

The Newton polyhedron of a function f: C^{g+1} → C can be used to define fans and a support function. Definition. For a function f = ∑_{p∈N^{g+1}} f_{p}u^{p}, let supp f = {p ∈ N^{g+1} : f_{p} ≠ 0} and N_{+}(f) = boundary of the convex hull of ({supp f} + R_{+}^{g+1}) in R_{+}^{g+1}. We call N_{+}(f) the Newton polyhedron of the function f. Note that N_{+}(f) has finitely many compact faces and that its non-compact faces of dimension ≤ g are parallel to coordinate hyperplanes.

Define the function m by:

m(q) = inf_{p∈N_{+}(f)} < q, p >

and define an equivalence relation: two vectors are equivalent if and only if the same elements of N_{+}(f) minimize the inner product with each vector. In other words,

q ∼ q′
⇕
{p ∈ N_{+}(f) |< q, p >= m(q)} = {p ∈ N_{+}(f) |< q′, p >= m(q′)} (***)

The function m is homogenous and piecewise linear, thus the equivalence classes (***) form a fan; to each class (and corresponding set of p ∈ N_{+}(f)), we associate a cone σ which consists of those vectors whose inner product is minimized by (and only by) these p. We obtain a convex rational fan Σ_{0} in R_{+}^{g+1} with vertex 0. By construction, the function m is linear on each cone σ ∈ Σ, and it is easy to verify that it is strictly upper convex. The non-compact faces of the boundary N (f) of N_{+}(f) correspond to cones σ in the fan which are contained in a coordinate hyperplane of R^{g+1}. Moreover, given a subset J ⊂ {0, …, g}, we can consider the projection of the Newton polyhedron to the subspace R^{J} = {p ∈ R^{g+1} | p_{k} = 0 for k ∉ J} of R^{g+1}. If that projection contains the origin, then for all a ∈ Ř^{g+1} | a_{k} = 0 for k ∉ J, we have m(a) = 0, and therefore we may assume that the cone R_{+}^{J} is in the fan, so that the basis vectors of R_{+}^{J} are in the 1-skeleton of the fan.

By a theorem of Kempf, Mumford, et al. (see [TE], pp. 32-35, [Oda], p.23), any fan can be refined into a non-singular fan, still containing as faces the spaces R_{+}^{J} such that m(a) = 0 for a ∈ R_{+}^{J}. The function m is of course linear in each cone of the finer fan and upper convex (but not strictly so in general), and we are in the situation described above.

If we now consider several functions f_{j}; 1 ≤ j ≤ k, to each of them corresponds a Newton polyhedron, and therefore a fan Σ_{0}^{(j)} and a support function m_{j}.

Definition. Let Σ_{0} be the fan consisting of the intersections of the cones of the fans Σ_{0}^{(j)}. The fan Σ_{0} is the least fine common refinement of all the Σ_{0}^{(j)}.

It is also the fan associated in the manner we have just seen to a single Newton polyhedron, which is the Minkowski sum of the Newton polyhedra of the f_{j} and is also the Newton polyhedron of the product f_{1} … f_{k}.

Recall that the Minkowski sum N_{1} + N_{2} of two Newton polyhedra N_{1}, N_{2} is the convex domain spanned by vector sums {p_{1} + p_{2} | p_{1} ∈ N_{1}, p_{2} ∈ N_{2}}. It is a commutative and associative operation. If N_{k} is the Newton polyhedron of f_{k}, N_{1} + N_{2} is the Newton polyhedron of f_{1}f_{2}.

All the functions m_{j} are linear in each cone of Σ_{0}. Taking a non-singular refinement Σ of Σ_{0} as described above, we finally have a non-singular fan with support functions m_{j} and thus, since this fan is also a refinement of the “trivial” fan of R_{+}^{g+1}, we get a proper toric map π(Σ): Z(Σ) → C^{g+1} where Z(Σ) is a non singular toric variety, and each function f_{j} ∘ π(Σ) defines a map from Z(Σ) to C.

To study the effect of the modification π(Σ) on the functions f_{j}, we restrict ourselves to a chart Z(σ) of Z(Σ) corresponding to a cone σ whose primitive integral vectors are denoted by a = (a^{0}, …, a^{g}). We assume that they form a basis of the integral lattice, and we shall write σ =< a^{0}, …, a^{g} > for the convex cone spanned by the vectors (a^{0}, …, a^{g}). If we write the function

f_{j} = ∑_{p∈N^{g+1}} f_{p}^{(j)}u^{p} ,

where u^{p} = u_{0}^{p_{0}}u_{1}^{p_{1}} ⋯ u_{g}^{p_{g}}, then the composition

f_{j} ∘ π(a^{0}, …, a^{g}) = ∑_{p∈N^{g+1}} f_{p}^{(j)}y_{0}^{<a^{0},p>} ⋯ y_{g}^{<a^{g},p>},  (2)

where <, > is the standard inner product, can be written

\begin{aligned}f_{j} ∘ π(a) & = & y_{0}^{m_{j}(a^{0})} … y_{g}^{m_{j}(a^{g})} ∑_{p∈N^{g+1}} f_{p}^{(j)}y_{0}^{<a^{0},p>−m_{j}(a^{0})} ⋯ y_{g}^{<a^{g},p>−m_{j}(a^{g})} \\ & = & y_{0}^{m_{j}(a^{0})} … y_{g}^{m_{j}(a^{g})}f̃_{j}(y_{0}, …, y_{g}) . \end{aligned}

The function f̃_{j} is called the strict transform of f_{j}; it satisfies f̃_{j}(0) ≠ 0.

We use the following fact, which is obvious from the definitions: For each j, 1 ≤ j ≤ k, there is a unique p^{(j)} ∈ N_{+}(f_{j}) such that < a^{i}, p^{(j)} >= m_{j}(a^{i}), 0 ≤ i ≤ g, and the strict transforms take the form

f̃_{j} = f_{p^{(j)}}^{(j)} + ∑_{p∈N_{+}(f_{j})\{p^{(j)}}} f_{p}^{(j)}y_{0}^{<a^{0},p>−m_{j}(a^{0})} ⋯ y_{g}^{<a^{g},p>−m_{j}(a^{g})}.

Moreover we can compute the critical locus of critπ(Σ) of π(Σ), which locally is the critical locus of π(σ) for each σ =< a^{0}, …, a^{g} >∈ Σ. By direct inspection we find the following relation for the jacobian matrix jacπ(σ) of π(σ):

y_{0} ⋯ y_{g} jacπ(σ) = u_{0} … u_{g} det(a^{0}, …, a^{g}),

setting α_{j} = ( ∑_{i=0}^{g} a_{i}^{j}) − 1 this means since det(a^{0}, …, a^{g}) = ±1,

jacπ(σ) = ±y_{0}^{α_{0}} … y_{g}^{α_{g}}.

So the divisor y_{j} = 0 is contained in the critical locus of π(σ) if and only if a^{j} is not a coordinate vector.

Note that the divisor y_{j} = 0 is contained in π(σ)^{−1}(0) if and only if the vector a^{j} has all its coordinates different from zero.

5 Existence of a toric resolution

In this section, we study a germ at 0 of a complete intersection X ⊂ C^{g+1} defined by a set of equations {f_{1} = f_{2} = ⋯ = f_{k} = 0} with f_{j} ∈ C{u_{0}, …, u_{g}}, j = 1, … k. We assume that the series f_{j} have no constant term. We use the notations introduced above and consider a toric map of non-singular spaces π(Σ): Z(Σ) → C^{g+1}. We make our computations in a chart corresponding to a regular simplicial cone σ =< a^{0}, …, a^{g} > of Σ.

Definition. Given a family of functions {f_{j}}_{1≤j≤k} as above, and a toric morphism associated to a regular fan Σ such that all the support functions (m_{j})_{1≤j≤k} are linear in each cone σ of Σ, for each map π(σ): C^{g+1}(σ) → C^{g+1}, we will call the divisor y_{0}^{m(a^{0})} … y_{g}^{m(a^{g})}, where m(a) = ∑_{j=1}^{k} m_{j}(a), the toric exceptional divisor of π(σ) associated to the support functions m_{j}. This definition globalizes to π(Σ).

¿From what we have seen one deduces for each π(σ) with σ =< a^{0}, …, a^{g} > the following statements:

Proposition 5.1. The fiber π(σ)^{−1}(0) ⊂ C^{g+1} is the union of the intersections y_{i_{1}} = ⋯ = y_{i_{t}} = 0 over all minimal families J = (i_{1}, …, i_{t}) of indices such that for each ℓ , 0 ≤ ℓ ≤ g, there is an i ∈ J such that the ℓ-th coordinate of a^{i} is ≠ 0. In particular the divisorial part of π(σ)^{−1}(0) is the union of the divisors y_{i} = 0 for those i such that the vector a^{i} has no zero component. The critical locus crit(π(σ)) is the union of the divisors y_{s} = 0 for those s such that a^{s} is not a coordinate vector of Z^{g+1}.

For any set of functions {f_{j}}, we label the compact faces of N_{+}(f_{j}) by γ_{j}. Any compact face of the Newton polyhedron N_{+} = ∑_{j=1}^{k} N_{+}(f_{j}) which is the Minkowski sum of the Newton polyhedra of the {f_{j}} will be of the form γ = γ_{1} + ⋯ + γ_{k}. Each face γ_{j} in turn is of the form

γ_{j} = γ_{j}(I) = {p ∈ N_{+}(f_{j}) |< a^{h}, p >= m_{j}(a^{h}) for h ∈ I} ,

where I is a subset of {0, …, g}. If the face γ_{j}(I) is not compact, it contains a line parallel to a coordinate axis, say the ℓ^{th}, so that there is an ℓ, 0 ≤ ℓ ≤ g, such that for all h ∈ I we have a_{ℓ}^{h} = 0.

Definition. We call the set of functions {f_{j}}_{1≤j≤k} non-degenerate if for all compact faces γ = γ_{1} + ⋯ + γ_{k} of N_{+} = ∑_{j=1}^{k} N_{+}(f_{j}), denoting by f_{j}|_{γ_{j}}, the sum

f_{j}|_{γ_{j}} = ∑_{p∈γ_{j}} f_{p}^{(j)}u^{p},

the k × (g + 1) matrix

( \begin{matrix}∂_{u_{0}}f_{1}|_{γ_{1}} & · & · & · & ∂_{u_{g}}f_{1}|_{γ_{1}} \\ ∂_{u_{0}}f_{2}|_{γ_{2}} & · & · & · & ∂_{u_{g}}f_{2}|_{γ_{2}} \\ · & & & & · \\ · & & & & · \\ · & & & & · \\ ∂_{u_{0}}f_{k}|_{γ_{k}} & · & · & · & ∂_{u_{g}}f_{k}|_{γ_{k}} \end{matrix} )

has maximal rank k on (C^{∗})^{g+1}. This means that the equations

f_{1}|_{γ_{1}} = ⋯ = f_{k}|_{γ_{k}} = 0

define a non singular complete intersection in (C^{∗})^{g+1}. For example a single function f is non-degenerate if for each compact face γ of N_{+}(f), the (g + 1)-vector (∂_{u_{i}}f|_{γ})_{i} does not vanish in (C^{∗})^{g+1}; this is the original definition of Varchenko [V].

Definition. The morphism π : X̃ ⟶ X is a resolution if the following conditions hold:

• π is a proper morphism,

• X̃ is non-singular, and

• X̃ \ π^{−1}(SingX) ⟶ X \ SingX is an isomorphism.

If the morphism π is a toric morphism π(Σ), the last condition is a consequence of the inclusion

X̃ ∩ crit(π(Σ)) ⊂ π(Σ)^{−1}(SingX).

However, this condition is difficult to check if X is not itself a toric subvariety, which motivates the following definition of a toric pseudo-resolution, after recalling that the strict transform X̃ of X ⊂ C^{g+1} by π(Σ) is the closure in Z(Σ) of π(Σ)^{−1}(X) \ crit(π(Σ)).

Definition. A toric morphism π(Σ): Z(Σ) → C^{g+1} is a toric (embedded) pseudoresolution of a subvariety X ⊂ C^{g+1} if the strict transform X̃ of X by π(Σ) is smooth and transversal to a stratification of the critical locus of π(Σ).

Note that a toric pseudo-resolution is not necessarily a resolution of singularities in the usual sense since it may not induce an isomorphism

X̃ \ π(Σ)^{−1}(SingX) → X \ SingX.

A toric pseudo-resolution only induces an isomorphism

X̃ \ crit(π(Σ)) → X \ disc(π(Σ)),

where disc(π(Σ)) is the image by π(Σ) of crit(π(Σ)), and since X̃ is non singular, the inclusion SingX ⊂ disc(π(Σ)) holds. In the case of a single function f, Varchenko introduced in [V] the condition of being commode which is equivalent to asking that if a primitive vector a^{i} of a cone σ (of a fan compatible with the Newton polyhedron of f) is contained in a hyperplane (which means that if in the chart C^{g+1}(σ) the divisor y_{i} = 0 is not contained in π(σ)^{−1}(0)), then we have m(a^{i}) = 0; it follows then from the conditions satisfied by our fans that a^{i} must in fact be a basis vector, so that y_{i} = 0 is not in the critical locus. For commode functions, in fact, the toric exceptional divisor and the critical locus of π(Σ) both coincide set theoretically with π(Σ)^{−1}(0) so that a toric pseudo-resolution of f(u_{0}, …, u_{g}) = 0 is also a resolution in the usual sense.

For complete intersections, Oka introduced in [O3] a notion of convenient generalizing Varchenko’s definition. It is much too strong for our purposes. Our monomial curve is only 1-convenient in the sense of [O3], and the product of its equations is far from being commode in the sense of [V].

However, in the case of a curve, a toric embedded pseudo-resolution is an embedded resolution in the usual sense unless the curve is contained in a coordinate hyperplane.

5.1 The Inverse Image of X by the Morphism π(Σ).

Theorem 5.2. If the set of functions {f_{j}}_{1≤j≤k} defining the complete intersection X ⊂ C^{g+1} is non-degenerate at the origin, there exists a neighborhood U of 0 in C^{g+1} such that in π(Σ)^{−1}(U) the strict transform of X by π(Σ) is non-singular and transversal in Z(Σ) to the strata of a stratification of the divisor π(Σ)^{−1}(0) .

Proof. We consider for each π(σ) a natural stratification of C^{g+1}(σ) such that π(σ)^{−1}(0) is a union of strata, as follows: For each I ⊂ {0, …, g}, define S_{I} to be the constructible subset of C^{g+1}(σ) defined by y_{i} = 0 for i ∈ I, y_{i} ≠ 0 for i ∉ I. The sets S_{I} form a partition of C^{g+1}(σ) by non-singular varieties.

If the subset I is such that for each ℓ , 0 ≤ ℓ ≤ g, there is a j ∈ I such that the ℓ-th coordinate of a^{j} is ≠ 0 (see 5.1), the stratum S_{I} is contained in π(σ)^{−1}(0), and conversely, so that π(σ)^{−1}(0) is a union of strata. This stratification of each C^{g+1}(σ) is compatible with the chart decomposition of Z(Σ) and so gives a stratification of Z(Σ). Moreover, this stratification satisfies the Whitney conditions, so that to check the transversality of a non-singular subspace to every stratum in a neighborhood it is sufficient to check transversality to the strata contained in π(σ)^{−1}(0); as we have seen, these strata correspond to compact faces of the Newton polyhedra. Now we can compute in each chart C^{g+1}(σ) the restriction of each f̃_{j} to S_{I}:

f̃_{j}|_{S_{I}} = ∑_{p|<a^{i},p>=m_{j}(a^{i}) for i∈I} f_{p}^{(j)}y_{j_{1}}^{<a^{j_{1}} ,p>−m_{j}(a^{j_{1}} )} … y_{j_{ℓ}}^{<a^{j_{ℓ}} ,p>−m_{j}(a^{j_{ℓ}})},

where {j_{1}, …, j_{ℓ}} = {0, …, g} \ I. The set {p |< a^{i}, p >= m_{j}(a^{i}) for i ∈ I} is by definition a face γ_{j,I} of the Newton polyhedron N_{+}(f_{j}), and this face is compact by the assumption on I.

The function f̃_{j}|_{S_{I}} is the restriction to S_{I} of a function on C^{g+1}(σ) which is independent of the y_{i} ; i ∈ I. Moreover, this last function differs from the composition with π(σ) of the function

f_{j,γ_{j,I}} = ∑_{p∈γ_{j,I}} f_{p}^{(j)}u^{p}

by multiplication by a monomial in (y_{0}, …, y_{g}). Since on S_{I} the coordinates y_{k}; k ∉ I are ≠ 0, the behavior of the functions f̃_{j}|_{S_{I}} is determined by the behavior of the f_{j,γ_{j,I}} ∘ π(σ) at points where all the y_{i} are ≠ 0. It follows that if the jacobian determinant of the functions f_{1,γ_{1,I}} , … f_{k,γ_{k,I}} has a k × k minor which does not vanish on (C^{∗})^{g+1} then the functions f̃_{1}|_{S_{I}} , …, f̃_{k}|_{S_{I}} define a nonsingular subspace of codimension k in S_{I} (in particular it is empty if k > dimS_{I}). This implies that the strict transform of X by π(Σ) is non-singular and transversal to all the strata S_{I} in a neighborhood of π(σ)^{−1}(0); in particular it is transversal to the critical locus of π(σ). ⟲

Applying this to each chart of π(Σ) gives:

Corollary. If the set of functions {f_{j}}_{1≤j≤k} defining the complete intersection X ⊂ C^{g+1} is non-degenerate at 0, there exists a neighborhood U of 0 in C^{g+1} such that the strict transform X̃ → X of X by π(Σ) induces in π(Σ)^{−1}(U) an embedded toric pseudo-resolution of singularities of X ∩ U.

5.2 The Toric Resolution of the Monomial Curve

Let C^{Γ} be the monomial curve derived from C. In this section, we study the embedded resolution of the curve C^{Γ} by a toric morphism from two different viewpoints, which correspond to its parametric and equational presentations respectively.

First, the monomial curve is the closure in C^{g+1} of an orbit of C^{∗} described by

t ↦ (t^{\overline{β}^{0}}, t^{\overline{β}^{1}}, …, t^{\overline{β}^{g}}).

By the theory of toric varieties (see [O], Chap 1) this orbit is described combinatorially by the linear map Z → Z^{g+1} such that the image of 1 is the weight vector

w = (\overline{β}_{0}, \overline{β}_{1}, …, \overline{β}_{g}) ∈ Z^{g+1}.

This means that the corresponding morphism of semigroup algebras

C[u_{0}, …, u_{g}] → C[t]

obtained from the dual map Ž^{g+1} → Ž by restriction to N^{g+1} is the morphism of algebras which corresponds to our parametrization of the monomial curve.

We use the notations of §4. Now if we take a regular fan Σ in R_{+}^{g+1} which is compatible with the weight vector w, in the sense that w is contained in an edge of any simplicial cone it meets outside 0, we obtain a toric map

π(Σ): Z(Σ) → C^{g+1}.

To compute the strict transform of C^{Γ} by π(Σ), we need to find out first which charts Z(σ) of Z(Σ), corresponding to (g + 1)-dimensional simplicial cones σ of Σ, have an image which contains C^{Γ}. Seeking solutions of the form y_{i}(t) = c_{i}t^{α_{i}} + ⋯ to the equations y_{0}^{a_{i}^{0}}. . . . .y_{g}^{a_{i}^{g}} = t^{\overline{β}^{i}}, we see that the vector (α_{0}, …, α_{g}) must satisfy M.α = w where M = (a_{i}^{j}) is the matrix of the column vectors (a^{0}, …, a^{g}) which are the generators of the simplicial cone σ. Since we want the α’s to be positive, this implies that σ contains w, so that by our assumption that the fan is compatible with w, the vector w must be equal to one of the a^{j}. We see that we need to consider only those simplicial cones σ = < a^{0}, …, a^{g} >, where up to reordering we have w = a^{0}. Then, w is the image of the coordinate vector (1, 0, …, 0) by the linear map p(σ): Z^{g+1} → Z^{g+1} described by the matrix M, i.e., sending the i-th basis vector to a^{i}. The strict transform of C^{Γ} by the map π(σ) is again a monomial curve, but its weight vector is the pull back of w by p(σ), which is the basis vector (1, 0, …, 0); in fact, the simplicial cone σ is the image by p(σ) of the cone generated by the basis vectors. So the strict transform of C^{Γ} has the parametric representation

y_{0} = t, y_{1} = ⋯ = y_{g} = 1,

and it is indeed non singular and transversal to the exceptional divisor.

Remark. 1) The regular simplicial fan compatible with w is by no means unique; however, there are, at least in the case of plane branches, algorithms which produce such fans. Their description is beyond the intent of this text, but the reader can get some idea of the problem by looking at the example which concludes it.

2) This resolution process works for any monomial curve: the embedded toric resolution reduces entirely to the combinatorial problem of finding a regular simplicial fan Σ in R_{+}^{g+1} compatible with the weight vector. We can summarize this in:

Theorem 5.3. Let C^{Γ} ⊂ C^{g+1} be a monomial curve with weight vector w ∈ N^{g+1}. For any regular simplicial fan Σ in R_{+}^{g+1} compatible with w, the toric map π(Σ): Z(Σ) → C^{g+1} has the property that the strict transform of C^{Γ} is non singular and transverse to the exceptional divisor. In fact, it appears only in the charts corresponding to simplicial cones containing w and there it is defined parametrically, up to reordering of the coordinates, by y_{0} = t, y_{1} = ⋯ = y_{g} = 1. ⟲

This construction of the resolution is simpler than the following one, but not as easily adaptable to the study of the effect of the toric map π(Σ) on deformations of C^{Γ} such as our curve C ⊂ C^{g+1}. The other viewpoint is to explore the implications of the previous section on C^{Γ}. We reprove the result above from this viewpoint. As specified in Equations (**), C^{Γ} is defined by equations {f_{j} = 0}_{1≤j≤g}, where each f_{j} = u_{j}^{α} − M_{j} and M_{j} is a monomial in u_{0}, …, u_{j−1}. Note that suppf_{j} has exactly two elements, corresponding to the exponents of the two monomials in f_{j}, and each Newton polyhedron N_{+}(f_{j}) has only one compact face, which is a segment. For each function f_{j}, let Σ_{0}^{(j)} be the fan representing the equivalence classes derived from the function m_{j}. Let Σ be a non-singular fan obtained by refining Σ_{0} = ⋂_{j} Σ_{0}^{(j)}, and N_{+} be the corresponding Newton polyhedron. We refer to the compact face of the Newton polyhedron N_{+}(f_{j}) as γ_{j}.

Proposition 5.4. The curve defined by {f̃_{1} = f̃_{2} = ⋯ = f̃_{g} = 0} intersects the divisor π(σ)^{−1}(0) only in the charts C^{g+1}(σ), σ =< a^{0}, …, a^{g} >, in which for some b the vector a^{b} is the weight vector (\overline{β}_{0}, …, \overline{β}_{g}). If a^{b} is the weight vector, then (Π_{i=0}^{g}y_{i} = 0) ⋂{f̃_{1} = f̃_{2} = ⋯ = f̃_{k} = 0} is contained in {y_{b} = 0} but not in any other hyperplane y_{i} = 0 for i ≠ b.

Proof. Each strict transform f̃_{j} has only two terms

f̃_{j} = f_{p_{0}^{(j)}}^{(j)} + f_{p_{1}^{(j)}}^{(j)}y_{0}^{<a^{0},p_{1}^{(j)}>−m_{j}(a^{0})} … y_{g}^{<a^{g},p_{1}^{(j)}>−m_{j}(a^{g})}

since each equation f_{j} has only two terms. The only possibility for the g equations f̃_{j} = 0 to have a common root with y_{i} = 0 for some i is that < a^{i}, p_{1}^{(j)} >= m_{j}(a^{i}) =< a^{i}, p_{0}^{(j)} >. But this implies that a^{i} is constant on the Minkowski sum of the g segments constituting the Newton polyhedra of the f_{j}. By the structure of the equations, this sum is of dimension g; a^{i} is uniquely determined as the primitive normal vector of this face, and it has to be the weight vector a^{b}. Indeed, the monomial map π(σ): Z(σ) → C^{g+1} maps the strict transform C̃^{Γ} to C^{Γ}; if the coordinate y_{b} corresponds to the weight vector, we see that y_{b} must be a coordinate on the strict transform of C^{Γ}, and all other coordinates equal to 1. In other words, the equations in Z(σ) of the desingularization of C^{Γ} are (y_{i} = 1, for i ≠ b); they are the strict transforms of the equations of C^{Γ}, which have only ±1’s as coefficients. ⟲

Theorem 5.5. Let C^{Γ} be a monomial curve defined by {f_{j}}_{1≤j≤g} as above. The closure of the inverse image of {f_{1} = ⋯ = f_{g} = 0} ∩ (C^{∗})^{g+1} by the morphism π(Σ) is non-singular and transversal in Z(Σ) to the strata of π(Σ)^{−1}(0).

Proof. This follows directly from Theorem 5.2 if we can prove the fact that the family of functions {f_{j}}_{1≤j≤g} is non-degenerate.

But there is only one compact face for the Newton polyhedron ∑_{j=1}^{g} N_{+}(f_{j}); it is the (Minkowski) sum ∑_{j=1}^{g} γ_{j}. By Proposition 5.4, this compact face corresponds to the subset I = {j} ⊂ {0, …, g} such that a^{j} is the weight vector. Moreover, for each j, the function f_{j} is equal to f_{j,γ_{j}}, so that we only have to check that the equations of the monomial curve define a non-singular complete intersection in (C^{∗})^{g+1} and we have

df_{1} ∧ ⋯ ∧ df_{g} = n_{1} … n_{g}u_{1}^{n_{1}−1} … u_{g}^{n_{g}−1}du_{1} ∧ ⋯ ∧ du_{g} + ⋯

so the differential form does not vanish outside of the coordinate hyperplanes: the set of equations f_{1}, …, f_{g} is non-degenerate. Note that unlike the first proof, this one encounters a (minor) difficulty if we work over a field of characteristic dividing one of the n_{i}. ⟲

6 Simultaneous Resolution

We will now show that the toric morphism not only resolves C^{Γ}, but simultaneously resolves all curves in the miniversal deformation with constant semigroup − χ_{u} of C^{Γ} which we saw in section 3. Definition. Let f : (X, 0) → (Y, 0) be a flat map with reduced fibres and Y reduced. We say (see [T2] ) that f admits a very weak simultaneous resolution if, for all sufficiently small representatives, there exists a proper morphism π : X̃ → X such that:

1. The composition q = f ∘ π : X̃ → Y is an analytic submersion, i.e., q is flat, and for all y ∈ Y, the fiber X̃(y) = q^{−1}(y) is non-singular.

2. For all y ∈ Y, the induced morphism X̃(y) → X(y) is a resolution of the singularities of X(y). Let Y_{1} be the image of a section σ : Y → X.

We say that f admits a weak simultaneous resolution along Y_{1} if it also satisfies the condition:

1. The morphism q_{Y_{1}} : (π^{−1}(Y_{1}))_{red} → Y induced by q is locally topologically a fibration, in the sense that every point x̃ ∈ π^{−1}(Y_{1}), has an open neighborhood U ∈ (π^{−1}(Y_{1}))_{red} such that U_{Y_{1}} ≃ V × Ỹ_{0}, where V is an open set in Y, Ỹ_{0} is an open neighbourhood of x̃ in the fibre of q_{Y_{1}} passing through x̃, and ≃ is a Y -homeomorphism.

We say that f admits a strong simultaneous resolution along Y_{1} if it also satisfies the condition:

The morphism q_{Y_{1}} : (π^{−1}(Y_{1})) → Y induced by q is locally analytically trivial (before reduction), in the sense that every point x̃ ∈ π^{−1}(Y_{1}), has an open neighborhood U ∈ (π^{−1}(Y_{1})) such that U_{Y_{1}} ≃ V × Ỹ_{0}, where V is an open set in Y, Ỹ_{0} is an open neighbourhood of x̃ in the fibre of q_{Y_{1}} passing through x̃, and ≃ is a Y -isomorphism.

An embedded resolution for − χ ⊂ C^{N} is a bimeromorphic map π: Z → C^{N} where Z is smooth, the exceptional locus E of π is a divisor with normal crossings in Z and the strict transform of − χ is smooth and transversal to the canonical stratification of E. As usual, C^{N} stands for an open subset of C^{N}.

Recall from Section 3 that the miniversal deformation of the curve C^{Γ} yields a family of curves, p: −χ_{u} → C^{τ_{−}}, embedded in C^{g+1} × C^{τ_{−}} in such a way that p is induced by the second projection, and where − χ_{u} is defined by the equations

F_{1} = f_{1} + ∑_{r=1}^{τ_{−}} v_{r}ϕ_{r,1}(u_{0}, …, u_{g}) = 0
·  ·  ·
·  ·  ·
·  ·  ·
F_{g} = f_{g} + ∑_{r=1}^{τ_{−}} v_{r}ϕ_{r,g}(u_{0}, …, u_{g}) = 0

where the ϕ_{r,j} are polynomials and each monomial in (u_{0}, …, u_{g}) appearing in ϕ_{r,j} is of weight > n_{j}\overline{β}_{j} when each u_{k} is given the weight \overline{β}_{k}; the equation f_{j} is then homogeneous of weight n_{j}\overline{β}_{j}. We have − χ_{u}(0) ≃ C^{Γ}. Let π(Σ): Z(Σ) → C^{g+1} be the toric modification associated to a regular fan in R_{+}^{g+1} compatible with the sum of the Newton polyhedra of the equations f_{j} of C^{Γ}. By construction, C^{Γ} is resolved by the morphism π(Σ).

Theorem 6.1. If Σ is a regular fan compatible with the Newton polyhedra of the equations (∗∗) of C^{Γ}, the morphism

Π(Σ) = π(Σ) × Id_{C^{τ_{−}}} : Z(Σ) × C^{τ_{−}} → C^{g+1} × C^{τ_{−}}

induces by restriction to the strict transform ˜− χ_{u} a resolution of singularities of − χ_{u}, which is a strong simultaneous resolution for p: −χ_{u} → C^{τ_{−}} with respect to the subspace Y_{1} = {0} × C^{τ_{−}}. In addition, it is an embedded resolution for − χ ⊂ C^{g+1} × C^{τ_{−}}.

Proof. Consider in a chart C^{g+1}(σ) × C^{τ_{−}} the composition

F_{j} ∘ Π(σ) = f_{j} ∘ Π(σ) + ∑_{r=1}^{τ_{−}} v_{r}(ϕ_{r,j} ∘ π(σ)).

Since for each support function m(a) and fixed q the function < a, q > /m(a) is a continuous function of a, and since the ϕ_{j,r} are polynomials, after possibly refining the fan Σ we may assume that for each cone σ =< a^{b}, a^{1}, …, a^{g} > containing the weight vector a^{b}, for each j, 1 ≤ j ≤ g, and for each monomial appearing in one of the polynomials ϕ_{j,r}, the inequality < a^{b}, q > > m_{j}(a^{b}), which is equivalent to the weight inequalities mentioned above, implies the inequalities < a^{s}, q > ≥ m_{j}(a^{s}) for 1 ≤ s ≤ g. We can therefore write the composition F_{j} ∘Π(σ) as follows:

F_{j} ∘ Π(σ) = y_{0}^{m_{j}(a^{0})} … y_{g}^{m_{j}(a^{g})}(f̃_{j} + ∑_{r=1}^{τ_{−}} v_{r}ϕ̃_{r,j}),

where ϕ̃_{r,j} = ∑_{q} ϕ̃_{r,j,q}y_{0}^{<a^{b},q>−m_{j}(a^{b})} … y_{g}^{<a^{g},q>−m_{j}(a^{g})}. From this follows, since it is true for the f̃_{j}, that at least for small ∥v∥ the F̃_{j} = f̃_{j} + ∑_{r=1}^{τ_{−}} v_{r}ϕ̃_{r,j} define a non-singular complete intersection ˜− χ_{u} in Z(σ) × C^{τ_{−}}: the strict transform of − χ_{u}; moreover, each fiber over a point v ∈ C^{τ_{−}} for sufficiently small ∥v∥ is the resolution of the corresponding fiber of − χ_{u}(v). Let us now consider Ỹ_{1} = (π(σ) × Id_{C^{τ_{−}}} | ˜ −χ_{u})^{−1}(Y_{1}). Its equations are y_{b} = 0, F̃_{j} = 0; by the Implicit

Function theorem, it is non-singular and admits at least for small ∥v∥ the coordinates v_{1}, …, v_{τ_{−}}; the morphism Ỹ_{1} → Y_{1} is not only a homeomorphism, but a local analytic isomorphism. Since there is an equivariant action of C^{∗}, all this is in fact true for all v. Let us now consider a cone σ not adjacent to the weight vector. Let u^{q} be a monomial appearing as a ϕ_{r,j}, set G_{j} = f_{j} + v_{r}u^{q}, and let K_{j} ⊂ {0, … g} be the set of those indices s such that < a^{s}, q > < m_{j}(a^{s}). Set n_{j}(a^{s}) =< a^{s}, q > for s ∈ K_{j}. Then G_{j} ∘ Π(σ) equals

Π_{s∉K_{j}}y_{s}^{m_{j}(a^{s})}Π_{s∈K_{j}}y_{s}^{n_{j}(a^{s})}(f̃_{j}Π_{s∈K_{j}}y_{s}^{m_{j}(a^{s})−n_{j}(a^{s})} + v_{r}Π_{s∉K_{j}}y_{s}^{<a^{s},q>−m_{j}(a^{s})}).

By our choice of σ we know that the f̃_{j} do not vanish together on the exceptional divisor. The common zeroes of the strict transforms

G̃_{j} = f̃_{j}Π_{s∈K_{j}}y_{s}^{m_{j}(a^{s})−n_{j}(a^{s})} + v_{r}Π_{s∉K_{j}}y_{s}^{<a^{s},q>−m_{j}(a^{s})}

constitute in the chart C^{g+1}(σ) × C^{τ_{−}} the strict transform ˜− χ_{u} of the space − χ_{u}, therefore this strict transform meets the exceptional divisor only for v_{r} = 0 and then it coincides with the union of some components in C^{g+1}(σ) × {0} of the exceptional divisor of π(σ). However, these components do not meet the strict transform (see 5.4) of the curve C^{Γ}, and therefore the inverse image of {0} in ˜− χ_{u} is not connected. Since − χ_{u} is the total space of an equisingular deformation of C^{Γ} it is analytically irreducible at the origin and by Zariski’s main theorem the inverse image of {0} must be connected. We therefore obtain a contradiction, which shows that for each cone σ =< a^{0}, …, a^{g} >∈ Σ all the monomials u^{q} appearing in the miniversal equisingular deformation of C^{Γ} satisfy the inequalities < a^{s}, q > ≥ m_{j}(a^{s}) for 0 ≤ s ≤ g. We are in the situation described at the beginning of the proof, and this shows that the map ˜− χ_{u} → − χ_{u} induced by the toric modification Z(Σ) × C^{τ_{−}} → C^{g+1} × C^{τ_{−}} is a weak embedded simultaneous resolution along Y_{1}.

Finally we remark that a refinement Σ′ of our original regular fan corresponds to a birational toric map Z(Σ′) → Z(Σ) which is an isomorphism outside the exceptional divisor. Moreover, in a chart corresponding to a cone σ adjacent to the weight vector, it is an isomorphism outside Π_{s=1}^{g}y_{s} = 0; this open set contains the strict transform of C^{Γ} since we saw that it meets only y_{0} = 0 (see 5.4). It follows from this that if the strict transform of − χ_{u} in Z(Σ′)×C^{τ_{−}} is a simultaneous resolution of − χ_{u}, such was already the case for the strict transform in Z(Σ)×C^{τ_{−}}. This concludes the proof. ⟲

Corollary. For a plane branch (C, 0) with g characteristic exponents, with semi group Γ =< \overline{β}_{0}, …, \overline{β}_{g} > and given parametrically by x = x(t), y = y(t), with ν(x) = \overline{β}_{0}, ν(y) = \overline{β}_{1}, let ξ_{i}(t) ∈ O_{C,0} = C\{x(t), y(t)\} ⊂ C\{t\} for 2 ≤ i ≤ g be series such that

(ν(x), ν(y), ν(ξ_{2}), …, ν(ξ_{g})) = (\overline{β}_{0}, …, \overline{β}_{g}),

where ν is the t-adic valuation. Then the embedding (C, 0) → (C^{g+1}, 0) given by

u_{0} = x(t), u_{1} = y(t), u_{2} = ξ_{2}(t), …, u_{g} = ξ_{g}(t)

has the property that its image is resolved by one toric modification.

7 The transforms of plane curves

Let us return to the construction of the miniversal constant semigroup deformation in §3. It is easy to check that the vectors ϕ_{j} ∈ C[C^{Γ}]^{g} , 1 ≤ j ≤ g − 1, containing u_{j+1} at the j-th line and zero elsewhere, are independent modulo N with the notations of §3. In the miniversal constant semigroup deformation, the plane curves appear as those for which the coefficients of all these vectors are ≠ 0 (see [T1]). For each point v_{0} in the corresponding open set of C^{τ_{−}}, the equation of the corresponding plane branch is obtained as follows, in a small neighborhood of 0 :

One uses the first equation F_{1} to express u_{2} as a series in u_{0}, u_{1}, using the implicit function theorem. Then one substitutes this value in F_{2} and uses this equation to express u_{3} as a function of u_{0}, u_{1}, and so on. Finally the equation F_{g−1} = 0 allows us to express u_{g} as a series in u_{0}, u_{1}, and at this point we have the equations of a non-singular surface S_{v_{0}} containing our branch, thus explicitly shown to be planar.

The next step is to consider the strict transform S̃_{v_{0}} of S_{v_{0}} in C^{g+1}×{v_{0}} and the induced map S̃_{v_{0}} → S_{v_{0}} induced by Π(Σ). One shows that, in a neighborhood of the point of the exceptional divisor picked by the strict transform of C, it factors through the toric map S_{v_{0}}^{1} → S_{v_{0}} which on S_{v_{0}} with the coordinates u_{0}, u_{1} resolves

the singularities of the branch u_{1}^{n_{1}} − u_{0}^{ℓ_{1}^{(0)}} = 0. One then takes the strict transform of this equation as a new coordinate on S_{v_{0}}^{1} and then the strict transform of C appears as a branch on S_{v_{0}}^{1} with one less characteristic exponent; its equations are the strict transforms of F_{2}, …, F_{g} on S_{v_{0}}^{1}, in the new coordinates. Again one must look at the toric map (in the new coordinates) which resolves the first characteristic pair and show that it is dominated by S̃_{v_{0}}, and so on; finally we have factored the map S̃_{v_{0}} → S_{v_{0}} into the composition of g toric maps. This is similar to the process described by Spivakovsky in [S] of approximating a given valuation by a sequence of monomial valuations. However we see that if we allow a change in ambient space, the uniformization of a single monomial valuation, namely the t-adic valuation on the monomial curve, gives the uniformization of the t-adic valuation of C.

On the surface S, each of the coordinates u_{i}, 2 ≤ i ≤ g is expressed as a series in u_{0}, u_{1}. For each i, 2 ≤ i ≤ g, the equation u_{i} = 0 on S defines a curve with i characteristic exponents having at the origin maximal contact with our original plane branch; this is a consequence of what we have seen in Corollary 6, and gives a geometric construction of curves (singular or not) having maximal contact in the sense of [L-J] with our plane branch; they are the curves on S_{v_{0}} defined by the equations u_{i} = 0, i ≥ 1. Remark that for i ≥ 2 they can also be written f_{i−1}(u_{0}, …, u_{g}) + higher degree, and with a little more work we find the structure of the approximate roots.

The construction just outlined needs to be detailed but we will not do it here since it is not part of the main purpose of this text. It is performed in the first non trivial special case (two characteristic exponents (3/2, 7/4), i.e., semigroup < 4, 6, 13 >) at the end of the example to which we now turn.

8 Appendix : An example

Example. Suppose we have a plane branch C with semigroup < 4, 6, 13 >. The corresponding monomial curve C^{Γ} is described (see [T]) by the polynomials

\begin{aligned}f_{1} & = & u_{1}^{2} − u_{0}^{3} & = & 0 \\ f_{2} & = & u_{2}^{2} − u_{0}^{5}u_{1} & = & 0. \end{aligned}

As specified above, we construct N_{+}(f_{1}) and N_{+}(f_{2}), the Newton polygons, by the convex hull of the union of positive quadrants beginning at points corresponding to the exponents of the {u_{i}}’s . Thus,

N_{+}(f_{1}) = convex hull of {{(3, 0, 0), (0, 2, 0)} + R_{+}^{3}}
N_{+}(f_{2}) = convex hull of {{(5, 1, 0), (0, 0, 2)} + R_{+}^{3}}

and the corresponding fans are defined, respectively, by

Σ_{0}^{(1)} = the first quadrant of R^{3} cut by the plane 3x = 2y, and
Σ_{0}^{(2)} = the first quadrant of R^{3} cut by the plane 5x + y = 2z.

Then the equivalence class fan of f_{1} and f_{2} together is the intersection

Σ_{0} = Σ_{0}^{(1)} ∩ Σ_{0}^{(2)}

which has four maximal dimension cones:

1. ⟨(2, 3, 0), (0, 1, 0), (4, 6, 13), (0, 2, 1)⟩

2. ⟨(0, 2, 1), (0, 0, 1) (4, 6, 13)⟩

3. ⟨(2, 0, 5), (1, 0, 0), (2, 3, 0), (4, 6, 13)⟩

4. ⟨(0, 0, 1), (2, 0, 5), (4, 6, 13)⟩

We build here by an inductive method a refinement which resolves the fan Σ_{0} = Σ_{0}^{(1)} ∩ Σ_{0}^{(2)}. Notice that each maximal dimension cone spans Z^{3} (since the matrix of the spanning skeleton is unimodular). For each cone, we have listed the corresponding morphism π(σ) in terms of the resulting {u_{i}}, the composition f_{i} ∘ π(σ)(y_{0}, y_{1}, y_{2}) for i = 1, 2, and the exceptional divisor π(σ)^{−1}(0). We show that the two surfaces are transverse to each other and to the exceptional divisor. We start from a refinement for the fan in two dimensions associated to the first equation u_{1}^{2} − u_{0}^{3} = 0. The weight vector is (2, 3) and from the geometric interpretation of the continued fraction expansion we find that a regular fan subdividing it is composed of the following four 2-dimensional cones and their faces:

1. σ_{1}^{(2)} =⟨(1, 0), (1, 1)⟩

2. σ_{2}^{(2)} =⟨(1, 1), (2, 3)⟩

3. σ_{3}^{(2)} =⟨(2, 3), (1, 2)⟩

4. σ_{4}^{(2)} = ⟨(1, 2), (0, 1)⟩ Now we have to lift the vectors of the 1-skeleton to R^{3} in such a way that they form the 1-skeleton of a regular fan subdividing Σ_{0}. It suffices to show that in each cone of this fan both functions m_{1} and m_{2} are linear, i.e., that all linear forms take their minimum at a vertex of the sum N of the Newton polyhedra of f_{1} and f_{2}. This Newton polyhedron has four vertices: (3, 0, 2), (8, 1, 0), (0, 2, 2), (5, 3, 0), so we are especially interested in finding four cones having the weight vector as one of their faces.

Let us begin by lifting the weight vector (2, 3) : We seek integral vectors of the form (2k, 3k, z) with k > 0, z > 0; we know that they take their minimum value on the compact segment of the Newton polyhedron of f_{1}. On N_{+}(f_{2}) they take their minimum at ((5, 1, 0) if 13k ≤ 2z, at (0, 0, 2) if 13k ≥ 2z, at both if 13k = 2z. Remembering that we seek primitive vectors, we find three natural possibilities: k = 1, z = 6, giving the vector a^{1} = (2, 3, 6), which takes its minimum at (0, 0, 2), k = 1, z = 7 giving a^{2} = (2, 3, 7), which takes its minimum at (5, 1, 0), and of course the weight vector a^{0} = (4, 6, 13) itself. We remark that a^{0} = a^{1} + a^{2}. Let us now try to lift (1, 1); we seek vectors of the form (k, k, z); on N_{+}(f_{1}) they take their minimum value at (0, 2, 0), and on N_{+}(f_{2}) at (5, 1, 0) if 3k ≤ z, at (0, 0, 2) if 3k ≥ z. In this case we can take k = 1, z = 3, which gives the vector a^{3} = (1, 1, 3) which takes its minimum on the compact segment of N_{+}(f_{2}). Finally take (1, 2); we seek vectors (k, 2k, z) and there is one taking its minimum on the compact face of N_{+}(f_{2}): it corresponds to k = 1, z = 3, giving the vector a′^{4} = (1, 2, 3). We remark that a^{0} = a^{2} + a^{3} + a′^{4}, and it is therefore tempting to take a′^{4} as our fourth vector. However, it is not on the hyperplane 5x + y = 2z so that the support function m_{2} will not be linear on a cone spanned by {a^{0}, a^{2}, a′^{4}}. A better choice is to take a^{4} = a^{0} − a^{3} = (3, 5, 10).

Now we check that the four cones (which we have decorated with the points of the Newton polyhedra of f_{1} and f_{2} where the support functions m_{1}(a), m_{2}(a) take their minimum for a ∈ σ_{i}):

1. σ_{1}^{(3)} = ⟨a^{0}, a^{1}, a^{3}⟩; p_{0}^{(1)} = (0, 2, 0); p_{0}^{(2)} = (0, 0, 2)

2. σ_{2}^{(3)} =⟨a^{0}, a^{2}, a^{3}⟩; p_{0}^{(1)} = (0, 2, 0); p_{0}^{(2)} = (5, 1, 0)

3. σ_{3}^{(3)} =⟨a^{0}, a^{2}, a^{4}⟩; p_{0}^{(1)} = (3, 0, 0); p_{0}^{(2)} = (5, 1, 0)

4. σ_{4}^{(3)} =⟨a^{0}, a^{1}, a^{4}⟩; p_{0}^{(1)} = (3, 0, 0); p_{0}^{(2)} = (0, 0, 2)

are unimodular and such that their union is a neighborhood of R_{+}a^{0} in R^{3}; the second fact is obvious since the weight vector is in the interior of the convex hull of the σ_{i} , 1 ≤ i ≤ 4, and to check the first it suffices to check that one of them is unimodular. Now we can complete this subfan into a regular fan of R_{+}^{3}, either by the same method or by invoking a general theorem, but we do not much care about it, since we have seen that the only charts where something interesting happens are those which correspond to a cone containing the weight vector in its 1-skeleton, and we have those above. Moreover, the projections in R^{2} of the σ_{i} form part of a regular subfan of the σ_{i}^{(2)}.

Let us now study the behavior of f_{1}, f_{2} under the monomial maps corresponding to the σ_{i}^{(3)} , 1 ≤ i ≤ 4. For economy we write σ_{i} for σ_{i}^{(3)}.

1. The cone σ_{1} spanned by {(4, 6, 13), (2, 3, 6), (1, 1, 3)}

π(σ_{1}) : u_{0} = y_{0}^{4}y_{1}^{2}y_{2}
u_{1} = y_{0}^{6}y_{1}^{3}y_{2}
u_{2} = y_{0}^{13}y_{1}^{6}y_{2}^{3}

Then,

\begin{aligned}f_{1} ∘ π(σ_{1})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{12}y_{1}^{6}y_{2}^{2}(1 − y_{2}) \\ f_{2} ∘ π(σ_{1})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{26}y_{1}^{12}y_{2}^{6}(1 − y_{1}) \end{aligned}

and

π(σ_{1})^{−1}(0) = {y_{0} = 0} ∪ {y_{1} = 0} ∪ {y_{2} = 0}.

Here the strict transforms are

\begin{aligned}f̃_{1} & = & 1 − y_{2} \\ f̃_{2} & = & 1 − y_{1}. \end{aligned}

It is clear that the equations f̃_{1} = f̃_{2} = 0 define a non-singular complete intersection meeting the exceptional divisor transversally at the point y_{0} = 0, y_{1} = y_{2} = 1. To make the same computation for the other charts π(σ_{i}) is in fact superfluous, since by construction of the toric modification, we will only observe the same phenomenon in a different chart. For verification’s sake let us compute for π(σ_{3}):

2. The cone σ_{3} spanned by {(4, 6, 13), (2, 3, 7), (3, 5, 10)}

π(σ_{3}) : u_{0} = y_{0}^{4}y_{1}^{2}y_{2}^{3}
u_{1} = y_{0}^{6}y_{1}^{3}y_{2}^{5}
u_{2} = y_{0}^{13}y_{1}^{7}y_{2}^{10}

Then,

\begin{aligned}f_{1} ∘ π(σ_{3})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{12}y_{1}^{6}y_{2}^{9}(y_{2} − 1) \\ f_{2} ∘ π(σ_{3})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{26}y_{1}^{13}y_{2}^{20}(y_{1} − 1) \end{aligned}

And indeed it is the same situation viewed in another chart.

The miniversal deformation with constant semigroup of C^{Γ} is computed in [T1]: here τ_{−} = 2 and − χ_{u} is defined in C^{3} × C^{2} by the equations

\begin{aligned}F_{1} & = & u_{1}^{2} − u_{0}^{3} + v_{1}u_{2} + v_{2}u_{0}u_{2} = 0 \\ F_{2} & = & u_{2}^{2} − u_{0}^{5}u_{1} = 0 \end{aligned}

If we compose F_{1} and F_{2} with Π(σ_{1}) = π(σ_{1})×Id_{C^{2}} : C^{g+1}(σ_{1})×C^{2} → C^{g+1}×C^{2} we get

\begin{aligned}F_{1} ∘ Π(σ_{1})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{12}y_{1}^{6}y_{2}^{2}(1 − y_{2} + v_{1}y_{0}y_{2} + v_{2}y_{0}^{5}y_{1}^{2}y_{2}^{2}) \\ F_{2} ∘ Π(σ_{1})(y_{0}, y_{1}, y_{2}) & = & y_{0}^{26}y_{1}^{13}y_{2}^{20}(1 − y_{1}) \end{aligned}

The strict transform of − χ_{u} is defined in this chart by

\begin{aligned}F̃_{1}(y_{0}, y_{1}, y_{2}) & = & 1 − y_{2} + v_{1}y_{0}y_{2} + v_{2}y_{0}^{5}y_{1}^{2}y_{2}^{2} = 0 \\ F̃_{2}(y_{0}, y_{1}, y_{2}) & = & 1 − y_{1} = 0 \end{aligned}

It is indeed a simultaneous resolution for − χ_{u}. There remains to check that the strict transform of − χ_{u} does not meet the charts corresponding to cones that are not adjacent to the weight vector. This depends on our construction of a regular fan and will not be done here.

Let us now consider the plane branch with equations in C^{3} (for v = v_{0} ≠ 0):

\begin{aligned}F_{1}(v_{0}; u_{0}, u_{1}, u_{2}) & = & u_{1}^{2} − u_{0}^{3} − v_{0}u_{2} = 0 \\ F_{2}(u_{0}, u_{1}, u_{2}) & = & u_{2}^{2} − u_{0}^{5}u_{1} = 0 \end{aligned}

It lies on the non-singular surface S_{v_{0}} with equation F_{1}(v_{0}; u_{0}, u_{1}, u_{2}) = 0. Let S̃_{v_{0}} be the strict transform of S_{v_{0}} by the toric map Z(Σ) → C^{g+1}. As before we need to examine the situation only in a chart Z(σ) where σ is adjacent to the weight vector. We take a^{0} = (4, 6, 13), a^{1} = (2, 3, 7), a^{2} = (3, 5, 10). So π(σ) is described as:

π(σ) : u_{0} = y_{0}^{4}y_{1}^{2}y_{2}^{3}
u_{1} = y_{0}^{6}y_{1}^{3}y_{2}^{5}
u_{2} = y_{0}^{13}y_{1}^{7}y_{2}^{10}

and we have

(u_{1}^{2} − u_{0}^{3} + v_{0}u_{2}) ∘ π(σ) = y_{0}^{12}y_{1}^{6}y_{2}^{9}(1 − y_{2} + v_{0}y_{0}y_{1}y_{2})

so that the equation of S̃_{v_{0}} in this chart is

y_{2}(1 − v_{0}y_{0}y_{1}) = 1,

and the mapping S̃_{v_{0}} → S is described by

\begin{aligned}u_{0} & = & y_{0}^{4}y_{1}^{2}(\frac{1}{1−v_{0}y_{0}y_{1}} )^{3} \\ u_{1} & = & y_{0}^{6}y_{1}^{3}(\frac{1}{1−v_{0}y_{0}y_{1}} )^{5} \end{aligned}

All computations are made in a neighborhood of the exceptional divisor, i.e., for |y_{0}| small.

Now the toric map S_{v_{0}}^{1} → S_{v_{0}} which resolves the plane branch with one characteristic pair u_{1}^{2} − u_{0}^{3} = 0 has a chart:

\begin{aligned}u_{0} & = & x_{0}x_{1}^{2} \\ u_{1} & = & x_{0}x_{1}^{3} \end{aligned}

and we can check that there is a factorization S̃_{v_{0}} → S_{v_{0}}^{1} given as follows in the coordinates y_{0}, y_{1} on S̃_{v_{0}}:

\begin{aligned}x_{0} & = & 1 − v_{0}y_{0}y_{1} \\ x_{1} & = & y_{0}^{2}y_{1}(\frac{1}{1−v_{0}y_{0}y_{1}} )^{2} \end{aligned}

and we can view this factorization itself as composed of the maps

\begin{aligned}x_{0} & = & w_{0} \\ x_{1} & = & w_{0}^{2}w_{1} \end{aligned}

and

\begin{aligned}w_{0} & = & 1 − v_{0}y_{0}y_{1} \\ w_{1} & = & y_{0}^{2}y_{1} \end{aligned}

Now this last map is monomial after a change of the coordinates w_{0}, w_{1} and moreover the substitution of the w’s in the x’s gives a monomial map S_{v_{0}}^{2} → S_{v_{0}} described as follows

\begin{aligned}u_{0} & = & w_{0}^{5}w_{1}^{2} \\ u_{1} & = & w_{0}^{7}w_{1}^{3} \end{aligned}

which is again a monomial map, so that we have factorized our map S̃_{v_{0}} → S_{v_{0}} as a composition of two monomial maps, up to a change of variables (essentially a translation w_{0} ↦ w_{0} −1). Note the fact that it was convenient to refine part of the fan of the plane branch u_{1}^{2} − u_{0}^{3} = 0 from (1, 1), (2, 3) to (5, 7), (2, 3) before writing the translation, but that we could also have made a change in the variables y_{0}, y_{1} (for |y_{0}| small) to bring directly the map S̃_{v_{0}} → S_{v_{0}}^{1} into monomial form.

Finally, note that the strict transform on S̃_{v} of our plane curve C_{v_{0}} ⊂ S_{v_{0}} has equation y_{1} − 1 = 0 In the general case, since we do not deform the last equation f_{g} of the monomial curve, the last equation of the strict transform of C_{v_{0}}, in the proper chart will still be of the form y_{k} − 1 = 0.

References

[C] D. Cox, Toric varieties and toric resolutions. This volume.

[F] W. Fulton, Introduction to Toric Varieties. Princeton University Press, 1993.

[TE] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. Lecture Notes in Math. 339, Springer-Verlag 1973.

[L-J] M. Lejeune-Jalabert, Thèse d’Etat, Université Paris 7, 1973.

[L-T] M. Lejeune-Jalabert, B. Teissier, Cloture intégrale des idéaux et équisingularité. In: Séminaire sur les singularités, Ecole Polytechnique, 1974-75. Institut Fourier, Université de Grenoble.

[M] M. Merle, Polyèdre de Newton, Eventail et Désingularisation, d’après Varchenko. In Séminaire sur les Singularités des Surfaces, Palaiseau, France, 1976-1977, Edited by M. Demazure, H. Pinkham, B. Teissier, Lecture Notes in Math. 777, Springer-Verlag 1980.

[Oda] T. Oda, Convex bodies and algebraic geometry, Ergeb. der Math., 3. Folge, Band 15, Springer-Verlag 1988.

[O1] M. Oka, Geometry of plane curves via toroidal resolution. In: Algebraic Geometry and Singularities, A. Campillo et al., editors, Progress in Math., No. 134, Birkhaüser, 95-121.

[O2] M. Oka, Canonical stratification of non degenerate complete intersection varieties. Jour. Math. Soc. Japan, 1990, 42, 397-422.

[O3] M. Oka, Non degenerate complete intersection singularities. Actualités Mathématiques, Hermann, Paris, 1997.

[S] M. Spivakovsky, Resolution of singularities. Preprint, University of Toronto, 1994.

[T1] B. Teissier, Appendix in [Z].

[T2] B. Teissier, Résolution simultanée II. In: Séminaire sur les Singularités des Surfaces, Palaiseau, France, 1976-1977, Edited by M. Demazure, H. Pinkham, B. Teissier, Lecture Notes in Mathematics 777, Springer-Verlag, 1980.

[T3] B. Teissier, The hunting of invariants in the geometry of discriminants. In: Real and Complex Singularities, Nordic Summer School, Oslo 1976, Edited by Per Holm, Noordhoff and Sijthoff, 1977.

[V] A.N. Varchenko, Zeta function of the monodromy and Newton’s diagram. Invent. Math. 37 (1976), 253-262.

[Z] Oscar Zariski. Le Problème des Modules pour les Branches Planes. Hermann, Paris, 1986.

Dept. of Mathematics M.I.T. Cambridge, MA 02139, USA goldin@math.mit.edu Laboratoire de Mathématiques de l’E.N.S., URA 762 du C.N.R.S., Ecole Normale Supérieure de Paris 45 Rue d’Ulm 75005 Paris, France teissier@dmi.ens.fr