# Groebner fan and embedded resolutions of ideals on toric varieties

F. Aroca, M. Gómez-Morales, H. Mourtada

(February 2022)

###### Abstract

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal
on a toric variety, extending the two existing notions for ideals on affine
spaces. We prove, without assumptions on the characteristic of the base
fields, that the ”Groebner fan” of such an ideal is actually a polyhedral fan
and that a sub-variety defined by a Newton non-degenerate ideal on a toric
variety $X_{\sigma}$ admits a toric embedded resolution of singularities
$Z\longrightarrow X_{\sigma}.$

Keywords– Embedded resolution of singularities, Newton non-degenerate ideals,
Groebner fan, Toric varieties, Tropical Geometry.

2020 Math. Sub. Class– 14B05, 14M25, 13F65, 14T90.

##  1 Introduction

Resolution of singularities is a fundamental tool in Algebraic Geometry, which
also has applications in several other domains like Differential Geometry,
Analysis and Number Theory. While its existence when working over fields of
characteristic zero was proved by Hironaka in 1968 [Hi], in positive
characteristics this (and its super local couterpart: local uniformization)
remains a widely open problem, e.g. [CP, CP2, BV, Surf, KFV, T1, Cudim3, CuM,
NoS, Sa].

In characteristic zero, a huge amount of work has been done to simplify
Hironaka’s proof and to make it functorial [V, BM1, EH]. The majority of this
work follows one philosophy which is about making ”very” small improvements to
the singularity by blowing up smooth centers. See also [Mc, ATW] for
resolution of singularities using a sequence of weighted blowing ups.

In this article, we are concerned with toric resolutions of singularities
which, when they exist, are obtained with one toric morphism. Such resolutions
are then very useful in applications and allow to effectively compute many
important invariants of singularities, e.g. [Va, DL2, AGL, LMS, RDP, MP,
CPPT]. For hypersufaces in affine spaces, the so called condition of being
Newton non-degenerate, introduced in the seventies by Khovanski, Kuchnirenko
and Varchenko [Kou, Va], guaranties the existence of toric resolution of
singularities. Recently, this was generalized to higher codimensions in [Te1,
AGS] via the equivalent notions (but expressed in different languages) of
Schon varieties and Newton non-degenerate ideals.

In this article, we extend this last notion to ideals on toric varieties and
prove that their associated varieties admit toric embedded resolutions of
singularities in a sense that will be explained below. Here, it is worth
noticing that given a sub-variety $Y$ of a normal toric variety $X_{\sigma},$
in general there exists no toric embedded resolution $Z\longrightarrow
X_{\sigma}$ of $Y\subset X_{\sigma};$ this can already be noticed for curves
embedded in the plane [GT, dFGPM, LMR, gen]. But there is a conjecture of
Teissier that a toric resolution exists after embedding in a higher
dimensional affine space [T1, gen, hdr, GP1]. We insist here that in the
context of this article the embedding is fixed. We now explain the results of
this article; the reader who is unfamiliar with toric geometry may find a
reminder on this subject in section 2.

Since the problem is of local nature, we will be considering affine normal
toric varieties. Let K be a perfect field. Let $N$ be a free abelian group of
rank $n,$ i.e. $N\simeq\textbf{Z}^{n}.$ Let $M$ be the dual lattice of $N,$
$N=\mathrm{Hom}_{\textbf{Z}}(M,\textbf{Z})$ and let $\sigma\subset
N_{\textbf{R}}=N\otimes_{\textbf{Z}}\textbf{R}$ be a strongly convex rational
polyhedral cone (i.e $,\sigma\cap-\sigma=\\{0\\}\$).$ Let
$\sigma^{\vee}\subset M_{\textbf{R}}=M\otimes_{\textbf{Z}}\textbf{R}$ be the
dual cone of $\sigma;$ then the normal toric variety associated with $\sigma$
is by definition $X_{\sigma}=\textrm{Spec}\,{\mathscr{R}_{\sigma}}$ where
$\mathscr{R}_{\sigma}=\textbf{K}[\sigma^{\vee}\\!\cap M];$ note that by
definition of $X_{\sigma},$ the torus
$\textbf{T}=\textrm{Spec}\,{\textbf{K}[M]}$ is dense in $X_{\sigma}.$

Let $f\in\mathscr{R}_{\sigma};$ hence we can write

$f=\sum_{\alpha\in\sigma^{\vee}\\!\cap M}a_{\alpha}\textbf{x}^{\alpha},$

where $\textbf{x}^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$. Since
$\sigma$ (and hence $\sigma^{\vee}$) is strongly convex, the expression of $f$
in this form is unique. Note that the support of $f,$
$\mathrm{Supp}(f):=\\{\alpha\in\sigma^{\vee}\\!\cap M;a_{\alpha}\not=0\\}$ is
a finite set. We define the valuation associated with $v\in N_{\textbf{R}}$,
$\nu_{v}:\mathscr{R}_{\sigma}\longrightarrow\textbf{Z},$ by

$\nu_{v}(f)=\mathrm{min}\\{v\cdot\alpha\leavevmode\nobreak\
;\alpha\in\mathrm{Supp}(f)\\}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{for any}\leavevmode\nobreak\
\leavevmode\nobreak\ f\in\mathscr{R}_{\sigma}\setminus\\{0\\}.$

Here, $v\cdot\alpha$ stands for the action of $v\in N_{\textbf{R}}$ on
$\alpha\in M$ being defined by the scalar product; in which we set
$\nu_{v}(0):=\infty.$ The initial form of $f\not=0$ with respect to $v$ is
then defined by

$\mathrm{I}n_{v}(f)=\sum_{\\{\alpha\in\mathrm{Supp}(f);v\cdot\alpha=\nu_{v}(f)\\}}a_{\alpha}\textbf{x}^{\alpha},$

and the initial form of $0$ is defined as $0$. Both objects $\nu_{v}(f)$ and
$\mathrm{I}n_{v}(f)$ are well defined, thanks to the uniqueness of the
expression of $f$ in $\textbf{K}[\sigma^{\vee}\\!\cap M],$ as mentioned above.
Let $\mathscr{J}\subset R_{\sigma}$ be an ideal. For $v\in\sigma,$ we define
the initial ideal $\mathrm{I}n_{v}(\mathscr{J})$ of $\mathscr{J}$ with respect
to $v$ as follows:

$\mathrm{I}n_{v}(\mathscr{J})=\\{\mathrm{I}n_{v}(f);f\in\mathscr{J}\\}.$

Notice that $\mathrm{I}n_{v}(\mathscr{J})$ is an ideal. The ideal
$\mathscr{J}$ defines an equivalence relation $\sim$ on the cone $\sigma$ as
follows: let $v_{1},v_{2}\in\sigma;$ then we set

$v_{1}\sim v_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{if}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{I}n_{v_{1}}(\mathscr{J})=\mathrm{I}n_{v_{2}}(\mathscr{J}).$

We prove the following (see 3.5):

######  Theorem 1.1.

The closures (with respect to the topology of $\textbf{R}^{n}$) of the
equivalence classes of the relation $\sim$ defines a polyhedral fan which is a
subdivision of $\sigma.$

The fan in theorem 1.1 is called the Groebner fan; it generalizes the well
known Groebner fan of ideals in polynomial rings cite [MaRo, As, FJT, BaTa].
Using [FJT], we actually have an algorithm to compute this fan.

We will use this notion of Groebner fan to produce embedded resolutions of
singularities of sub-varieties of toric varieties defined by “non-degenerate
ideals” which are defined as follows:

######  Definition 1.2.

Let $\mathscr{J}\subset\mathscr{R}_{\sigma}$ be an ideal. We assume that the
closed orbit $O$ (which correspond to the cone $\sigma$) belongs to
$V(\mathscr{J}).$ We say that $\mathscr{J}$ is Newton non-degenerate at $O$
if, for every $v\in\sigma,$ we have

$\mathrm{Sing}(V(\mathrm{I}n_{v}(\mathscr{J}))\cap\textbf{T}=\emptyset;$

here $\mathrm{Sing}(V(\mathrm{I}n_{v}(\mathscr{J}))$ denotes the singular
locus of the sub-variety of $X_{\sigma}$ defined by the ideal
$\mathrm{I}n_{v}(\mathscr{J}).$

Another main result of this article is that a non-degenerate ideal admits a
toric embedded resolution in the following sense.

######  Theorem 1.3.

Let $\mathscr{J}\subset\mathscr{R}_{\sigma}$ be a Newton non-degenerate ideal
and let $Y=V(\mathscr{J})\subset X_{\sigma}.$ Let $\Sigma$ be a regular
subdivision of $\sigma$ which is compatible with the Groebner fan of
$\mathscr{J}.$ Then the associated toric morphism
$\pi_{\Sigma}:X_{\Sigma}\longrightarrow X_{\sigma}$ is a proper birational
morphism, in which irreducible components of the total transform
$\pi_{\Sigma}^{-1}(Y)$ are smooth (in particular, the strict transform of $Y$
is smooth) and meet transversely.

Note that the variety $X_{\Sigma}$ is smooth since it is associated with a
regular fan. The main result of [AGS] is the special case of theorem 1.3,
where $\sigma$ is considered to be $\textbf{R}_{+}^{n}.$ In the special case
where $Y$ is a hypersurface, a version of theorem 1.3 was proved in [St], see
also [NS]. The full generality of theorem 1.3 is used in [MS2] (see also
[MS1]).

######  Remark 1.4.

The embedded resolution of singularities in theorem 1.3 should be understood
as an embedded resolution of singularities of the pair $Y\subset X_{\sigma}$
in a small neighbourhood of $O.$

Acknowledgements. Hussein Mourtada would like to thank the UNAM, Mexico City,
for its hospitality during the preparation of this article.

##  2 The toric context

We refer to [Fu, Cox] for the basics on normal toric varieties. As in the
intro-duction K is a perfect field; $N$ is a free abelian group of rank $n,$
i.e. $N\simeq\textbf{Z}^{n};$ $M\simeq\textbf{Z}^{n}$ is the dual lattice of
$N,$ $N=\mathrm{Hom}_{\textbf{Z}}(M,\textbf{Z})$ and $\sigma\subset
N_{\textbf{R}}=N\otimes_{\textbf{Z}}\textbf{R}\simeq\textbf{R}^{n}$ is a
strongly convex rational polyhedral cone (of dimension $n$) which is generated
by $v^{1},\ldots,v^{r}$, i.e.,

$\sigma=\\{\lambda_{1}v^{1}+\cdots+\lambda_{r}v^{r};\lambda_{i}\in\textbf{R}_{\geq
0}\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\
\leavevmode\nobreak\ i=1,\ldots,r\\}.$

We denote by $\sigma^{\vee}\subset
M_{\textbf{R}}=M\otimes_{\textbf{Z}}\textbf{R}\simeq\textbf{R}^{n}$ the dual
cone of $\sigma;$ the normal toric variety associated with $\sigma$ is by
definition $X_{\sigma}=\textrm{Spec}\,{\mathscr{R}_{\sigma}}$ where
$\mathscr{R}_{\sigma}=\textbf{K}[\sigma^{\vee}\\!\cap M].$ Let
$\alpha^{1},\ldots,\alpha^{s}$ be a minimal set of generators of
$\sigma^{\vee}\\!\cap M$ as a semigroup. If we write
$\textbf{K}[M]=\textbf{K}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}],$ then we have

$\mathscr{R}_{\sigma}=\textbf{K}[\textbf{x}^{\alpha^{1}},\ldots,\textbf{x}^{\alpha^{s}}];$

here we use the notation
$\textbf{x}^{\alpha^{i}}:=x_{1}^{\alpha^{i}_{1}}\cdots
x_{n}^{\alpha^{i}_{n}}.$ This allows to embed
$X_{\sigma}\hookrightarrow\textbf{A}^{s}$ as follows: let

$\Psi:\textbf{K}[y_{1},\ldots,y_{s}]\longrightarrow\mathscr{R}_{\sigma}=\textbf{K}[\textbf{x}^{\alpha^{1}},\ldots,\textbf{x}^{\alpha^{s}}]$

be the $\textbf{K}-$algebra morphism defined by
$\Psi(y_{i})=\textbf{x}^{\alpha^{i}}.$ Let $\mathscr{M}$ be the $n\times s$
matrix whose columns are the $\alpha^{i}\,$’s, i.e.
$(\mathscr{M})_{k,l}=\alpha^{l}_{k}.$ For $\gamma\in\textbf{Z}_{\geq 0}^{s},$
we use the notation $\textbf{y}^{\gamma}:=y_{1}^{\gamma_{1}}\cdots
y_{s}^{\gamma_{s}};$ an element
$h\in\textbf{K}[\textbf{y}]:=\textbf{K}[y_{1},\ldots,y_{s}]$ is expressed as
follows:

$h(\textbf{y})=\sum_{\gamma\in\textbf{Z}_{\geq
0}^{s}}a_{\gamma}\textbf{y}^{\gamma}=\sum_{\gamma\in\textrm{Supp}\,(h)}a_{\gamma}\textbf{y}^{\gamma},$

where $a_{\gamma}\in\textbf{K}$ is nonzero for a finite number of $\gamma.$
The morphism $\Psi,$ being a K-algebra morphism, satisfies

$\Psi(h)=\sum_{\gamma\in\textrm{Supp}\,(h)}a_{\gamma}\textbf{x}^{\mathscr{M}\gamma}.$
(1)

Notice that the morphism $\Psi$ is surjective and hence the ideal
$I_{\sigma}:=\mathrm{Ker}(\Psi)$ is the defining ideal of $X_{\sigma}$ in
$\textbf{A}^{s}.$ The affine space $\textbf{A}^{s}$ has the structure of a
toric variety which is associated with the cone $\textbf{R}_{+}^{s}.$ We embed
$\sigma$ in $\textbf{R}_{+}^{s}$ via the following linear map:

$\phi:\textbf{R}^{n}\longrightarrow\textbf{R}^{s},$
$v\mapsto\phi(v)=(v\cdot\alpha^{1},\ldots,v\cdot\alpha^{s}).$

The linear map $\phi$ is injective; indeed, since the vector space
$\textbf{R}^{n}$ is generated by $\alpha^{1},\ldots,\alpha^{s}$ (the cone
$\sigma$ being of full dimension and hence so is $\sigma^{\vee}$), the kernel
of $\phi$ is reduced to $\\{0\\}.$ Note that since the $\alpha^{i}\,$’s are in
$\sigma^{\vee},$ we have

$\phi(\sigma)\subset\textbf{R}_{+}^{s}.$

For $\omega\in\textbf{R}_{+}^{s},h\in\textbf{K}[\textbf{y}]$ and
$J\subset\textbf{K}[\textbf{y}]$ an ideal, the objects
$\nu_{\omega}(h),\mathrm{I}n_{\omega}(h)$ and $\mathrm{I}n_{\omega}(J)$ are
definied in the same way as in the introduction; this is the special case
$\sigma=\textbf{R}_{+}^{s}.$

In the sequel, we will use the notation $v$ for a vector in a given $\sigma$
and $\omega$ for a vector in $\textbf{R}_{+}^{s};$ we will use the notation
$\mathscr{J}$ for an ideal in $\mathscr{R}_{\sigma}$ and $J$ for an ideal in
$\textbf{K}[\textbf{y}].$

##  3 Proofs of the main results

We keep the notations of the precedent sections. Let $v\in\sigma$ and set
$\omega=\phi(v).$ Let $\mathscr{J}$ be an ideal of $\mathscr{R}_{\sigma};$
then $J:=\Psi^{-1}(\mathscr{J})$ is an ideal of $\textbf{K}[\textbf{y}].$ In
order to determine a relation between $\mathrm{I}n_{v}(\mathscr{J})$ and
$\mathrm{I}n_{\omega}(J),$ we begin by comparing, for
$h\in\textbf{K}[\textbf{y}]$ and $f=\Psi(h);$ $\nu_{w}(h)$ with $\nu_{v}(f)$
and $\mathrm{I}n_{\omega}(h)$ with $In_{v}(f).$

######  Lemma 3.1.

Let $v\in\sigma$ and set $\omega=\phi(v).$ Let $h\in\textbf{K}[\textbf{y}]$
and $f=\Psi(h).$ We have

$\nu_{\omega}(h)\leq\nu_{v}(f).$

Moreover, the inequality is strict if and only if
$\Psi(\mathrm{I}n_{\omega}(h))=0.$

###### Proof.

For a monomial $\textbf{y}^{\gamma}=y_{1}^{\gamma_{1}}\cdots
y_{s}^{\gamma_{s}},$ notice that

$\displaystyle\nu_{\omega}(\textbf{y}^{\gamma})$
$\displaystyle=\omega\cdot\gamma=\sum_{i=1}^{s}\omega_{i}\gamma_{i}=\sum_{i=1}^{s}\left(\sum_{j=1}^{n}v_{j}\alpha^{i}_{j}\right)\gamma_{i}=\sum_{j=1}^{n}v_{j}\left(\sum_{i=1}^{s}\alpha^{i}_{j}\gamma_{i}\right)$
$\displaystyle=v\cdot(\mathscr{M}\gamma)=\nu_{v}(\textbf{x}^{\mathscr{M}\gamma})=\nu_{v}(\Psi(\textbf{y}^{\gamma})).$
(2)

If we write

$h=\sum_{\\{\gamma\in\textrm{Supp}\,(h);\leavevmode\nobreak\
\leavevmode\nobreak\
\omega\cdot\gamma=\nu_{\omega}(h)\\}}a_{\gamma}\textbf{y}^{\gamma}+\sum_{\\{\gamma\in\textrm{Supp}\,(h);\leavevmode\nobreak\
\leavevmode\nobreak\
\omega\cdot\gamma>\nu_{\omega}(h)\\}}a_{\gamma}\textbf{y}^{\gamma},$

it follows from the formulas (1) and (2) that

$f=\sum_{\\{\gamma\in\textrm{Supp}\,(h);\leavevmode\nobreak\
\leavevmode\nobreak\
v\cdot(\mathscr{M}\gamma)=\nu_{\omega}(h)\\}}a_{\gamma}\textbf{x}^{\mathscr{M}\gamma}+\sum_{\\{\gamma\in\textrm{Supp}\,(h);\leavevmode\nobreak\
\leavevmode\nobreak\
v\cdot(\mathscr{M}\gamma)>\nu_{\omega}(h)\\}}a_{\gamma}\textbf{x}^{\mathscr{M}\gamma}.$

In particular, we have that $\nu_{v}(f)=\nu_{w}(h)$ if

$\sum_{\\{\gamma\in\textrm{Supp}\,(h);v\cdot\mathscr{M}\gamma=\nu_{\omega}(h)\\}}a_{\gamma}\textbf{x}^{\mathscr{M}\gamma}=\Psi(\mathrm{I}n_{\omega}(h))\not=0,$
(3)

while the strict inequality $\nu_{v}(f)>\nu_{w}(h)$ holds if not; note that
the equality in (3) may happen because the linear morphism defined by
$\mathscr{M}$ is not injective in general. Moreover, if
$\nu_{v}(f)=\nu_{w}(h),$ we have

$\Psi(\mathrm{I}n_{\omega}(h))=\mathrm{I}n_{v}(f)$ (4)

∎

Recall that the local tropical variety [PPS] (see also [trop]) at the origin
($O$ of $\textbf{A}^{s}$) of an ideal $I\subset\textbf{K}[\textbf{y}]$
verifying $O\in V(I)$ is given by

$\textrm{Trop}\,(I):=\\{\omega\in\textbf{R}_{+}^{s}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{such that}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{I}n_{\omega}{I}\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{does
not contain monomials}\\}.$

We determine in the following proposition the local tropical variety of
$I_{\sigma}\subset\textbf{K}[\textbf{y}];$ a result which is also of
independent interest. We prove it here because it makes use of formula (2)
that we have just showed.

######  Proposition 3.2.

  1. 1.

Let $\omega\in\mbox{Trop(I)}.$ We have

$\mathrm{I}n_{\omega}(I_{\sigma})=I_{\sigma}.$

  2. 2.

We have

$\phi(\sigma)=\textrm{Trop}\,(I_{\sigma}).$

###### Proof.

  1. 1.

By the main result of [KVA], to determine $\mathrm{I}n_{\omega}(I_{\sigma}),$
it is sufficient to determine a Groebner basis of
$I_{\sigma}\subset\textbf{K}[\textbf{y}]$ with respect to a total monomial
order which refines the preorder (i.e. partial order) defined by $\omega.$
Now, it is direct to see that a reduced Groebner basis of a binomial ideal
(which is the case for $I_{\sigma}$) is a binomial ideal [ES]: this follows
from the fact that the $S-$polynomial of two binomials is again a binomial.
So, let $(h_{1},\ldots,h_{l})=I_{\sigma}$ be such a Groebner basis of
$I_{\sigma};$ then, the $h_{i}\,$’s are binomials and
$\mathrm{I}n_{\omega}(I_{\sigma})=\left(\mathrm{I}n_{\omega}(h_{1}),\ldots,\mathrm{I}n_{\omega}(h_{l})\right)$
by [KVA]. But since $\omega\in\mbox{Trop(I)},$ $\mathrm{I}n_{\omega}(h_{i})$
cannot be a monomial for $i=1,\ldots,l$; hence we have
$\mathrm{I}n_{\omega}(h_{i})=h_{i},$ for $i=1,\ldots,l,$ since this latter is
a binomial. We deduce that $\mathrm{I}n_{\omega}(I_{\sigma})=I_{\sigma}.$

  2. 2.

It follows from formula (2) that

$\phi(\sigma)\subset\mbox{Trop}(I_{\sigma}),$

where, recall that $I_{\sigma}=Ker(\Psi)$ is the defining ideal of
$X_{\sigma}$ in $\textbf{A}^{s}.$ Indeed, if $\omega=\phi(v)\in\phi(\sigma),$
since the $\alpha^{i}\,$’s are in $\sigma^{\vee},$ the components of $\omega$
are positive or zero. Moreover, a binomial
$\textbf{y}^{\gamma}-\textbf{y}^{\gamma^{\prime}}\in I_{\sigma}$ if and only
if $\Psi(\textbf{y}^{\gamma})=\Psi(\textbf{y}^{\gamma^{\prime}}).$ Hence, by
formula (2), we have

$\nu_{\omega}(\textbf{y}^{\gamma})=\nu_{v}(\Psi(\textbf{y}^{\gamma}))=\nu_{v}\left(\Psi(\textbf{y}^{\gamma^{\prime}})\right)=\nu_{\omega}(\textbf{y}^{\gamma^{\prime}});$

we deduce that $\omega\in\mbox{Trop}(I_{\sigma}).$ Now, by what we have just
said, $\omega$ belongs to $\mbox{Trop}(I_{\sigma})$ if
$\omega\in\textbf{R}_{+}^{s}$ and
$\omega\cdot\gamma=\omega\cdot\gamma^{\prime}$ for every $\gamma$ and
$\gamma^{\prime}$ satisfying $\mathscr{M}\gamma=\mathscr{M}\gamma^{\prime}.$
This implies that $\mbox{Trop}(I_{\sigma})$ is the intersection of the
orthogonal $(Ker(\mathscr{M}))^{\perp}$ of $Ker(\mathscr{M})$ with
$\textbf{R}_{+}^{s}.$ Since the cone $\sigma^{\vee}$ is of maximal dimension,
we know that the vector space generated by the $\alpha^{i}\,$’s is of
dimension $n;$ hence $\mathscr{M}$ is of rank $n,$ therefore
$Ker(\mathscr{M})$ is of dimension $s-n$ and $(Ker(\mathscr{M}))^{\perp}$ is
of dimension $n.$ So, because we have proved that
$\phi(\sigma)\subset\mbox{Trop}(I_{\sigma})\subset(Ker(\mathscr{M}))^{\perp},$
the vector space generated by $\phi(\sigma)$ and $(Ker(\mathscr{M}))^{\perp}$
are equal; The intersection of this vector space with $\textbf{R}_{+}^{s}$ is
$\phi(\sigma)\subset\mbox{Trop}(I_{\sigma}).$

∎

######  Proposition 3.3.

Let $v\in\sigma$ and set $\omega=\phi(v).$ Let
$f\in\mathscr{R}_{\sigma}\setminus\\{0\\}.$ We have that

$\mbox{Max}\\{\nu_{\omega}(h);\,h\in\Psi^{-1}(f)\\}$

is finite and for $h\in\Psi^{-1}(f)$ realizing this finite maximum we have

$\nu_{\omega}(h)=\nu_{v}(f)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Psi(\mathrm{I}n_{\omega}(h))=\mathrm{I}n_{v}(f).$

###### Proof.

By lemma (3.1), we have the fact that

$\mbox{Max}\\{\nu_{\omega}(h);\leavevmode\nobreak\ \leavevmode\nobreak\
h\in\Psi^{-1}(f)\\}\,\leq\;\nu_{v}(f);$

hence this maximum is finite because $\nu_{v}(f)<\infty,$ for $f$ nonzero. Let
$g\in\Psi^{-1}(f)$ be such that
$\nu_{\omega}(g)=\mbox{Max}\\{\nu_{\omega}(h);\leavevmode\nobreak\
\leavevmode\nobreak\ h\in\Psi^{-1}(f)\\}.$ Suppose (by contradiction) that
$\nu_{\omega}(g)<\nu_{v}(f);$ by lemma (3.1), this implies that
$\Psi(\mathrm{I}n_{v}(g))=0;$ hence $\Psi(g-\mathrm{I}n_{v}(g))=f.$ At the
same time we have

$\nu_{\omega}(g-\mathrm{I}n_{\omega}(g))>\nu_{\omega}(g);$

this contradicts the fact that
$\nu_{\omega}(g)=\mbox{Max}\\{\nu_{\omega}(h);\leavevmode\nobreak\
\leavevmode\nobreak\ h\in\Psi^{-1}(f)\\}.$ Hence we have
$\nu_{\omega}(g)=\nu_{v}(f);$ by (4) this gives
$\Psi(\mathrm{I}n_{\omega}(g))=\mathrm{I}n_{v}(f).$

∎

Let $\mathscr{J}$ be an ideal of $\mathscr{R}_{\sigma};$ then
$J:=\Psi^{-1}(\mathscr{J})$ is an ideal of $\textbf{K}[\textbf{y}].$ We are
now ready to compare $\mathrm{I}n_{v}(\mathscr{J})$ and
$\mathrm{I}n_{\omega}(J)$ and deduce the structure of the Groebner fan of
$\mathscr{J}$ from the Greobner fan of $J.$

######  Proposition 3.4.

Let $\mathscr{J}$ be an ideal of $\mathscr{R}_{\sigma}$ and let
$J:=\Psi^{-1}(\mathscr{J})$ be an ideal of $\textbf{K}[\textbf{y}].$ Let
$v\in\sigma$ and let $\omega=\phi(v).$ We have

$\mathrm{I}n_{v}(\mathscr{J})=\Psi(\mathrm{I}n_{\omega}(J)).$

###### Proof.

Let $F\in\mathrm{I}n_{v}(\mathscr{J}),F\not=0;$ there exists $f\in\mathscr{J}$
such that $F=\mathrm{I}n_{v}(f).$ Let $g\in\Psi^{-1}(f)$ be such that
$\nu_{\omega}(g)=\mbox{Max}\\{\nu_{\omega}(h);\leavevmode\nobreak\
\leavevmode\nobreak\ h\in\Psi^{-1}(f)\\}.$ By proposition 3.3, we have
$F=\Psi(\mathrm{I}n_{\omega}(g));$ hence $F\in\Psi(\mathrm{I}n_{\omega}(J))$
and we have the inclusion
$\mathrm{I}n_{v}(\mathscr{J})\subset\Psi(\mathrm{I}n_{\omega}(J)).$

Let $G\in\mathrm{I}n_{\omega}(J);$ then $G=\mathrm{I}n_{\omega}(g),g\in J.$ By
lemma 3.1, we have either $\Psi(G)=0\in\mathrm{I}n_{v}(\mathscr{J})$, or we
have $\nu_{\omega}(g)=\nu_{v}(\Psi(g));$ and thus by (4) this gives
$\Psi(G)=\mathrm{I}n_{v}(\Psi(g))\in\mathrm{I}n_{v}(\mathscr{J}).$ We deduce
the reverse inclusion and the proposition. ∎

One of the two main results of this paper is the following:

######  Theorem 3.5.

Let $v,v^{\prime}\in\sigma$ and let $\omega=\phi(v),\leavevmode\nobreak\
\omega^{\prime}=\phi(v^{\prime}).$ Let $\mathscr{J}\in\mathscr{R}_{\sigma}$
and let $J:=\Psi^{-1}(\mathscr{J}).$ We have

$\mathrm{I}n_{v}(\mathscr{J})=\mathrm{I}n_{v^{\prime}}(\mathscr{J})\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if and only if}\leavevmode\nobreak\
\leavevmode\nobreak\
\mathrm{I}n_{\omega}(J)=\mathrm{I}n_{\omega^{\prime}}(J).$ (5)

In particular the equivalence classes of the relation $\sim$ defines a
polyhedral fan which is a subdivision of $\sigma$ and which is obtained from
the Groebner fan of $J$ intersected with $\sigma.$

###### Proof.

The reverse implication of $(\ref{fan})$ is a direct consequence of
proposition 3.4. Let us prove the direct implication. For that, we suppose
that $\mathrm{I}n_{\omega}(J)\not=\mathrm{I}n_{\omega^{\prime}}(J).$ Since,
$I_{\sigma}=Ker(\Psi)\subset J$, we have by proposition 3.2 (parts 1 and 2)

$\mathrm{I}n_{\omega}(I_{\sigma})=I_{\sigma}\subset\mathrm{I}n_{\omega}(J)\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{I}n_{\omega^{\prime}}(I_{\sigma})=I_{\sigma}\subset\mathrm{I}n_{\omega^{\prime}}(J).$

So we can choose basis of the forms

$\mathrm{I}n_{\omega}(I_{\sigma})\subset\mathrm{I}n_{\omega}(J)=(G_{1},\ldots,G_{l},H_{1},\ldots,H_{k})$
$\mathrm{I}n_{\omega^{\prime}}(I_{\sigma})\subset\mathrm{I}n_{\omega^{\prime}}(J)=(G^{\prime}_{1},\ldots,G^{\prime}_{l^{\prime}},H_{1},\ldots,H_{k}),$

where $I_{\sigma}=(H_{1},\ldots,H_{k}).$

Now, since we assumed that
$\mathrm{I}n_{\omega}(J)\not=\mathrm{I}n_{\omega^{\prime}}(J),$ there exist an
$i$ such that $G_{i}\not\in\mathrm{I}n_{\omega^{\prime}}(J).$ We have
$\Psi(G_{i})\not\in\Psi(\mathrm{I}n_{\omega^{\prime}}(J)):$ indeed, otherwise
we will have $\Psi(G_{i}-G^{\prime})=0$ for some
$G^{\prime}\in\mathrm{I}n_{\omega^{\prime}}(J);$ this implies that
$G_{i}-G^{\prime}=H\in I_{\sigma}$ and hence
$G_{i}=G^{\prime}+H\in\mathrm{I}n_{\omega^{\prime}}(J),$ which is a
contradiction. We deduce, since
$\Psi(G_{i})\not\in\Psi(\mathrm{I}n_{\omega^{\prime}}(J))$, that
$\mathrm{I}n_{v}(\mathscr{J})\not=\mathrm{I}n_{v^{\prime}}(\mathscr{J}).$ So
we have proved that if
$\mathrm{I}n_{\omega}(J))\not=\mathrm{I}n_{\omega^{\prime}}(J))$ then
$\mathrm{I}n_{v}(\mathscr{J})\not=\mathrm{I}n_{v^{\prime}}(\mathscr{J});$ this
ends the proof.

∎

######  Remark 3.6.

We keep the notation of theorem 3.5. Using the main result of [FJT], we know
how to compute the Groebner fan of $J$ (i.e. there is an algorithm for this);
hence, applying theorem 3.5, we obtain an algorithm to compute the (toric)
Groebner fan of $\mathscr{J}.$ At the same time, in practice and for dimension
reasons, it should be much easier to compute directly the (toric) Groebner fan
of $\mathscr{J}$; indeed, in general $s$ is much greater than $n.$

We now are ready to prove theorem 1.3.

###### Proof of theorem 1.3..

For $\mathscr{J}\in\mathscr{R}_{\sigma}$ satisfying the Newton non-degeneracy
property, we will prove that
$J:=\Psi^{-1}(\mathscr{J})\subset\textbf{K}[\textbf{y}]$ is Newton non-
degenerate. For that, we need to prove that for any
$\omega\in\textbf{R}_{+}^{s},$ the singular locus of
$V(\mathrm{I}n_{\omega}(J))\subset V((y_{1}\cdots y_{s})).$ Thanks to
proposition 3.2, we only need to consider the case where
$\omega\in\phi(\sigma)$; otherwise, for
$\omega\not\in\phi(\sigma)=\textrm{Trop}\,(I_{\sigma}),$ since
$I_{\sigma}\subset J,$ we have that $\mathrm{I}n_{\omega}(J)$ contains
monomials and hence $V(\mathrm{I}n_{\omega}(J))\subset V((y_{1}\cdots
y_{s})),$ in particular its singular locus is. Now, assume that
$\omega\in\phi(\sigma);$ by proposition 3.4, we have that
$\Psi(\mathrm{I}n_{\omega}(J))=\mathrm{I}n_{v}(\mathscr{J});$ hence we have

$V(\mathrm{I}n_{v}(\mathscr{J}))=V(\mathrm{I}n_{\omega}(J))\subset
X_{\sigma}\subset\textbf{A}^{s}.$

We hence know, by the Newton non-degeneracy condition defined in 1.2 that the
singular locus is included in the complement of the torus
$(\textbf{K}^{*})^{n}$ in $X_{\sigma}.$ So we need to prove that the
complement of the torus $(\textbf{K}^{*})^{n}$ in $X_{\sigma}$ is sent by the
embedding $X_{\sigma}\subset\textbf{A}^{s}$ to the complement of the torus
$(\textbf{K}^{*})^{s}.$ Note that the embedding
$(\textbf{K}^{*})^{n}\hookrightarrow(\textbf{K}^{*})^{s}$ is given by

$(t_{1},\ldots,t_{n})\mapsto(\textbf{t}^{\alpha^{1}},\ldots,\textbf{t}^{\alpha^{s}}),$

where $\textbf{t}^{\alpha^{i}}=t_{1}^{\alpha^{i}_{1}}\cdots
t_{n}^{\alpha^{i}_{n}}.$ Now, we know (e.g. section 3.2 in [Cox]) that

$X_{\sigma}\setminus(\textbf{K}^{*})^{n}=\bigsqcup_{0\not=\tau\subseteq\sigma}\mathbb{O}_{\tau},$

where $\mathbb{O}_{\tau}$ is the orbit associated to $\tau;$ note that the
orbit $\mathbb{O}_{0}$ is equal to $(\textbf{K}^{*})^{n}.$ Then, by the
definition of $\mathbb{O}_{\tau}$ we know that if
$\alpha^{i}\not\in\tau^{\perp},$ any point $p$ which is in $\mathbb{O}_{\tau}$
will belong to $V(y_{i})\subset\textbf{A}^{s}.$ Hence, we deduce that $J$ is
Newton non-degenerate as an ideal in $\textbf{K}[\textbf{y}].$ By the main
theorem of [AGS], we know that any regular subdivision $\widetilde{\Sigma}$ of
the Groebner fan of $J$ induces a proper birational map
$Z_{\widetilde{\Sigma}}\longrightarrow\textbf{A}^{s}$ (where
$Z_{\widetilde{\Sigma}}$ is the toric variety associated with the fan
$\widetilde{\Sigma}$), which is an embedded resolution of
$V(Y)\subset\textbf{K}^{s}.$ Now, by theorem 3.5, the fan $\tilde{\Sigma}$
induces a regular subdivision of the (toric) Greobner fan of $\mathscr{J}$
(actually, one can go the other way and construct $\tilde{\Sigma}$ from a
regular subdivision of the -toric- Groebner fan of $\mathscr{J}).$ The
morphism $Z_{\widetilde{\Sigma}}\longrightarrow\textbf{A}^{s}$ is an embedded
resolution of $X_{\sigma}\subset\textbf{A}^{s}$ (see [GPT]) and the strict
transform of $X_{\sigma}$ is $X_{\Sigma}.$ The strict transform of
$V(\mathscr{J})\subset X_{\sigma}$ in $X_{\Sigma}$ is the same as the strict
transform $Y^{\prime}$ of $V(J)\subset\textbf{A}^{s}$ in
$Z_{\widetilde{\Sigma}}.$ In particular, this strict transform is smooth, and
since it is transverse to the irreducible components of the exceptional locus
of $Z_{\widetilde{\Sigma}}\longrightarrow\textbf{A}^{s}$, we deduce that
$Y^{\prime}$ is transverse to the exceptional locus of
$X_{\Sigma}\longrightarrow X_{\sigma}$, which is nothing but the transverse
intersection of the exceptional locus of $Z_{\widetilde{\Sigma}}$ with
$X_{\Sigma}$ and the theorem follows.

∎

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[Fuensanta Aroca] Universidad Nacional Autónoma de México, Unidad Cuernavaca,
A.P. 273-3, C.P. 62251, Cuernavaca, MOR, Mexico.
<EMAIL_ADDRESS>

[Mirna Gómez-Morales] Universidad Nacional Autónoma de México, Unidad
Cuernavaca, A.P. 273-3, C.P. 62251, Cuernavaca, MOR, Mexico.
<EMAIL_ADDRESS>

[Hussein Mourtada] Université de Paris, Sorbonne Université, CNRS, Institut de
Mathématiques de Jussieu- Paris Rive Gauche, F-75013 Paris, France.
<EMAIL_ADDRESS>