On the Lipschitz Saturation of Toric Singularities

(of proposition 1.4)

After a linear change of coordinates we can assume that the linear projection $\pi:(\mathbb{C}^{n},0)\to(\mathbb{C}^{m},0)$ is the projection on the first $m$ coordinates $(z_{1},\ldots,z_{n})\mapsto(z_{1},\ldots,z_{m})$. Let $J\subset\mathcal{O}_{X\times X,(0,0)}$ denote the ideal defining the diagonal of $X\times X$

$$J=\left<z_{1}-w_{1},\ldots,z_{n}-w_{n}\right>\mathcal{O}_{X\times X,(0,0)}$$

Denote $J_{\pi}=\left<z_{1}-w_{1},\ldots,z_{m}-w_{m}\right>\mathcal{O}_{X\times X,(0,0)}$. Then by [13, Proposition 8.5.11] the genericity of $\pi$ is equivalent to the equality of integral closures $\overline{J}=\overline{J_{\pi}}$ in $\mathcal{O}_{X\times X,(0,0)}$.

On the other hand, since $\pi:(X,0)\to(\pi(X),0)$ is a finite, generically 1-1 and surjective map, then the morphism

$$\pi^{*}:\mathcal{O}_{\pi(X),0}\to\mathcal{O}_{X,0}$$

is injective, makes $\mathcal{O}_{X,0}$ a finitely generated $\mathcal{O}_{\pi(X),0}$-module and induces an isomorphism of the corresponding field of fractions $Q(\mathcal{O}_{X,0})$. All these together imply that $\mathcal{O}_{X,0}$ and $\mathcal{O}_{\pi(X),0}$ have isomorphic integral closures in $Q(\mathcal{O}_{X,0})$ and so if $\eta:(\overline{X},0)\to(X,0)$ denotes the normalization of $(X,0)$ then the composition

$$(\overline{X},0)\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}(X,0)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}(\pi(X),0)$$

is a normalization of $(\pi(X),0)$.

Note that the ideal $I_{\Delta}$ of definition 1.1 is defined by the “coordinate functions” of the normalization map

$$(\overline{X},0)\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}(X,0)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}(\pi(X),0)$$

$$\underline{y}\mapsto\left(\eta_{1}(y),\ldots,\eta_{n}(y)\right)\mapsto\left(\eta_{1}(y),\ldots,\eta_{m}(y)\right),$$

that is (see [13, Section 8.5.2]),

$$I_{\Delta_{X}}=\left<\eta_{1}(y)-\eta_{1}(x),\ldots,\eta_{n}(y)-\eta_{n}(x)\right>\mathcal{O}_{\overline{X}\times\overline{X},(0,0)}$$

$$I_{\Delta_{\pi(X)}}=\left<\eta_{1}(y)-\eta_{1}(x),\ldots,\eta_{m}(y)-\eta_{m}(x)\right>\mathcal{O}_{\overline{X}\times\overline{X},(0,0)}.$$

To prove the desired result it is enough to prove the equality of the integral closures $\overline{I_{\Delta_{X}}}=\overline{I_{\Delta_{\pi(X)}}}$. But the germ map:

$$\eta\times\eta:\left(\overline{X}\times\overline{X},(0,0)\right)\longrightarrow\left(X\times X,(0,0)\right)$$

induces a morphism of analytic algebras:

$$(\eta\times\eta)^{*}:\mathcal{O}_{X\times X,(0,0)}\longrightarrow\mathcal{O}_{\overline{X}\times\overline{X},(0,0)}$$

such that:

$$\left<\left(\eta\times\eta\right)^{*}(J)\right>=I_{\Delta_{X}}$$

$$\left<\left(\eta\times\eta\right)^{*}(J_{\pi})\right>=I_{\Delta_{\pi(X)}}$$

and since $\overline{J}=\overline{J_{\pi}}$ in $\mathcal{O}_{X\times X,(0,0)}$ the result follows. ∎

The first isomorphism is proved in [15, Lemme 1] or [14, Lemma 1.1]. The proof given there is for normal toric varieties. However, the same proof holds in the non-normal case.

We prove the second isomorphism. Consider the following exact sequence:

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{I_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 68.18054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 68.18054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}[z_{1},\ldots,z_{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 118.52934pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 142.02246pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 142.02246pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}[\Gamma]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 191.05026pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 191.05026pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

Let $\mathfrak{m}=\langle z_{1},\ldots,z_{n}\rangle$. Taking completions with respect to $\mathfrak{m}$ we obtain the following exact sequence:

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\hat{I_{\Gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 65.05557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 65.05557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}[[z_{1},\ldots,z_{n}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 119.7655pt\raise 5.59027pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.42361pt\hbox{$\scriptstyle{\varphi_{f}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 144.45306pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 144.45306pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}[[\Gamma]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 199.03644pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 199.03644pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

On the other hand, $\hat{I_{\Gamma}}\cong I_{\Gamma}\mathbb{C}[[z_{1},\ldots,z_{n}]]$ [1, Proposition 10.15]. Denote $\varphi_{a}=\varphi_{f}|_{\mathbb{C}\{z_{1},\ldots,z_{n}\}}$. In the proof of [14, Lemma 1.1], alternatively [15, Lemme 1] it is shown that $\mbox{Im }\varphi_{a}=\mathbb{C}\{\Gamma\}$. It remains to prove that $\ker\varphi_{a}=I_{\Gamma}\mathbb{C}\{z_{1},\ldots,z_{n}\}$.

Since $\ker\varphi_{f}=I_{\Gamma}\mathbb{C}[[z_{1},\ldots,z_{n}]]$, it follows that $I_{\Gamma}\mathbb{C}\{z_{1},\ldots,z_{n}\}\subset\ker\varphi_{a}$. Let $F\in\mathbb{C}\{z_{1},\ldots,z_{n}\}$ be such that $0=\varphi_{a}(F)=\varphi_{f}(F)$. We conclude that $F\in\mathbb{C}\{z_{1},\ldots,z_{n}\}\cap I_{\Gamma}\mathbb{C}[[z_{1},\ldots,z_{n}]]=I_{\Gamma}\mathbb{C}\{z_{1},\ldots,z_{n}\}$ (these ideals are equal by [8, Exercise 8.1.5]). ∎

Let $\eta:\bar{X}\to X$ be the normalization. Recall that $\bar{X}$ is the toric variety defined by the semigroup $\check{\sigma}\cap\mathbb{Z}^{d}$ and that $\eta$ is induced by the inclusion of semigroups $\Gamma\subset\check{\sigma}\cap\mathbb{Z}^{d}$ [7, Proposition 1.3.8]. In particular, $\eta$ is a toric morphism.

Being the normalization, $\eta$ is an isomorphism on dense open sets. In addition, $\eta^{-1}(0)=0$. Indeed, let $q\in\bar{X}$ be such that $\eta(q)=0$. Recall that points in toric varieties correspond to homomorphisms of semigroups. Hence, the homomorphism corresponding to $\eta(q)$ sends every non-zero element of $\Gamma$ to 0. On the other hand, for every $m\in\check{\sigma}\cap\mathbb{Z}^{d}$ there is $k\geq 1$ such that $km\in\Gamma$. Since $\eta$ is induced by the inclusion $\Gamma\subset\check{\sigma}\cap\mathbb{Z}^{d}$, it follows that $q=0$.

By the previous paragraph, the induced germ of an analytic function $\eta:(\bar{X},0)\to(X,0)$ is finite and generically 1-1. On the other hand, it is known that $\bar{X}$ normal implies that $(\bar{X},0)$ normal [17, Satz 4]. By the uniqueness of normalization we conclude that $\eta:(\bar{X},0)\to(X,0)$ is the normalization of $(X,0)$.

Finally, $\mathcal{O}_{\bar{X},0}\cong\mathbb{C}\{\check{\sigma}\cap\mathbb{Z}^{d}\}$ follows using lemma 2.2. ∎

By a well-known theorem of Zariski [26], $\bar{X}$ irreducible and normal at $0$ implies that the completion of the local ring at 0 is an integral domain. Hence, $(\bar{X},0)$ is irreducible by lemma 2.2. Hence, the germ $(X,0)$ must also be irreducible. ∎

Let $R:=\mathbb{C}\{z_{1},\ldots,z_{n}\}$ denote the ring of convergent power series in $n$-variables over the field of complex numbers with maximal ideal $\mathfrak{m}=\left<z_{1},\ldots,z_{n}\right>$, and let $\mathcal{I}=\left<f_{1},\ldots,f_{m}\right>R$ where each $f_{j}\in A:=\mathbb{C}[z_{1},\ldots,z_{n}]\subset R$ is a homogeneous polynomial.

Let $K=\left<f_{1},\ldots,f_{m}\right>A$ be the ideal generated by the $f_{j}$’s in the polynomial ring $A$, so $\mathcal{I}=\left<K\right>R$. Suppose that $\mathcal{I}$ contains a power of $\mathfrak{m}$, i.e. there exists $k\geq 1$ such that $\mathfrak{m}^{k}\subset\mathcal{I}$, and let $s\in R$ be in the integral closure of $\mathcal{I}$. We can write

$$s=p+s^{\prime},$$

where $p\in A$ is a polynomial and $s^{\prime}\in\mathfrak{m}^{k}\subset\mathcal{I}\subset\overline{\mathcal{I}}$ is a series of order greater than or equal to $k$. Note that $p=s-s^{\prime}\in\overline{\mathcal{I}}$.

Since the ring extension $A_{\mathfrak{m}}\rightarrow R$ is faithfully flat ([11, Lemma B.3.4]) then $\overline{\mathcal{I}}\cap A_{\mathfrak{m}}=\overline{\left<K\right>A_{\mathfrak{m}}}$ ([16, Prop.1.6.2]), in particular $p/1\in\overline{\left<K\right>A_{\mathfrak{m}}}$. This means that in the localized polynomial ring $A_{\mathfrak{m}}$ we have an equation of integral dependence of the form:

$$\frac{p^{r}}{1}+\frac{b_{1}}{c_{1}}\frac{p^{r-1}}{1}+\cdots+\frac{b_{r-1}}{c_{r-1}}\frac{p}{1}+\frac{b_{r}}{c_{r}}=\frac{0}{1},$$

where $b_{j}\in K^{j}$ and $c_{j}\notin\mathfrak{m}$. Letting $u=c_{1}\cdots c_{r}\in A$ and multiplying this equation by the unit $\frac{u^{r}}{1}$ of $A_{\mathfrak{m}}$ we get the following equality in $A_{\mathfrak{m}}$:

$$\frac{(up)^{r}+\widetilde{b_{1}}(up)^{r-1}+\cdots+\widetilde{b_{r-1}}(up)+\widetilde{b_{r}}}{1}=\frac{0}{1}.$$

Since A is an integral domain we get an integral dependence equation in $A$

$$(up)^{r}+\widetilde{b_{1}}(up)^{r-1}+\cdots+\widetilde{b_{r-1}}(up)+\widetilde{b_{r}}=0,$$

that is, $up\in\overline{K}\subset A$. The inclusions $\mathfrak{m}^{k}\subset K\subset\overline{K}$ imply that $\overline{K}$ is an $\mathfrak{m}$-primary ideal of $A$ and since $u\notin\mathfrak{m}$ then $p\in\overline{K}$. Since $K$ is a homogeneous ideal in the polynomial ring $A$ then $\overline{K}$ is also homogeneous ([24, Prop. (f), pg 38]) and so each homogeneous component of $p$ is also in $\overline{K}\subset\overline{\mathcal{I}}$. This implies that $\overline{\mathcal{I}}$ is also generated by homogeneous polynomials.

Now let $\mathcal{I}\subset R$ be an arbitrary ideal generated by homogeneous polynomials, and let $s$ be in the integral closure $\overline{\mathcal{I}}$ as before. Let $s_{0}$ be the initial form of $s$, $s_{0}$ is the non-zero homogeneous component of $s$ of lowest degree. For all $j\in\mathbb{N}$, $s$ is in the integral closure of $\mathcal{I}+\mathfrak{m}^{j}$. We proved in the previous paragraph that the integral closure $\overline{\mathcal{I}+\mathfrak{m}^{j}}$ of $\mathcal{I}+\mathfrak{m}^{j}$ in $R$ is generated by homogeneous polynomials for all $j\in\mathbb{N}$. In particular, $s_{0}$ is in the integral closure $\overline{\mathcal{I}+\mathfrak{m}^{j}}$ for all $j$. But by [16, Corollary 6.8.5] this implies that $s_{0}$ is in the integral closure $\overline{\mathcal{I}}$. Now we start over with $s^{\prime}=s-s_{0}\in\overline{\mathcal{I}}$ and in this way we get that all homogeneous components of $s$ are in $\overline{\mathcal{I}}$ which is what we wanted to prove. ∎

Let $\Gamma\subset\mathbb{Z}^{d}$ be the semigroup defining $(X,0)$. Recall that $\check{\sigma}=\mathbb{R}_{\geq 0}\Gamma$ is a strongly convex cone. This fact, together with the condition of smooth normalization allows us to assume, up to a change of coordinates, that $\Gamma\subset\mathbb{N}^{d}$ and $\check{\sigma}\cap\mathbb{Z}^{d}=\mathbb{N}^{d}$ (see lemma 2.3).

Let $\mathcal{A}=\{a_{1},\ldots,a_{n}\}\subset\mathbb{N}^{d}$ be the minimal generating set of $\Gamma$ defining the toric singularity $(X,0)$. The normalization map can be realized as the monomial morphism:

	$\displaystyle n:(\mathbb{C}^{d},0)$	$\displaystyle\longrightarrow(X,0)$
	$\displaystyle\left(u_{1},\ldots,u_{d}\right)$	$\displaystyle\mapsto\left(u^{a_{1}},\ldots,u^{a_{n}}\right).$

The ideal $I_{\Delta}$ of definition 1.1 is a homogeneous, binomial ideal in the ring of convergent power series $\mathbb{C}\{x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\}$:

$$I_{\Delta}=\left<x^{a_{1}}-y^{a_{1}},\ldots,x^{a_{n}}-y^{a_{n}}\right>.$$

For any point $\tau=(t_{1},\ldots,t_{d})\in\left(\mathbb{C}^{*}\right)^{d}$ we have an automorphism

$$\varphi_{\tau}:\mathbb{C}\{x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\}\circlearrowleft$$

defined by $x_{j}\mapsto t_{j}x_{j}$ and $y_{j}\mapsto t_{j}y_{j}$, such that $\varphi_{\tau}\left(I_{\Delta}\right)=I_{\Delta}$.

Now let $f\in\overline{\mathcal{O}_{X,0}}\cong\mathbb{C}\{u_{1},\ldots,u_{d}\}$ be a homogeneous polynomial such that $f\in\mathcal{O}_{X,0}^{s}$. Then $f(x)-f(y)\in\overline{I_{\Delta}}$ and it satisfies an integral dependence equation of the form:

$$\left(f(x)-f(y)\right)^{m}+h_{1}(x,y)\left(f(x)-f(y)\right)^{m-1}+\cdots+h_{m}(x,y)=0,$$

where $h_{j}(x,y)\in I_{\Delta}^{j}$. By applying the morphism $\varphi_{\tau}$ to the previous equation we get:

$$\left(f(\tau x)-f(\tau y)\right)^{m}+h_{1}(\tau x,\tau y)\left(f(\tau x)-f(\tau y)\right)^{m-1}+\cdots+h_{m}(\tau x,\tau y)=0,$$

where $h_{j}(\tau x,\tau y)\in I_{\Delta}^{j}$, and so we get that $g_{\tau}:=f(t_{1}u_{1},\ldots,t_{d}u_{d})\in\mathcal{O}_{X,0}^{s}$.

Since $f$ is a homogeneous polynomial, say of order $k$, then we can write it in the form

$$f(u_{1},\ldots,u_{d})=\sum_{\alpha_{1}+\cdots+\alpha_{d}=k}b_{\alpha}u^{\alpha},$$

where $b_{\alpha}\in\mathbb{C}$ and $\alpha=(\alpha_{1},\ldots,\alpha_{d})$. Then we have an expression for $g_{\tau}$ of the form

	$\displaystyle g_{\tau}=f(t_{1}u_{1},\ldots,t_{d}u_{d})$	$\displaystyle=\sum_{\alpha_{1}+\cdots+\alpha_{d}=k}\tau^{\alpha}b_{\alpha}u^{\alpha}$
		$\displaystyle=\left<\left(\tau^{\alpha^{1}},\ldots,\tau^{\alpha^{N}}\right),\left(b_{\alpha^{1}}u^{\alpha^{1}},\ldots,b_{\alpha^{N}}u^{\alpha^{N}}\right)\right>.$

By choosing $\tau_{1},\ldots,\tau_{N}$ generic points in $\left(\mathbb{C}^{*}\right)^{d}$ we get

$$\begin{pmatrix}g_{\tau_{1}}\\ \vdots\\ g_{\tau_{N}}\end{pmatrix}=\begin{pmatrix}\tau_{1}^{\alpha^{1}}&\ldots&\tau_{1}^{\alpha^{N}}\\ \vdots&\cdots&\vdots\\ \tau_{N}^{\alpha^{1}}&\ldots&\tau_{N}^{\alpha^{N}}\end{pmatrix}\begin{pmatrix}b_{\alpha^{1}}u^{\alpha^{1}}\\ \vdots\\ b_{\alpha^{N}}u^{\alpha^{N}}\end{pmatrix},$$

where the $i$-th row of the matrix corresponds to the image of the point $\tau_{i}\in\left(\mathbb{C}^{*}\right)^{d}$ of the Veronese map $\nu_{k}:\mathbb{P}^{d-1}\to\mathbb{P}^{N-1}$ of degree $k$. Since the image of the Veronese map is a nondegenerate projective variety, in the sense that it is not contained in any hyperplane, then these $N$ points are in general position. This implies that the matrix is invertible, and so we have

$$\begin{pmatrix}b_{\alpha^{1}}u^{\alpha^{1}}\\ \vdots\\ b_{\alpha^{N}}u^{\alpha^{N}}\end{pmatrix}=\begin{pmatrix}\tau_{1}^{\alpha^{1}}&\ldots&\tau_{1}^{\alpha^{N}}\\ \vdots&\cdots&\vdots\\ \tau_{N}^{\alpha^{1}}&\ldots&\tau_{N}^{\alpha^{N}}\end{pmatrix}^{-1}\begin{pmatrix}g_{\tau_{1}}\\ \vdots\\ g_{\tau_{N}}\end{pmatrix}$$

In particular we have that $b_{\alpha^{j}}u^{\alpha^{j}}\in\mathcal{O}_{X,0}^{s}$ which is what we wanted to prove. ∎

(of theorem 3.2 )
We want to prove that there exists a finitely generated semigroup $\Gamma^{s}\subset\mathbb{N}^{d}$ such that

$$\mathcal{O}_{X,0}^{s}\cong\mathbb{C}\{\Gamma^{s}\}.$$

As we mentioned before, in this setting the ideal $I_{\Delta}$ is a homogeneous, binomial ideal in the ring of convergent power series $\mathbb{C}\{x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\}$, and by proposition 3.1 the ideal $\overline{I_{\Delta}}$ is also generated by homogeneous polynomials. This means that if a series $f\in\mathbb{C}\{u_{1}\ldots,u_{d}\}$ is in $\mathcal{O}_{X,0}^{s}$,

$$f=f_{m}+f_{m+1}+\cdots+f_{N}+\cdots,$$

then every homogeneous component $f_{j}$ of $f$ is in $\mathcal{O}_{X,0}^{s}$. But by lemma 3.3 this implies that every monomial of degree $j$ with a non-zero coefficient in $f_{j}$ is also in $\mathcal{O}_{X,0}^{s}$. Let $\Gamma^{s}$ be the semigroup defined by

$$\Gamma^{s}=\{\alpha\in\mathbb{N}^{d}\,|\,u^{\alpha}\in\mathcal{O}_{X,0}^{s}\}.$$

We have that

$$\mathcal{O}_{X,0}^{s}\subset\mathbb{C}\{\Gamma^{s}\}\subset\mathbb{C}\{u_{1},\ldots,u_{d}\}.$$

We have to prove that $\Gamma^{s}$ is finitely generated and the equality $\mathcal{O}_{X,0}^{s}=\mathbb{C}\{\Gamma^{s}\}$.

To begin with the latter, take $g\in\mathbb{C}\{\Gamma^{s}\}$. For every $k\geq 0$, write $g=g_{\leq k}+\widetilde{g_{k}}$ where $g_{\leq k}$ is the truncation of $g$ to degree $k$, and since it is a finite sum of monomials of $\mathcal{O}_{X,0}^{s}$, by definition of $\Gamma^{s}$, we have that $g_{\leq k}\in\mathcal{O}_{X,0}^{s}$. This means that

$$g_{\leq k}(x)-g_{\leq k}(y)\in\overline{I_{\Delta}}$$

and since $\widetilde{g_{k}}(x)-\widetilde{g_{k}}(y)\in\mathfrak{m}^{k+1}\mathbb{C}\{x,y\}$ we have that

$$g(x)-g(y)\in\overline{I_{\Delta}}+\mathfrak{m}^{k+1}\subset\overline{I_{\Delta}+\mathfrak{m}^{k+1}}.$$

In particular,

$$g(x)-g(y)\in\bigcap_{k\in\mathbb{N}}\overline{I_{\Delta}+\mathfrak{m}^{k+1}}=\overline{I_{\Delta}}\textrm{ by \cite[cite]{[\@@bibref{}{HS06}{}{}, Corollary 6.8.5]}}$$

then $g\in\mathcal{O}_{X,0}^{s}$ and we have the equality we wanted.

Now we prove that $\Gamma^{s}$ is a finitely generated semigroup. Let $\leq$ be a monomial order in $\mathbb{C}[u_{1},\ldots,u_{d}]$. We order the elements of $\Gamma^{s}$ and write them as $\{\beta^{j}\,|\,j\in\mathbb{N}\}$, where $0=\beta^{0}<\beta^{1}<\beta^{2}<\cdots$. Consider the following ascending chain of ideals in $\mathcal{O}_{X,0}^{s}$:

$$\langle u^{\beta^{1}}\rangle\subset\langle u^{\beta^{1}},u^{\beta^{2}}\rangle\subset\langle u^{\beta^{1}},u^{\beta^{2}},u^{\beta^{3}}\rangle\subset\cdots$$

Since $\mathcal{O}_{X,0}^{s}$ is Noetherian, we have $u^{\beta^{j}}\in\langle u^{\beta^{1}},\ldots,u^{\beta^{m}}\rangle$, for some $m\in\mathbb{N}$ and for all $j\in\mathbb{N}$. We claim that $\Gamma^{s}=\mathbb{N}(\beta^{1},\ldots,\beta^{m})$.