# Differentially fixed ideals in toric varieties

Lance Edward Miller, William D. Taylor, Janet Vassilev

###### Abstract.

This article concerns monomial ideals fixed by differential operators of
affine semi-group rings over $\mathbb{C}$. We give a complete characterization
of when this happens. Perhaps surprisingly, every monomial ideal is fixed by
an infinite set of homogeneous differential operators and is in fact
determined by them. This opens up a new tool for studying monomial ideals. We
explore applications of this to (mixed) multiplier ideals and other variants
as well as give examples of detecting ideal membership in integrally closed
powers and symbolic powers of squarefree monomial ideals.

##  1\. Introduction

In this article, we give a complete answer to the following question. What
proper monomial ideals in a complex normal affine semigroup ring are fixed by
a fixed homogeneous differential operator? That is to say, when does
$\delta(I)=I$ for $\delta$ a homogeneous differential operator and $I$ a
monomial ideal. This is inspired by a rough, but important, analogy between
$p^{-e}$-linear maps which can be thought of as potential Frobenius splittings
for rings of positive characteristic $p>0$ and differential operators which of
course can be considered in any characteristic, see for example [BJNB19]. We
note the main theorem of [HSZ14] characterizes all ideals fixed by a fixed
$p^{-e}$-linear map in the toric setting. In that case, there are finitely
such fixed ideals [BB09]. In contrast, we show every proper monomial ideal $I$
has infinitely many homogeneous differential operators fixing it. Moreover,
the ideal $I$ is completely determined by the differential operators fixing
it. Our results are also distinct from but in similar spirit to [Ciu20] which
where similar questions were addressed concerning ideals invariant under a
fixed derivation.

Our approach is based on the Saito-Travis description of differential
operators for such semigroup rings, [ST01] and the structures subsequently
used in [BCK+21a, BCK+21b]. This gives an explicit description of homogeneous
differential operators $\delta$ of multidegree $\mathbf{d}\in\mathbb{Z}^{d}$
on a semigroup ring $S$ defined by a cone of dimension $d$. Specifically, for
each $\mathbf{d}$, a homogeneous differential operator $\delta$ of degree
$\mathbf{d}$ has the form $\delta=x^{\mathbf{d}}f$ where $x^{\mathbf{d}}$ is a
monomial in $d$ variables and $f$ is a polynomial differential operator of a
prescribed form.

Before stating the main theorem, we establish a bit of notation. For such an
operator, a key role in the characterization of fixed ideals is played by the
monomial vanishing locus. Specifically, for $\delta=x^{\mathbf{d}}f$, as $f$
is a polynomial differential operator one can look at the vanishing locus of
the polynomial defining it. The set of lattice points on which $f$ vanishes is
the set $V_{\textnormal{mon}}(f)=\\{\mathbf{a}\in\mathbb{Z}^{d}\colon
f(\mathbf{a})=\mathbf{0}\\}$. The geometric information in
$V_{\textnormal{mon}}(f)$ that plays an essential role are the points in
$V_{\textnormal{mon}}(f)$ which are furthest along rays in the direction of
$-\mathbf{d}$. To obtain these points, set
$\nu(\mathbf{a}):=\inf\\{t\in\mathbb{R}\colon\mathbf{a}+t\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}$. For $\delta$ to fix any ideals, we show it is
necessary that $\nu(\mathbf{a})>-\infty$ for all $\mathbf{a}\in S$. When this
condition is satisfied, we pick out the furthest points along these rays via
the quantity
$\nu^{\prime}(\mathbf{a}):=\max\\{t\in[\nu(\mathbf{a}),\nu(\mathbf{a})+1)\colon\mathbf{a}+t\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}$ and consider the subset
$V_{\textnormal{mon}}^{\prime}(f):=\\{\mathbf{a}+\nu^{\prime}(\mathbf{a})\mathbf{d}\colon\mathbf{a}\in\mathbb{Z}^{d},\nu(\mathbf{a})>-\infty\\}\subset
V_{\textnormal{mon}}(f).$ Utilizing this, we give a complete description of
all ideals in $R$ are $\delta$-fixed for some homogeneous $\delta$, and what
ideals are fixed for a given $\delta$ in terms of the exponent set
$\operatorname{Exp}I=\\{\mathbf{a}\colon x^{\mathbf{a}}\in I\\}$. We state
these characterization as follows.

######  Theorem (cf. Theorem 3.6 and Corollary 4.7).

Let $R=\mathbb{C}[S]$ be an affine semigroup ring defined by a cone
$\sigma^{\vee}$, $I\subseteq R$ a proper monomial ideal, and
$\mathbf{d}\in\mathbb{Z}^{d}$ such that $-\mathbf{d}\in S$. Set
$\mathbf{e}\in\mathbb{Z}^{d}$ primitive such that $\mathbf{d}=q\mathbf{e}$ for
some $q\in\mathbb{N}$.

  * •

There exists a homogeneous differential operator $\delta$ of degree
$\mathbf{d}$ such that $I$ is $\delta$-fixed.

  * •

If $-\mathbf{d}\in\operatorname{int}\sigma^{\vee}$, then $\delta$ fixes a
monomial ideal $I$ if and only if

    1. (1)

$\nu(\mathbf{a})>-\infty$ for all $\mathbf{a}\in S$,

    2. (2)

for all $\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)$ and $i=0,\ldots,q-1$,
$\mathbf{a}-i\mathbf{e}\in V_{\textnormal{mon}}(f)$, and

    3. (3)

for every $\mathbf{a},\mathbf{b}\in V_{\textnormal{mon}}^{\prime}(f)$,
$\mathbf{a}-\mathbf{b}\notin S-\mathbf{e}$.

In this case, $\delta$ fixes only one ideal, namely the ideal $I$ with
$\operatorname{Exp}I=\\{\mathbf{a}\in S\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$.

It is perhaps surprising but this characterization shows that every monomial
ideal $I$ is completely determined by the non-empty set of differential
operators $D_{I}:=\\{\delta\colon\delta(I)=I\\}$. This opens up a new line of
study for monomial ideals.

As an application, we consider a flexible but familiar setting where ideals
are defined as the intersection of special polyhedral regions $P$. These
regions simple from the point of view of combinatorial geometry. Specifically,
any polyhedral region which is $\sigma^{\vee}$-closed in that they are closed
under additive translation by $\sigma^{\vee}$ defines naturally an ideal and
the usual correspondence between monomial ideals and their exponent sets shows
all ideals are defined by such regions. We quite broadly work out explicit
differential operators fixing such an ideal from the face data, i.e.,
hypersurfaces $\\{g_{\tau}\\}$ defining $P$ together with integers $M_{\tau}$
which detect membership in $P$. This allows us to freely compute differential
operations of various ideal theoretic constructions from their associated
combinatioral geometric operations on polyhedra, a technique which is well-
worn and pervades work on monomial ideals. In its most general form, the
primary application is the following theorem.

######  Theorem.

(cf. Theorem 5.3) Let $P_{1},\ldots,P_{n}$ be $\sigma^{\vee}$-closed polyhedra
with face data $\\{(g_{\tau},M_{\tau})\\}_{\tau\in\mathcal{S}_{i}}$ for
$i=1,\ldots,n$ and $\mathbf{0}\neq\mathbf{d}\in-S$.

  1. (1)

Let $I=\langle x^{\mathbf{a}}\mid\mathbf{a}\in\bigcup P_{i}\rangle$ and
$f=\sum_{i=1}^{n}\prod_{\tau\in\mathcal{S}_{i}}(g_{\tau}(\theta),\lceil
M_{\tau}-g_{\tau}(\mathbf{d})\rceil-1)!$. For any
$h\in(H_{\mathbf{d}})\cap(f)$, $\delta=x^{\mathbf{d}}h(\theta)$ fixes $I$, and
$x^{\mathbf{a}}\in I$ if and only if $\delta(x^{\mathbf{a}-\mathbf{d}})\neq
0$.

  2. (2)

Let $J=\langle
x^{\mathbf{a}}\mid\mathbf{a}\in\bigcup\operatorname{int}P_{i}\rangle$ and
$f=\sum_{i=1}^{n}\prod_{\tau\in\mathcal{S}_{i}}(g_{\tau}(\theta),\lfloor
M_{\tau}-g_{\tau}(\mathbf{d})\rfloor)!$. For any
$h\in(H_{\mathbf{d}})\cap(f)$, $\delta=x^{\mathbf{d}}h(\theta)$ fixes $J$, and
$x^{\mathbf{a}}\in J$ if and only if $\delta(x^{\mathbf{a}-\mathbf{d}})\neq
0$.

We draw out explicit consequences of this as many important constructions in
algebraic geometry and commutative algebra considered on monomial ideals come
down to essentially lattice point membership in such a polyhedral region.
Examples of this include the Blickle-Howald description of multiplier ideals.
In particular, we characterize all jumping numbers using the differential
operators fixing an a multiplier ideal.

######  Corollary.

(cf. Corollary 5.18) Let $c\geq 0$ a real number and $I$ a monomial ideal of
an affine semigroup ring $R$. Set $\mathbf{w}=(1,1,\ldots,1)$,
$X=\operatorname{Spec}R$, and $J(X,I^{c})=\langle
x^{\mathbf{a}}\mid\mathbf{a}+\mathbf{w}\in
c\cdot\operatorname{int}\operatorname{Newt}I\rangle$ the multiplier ideal of
the pair $(X,I^{c})$. Set $(g_{\tau},M_{\tau})_{\tau\in\mathcal{F}}$ a face
data for $\operatorname{Newt}I$.

  1. (1)

The differential operator
$\delta_{c}=x^{-\mathbf{w}}H_{-\mathbf{w}}(\theta)\prod_{\tau\in\mathcal{F}}(g_{\tau}(\theta),\lfloor
cM_{\tau}\rfloor)!$ fixes $J_{c}$.

  2. (2)

The jumping numbers of $I$ are exactly
$\left\\{\min_{\tau\in\mathcal{F}}\left\\{\frac{1}{M_{\tau}}g_{\tau}(\mathbf{a}+\mathbf{w})\right\\}\mid\mathbf{a}\in
S\right\\}$.

  3. (3)

The log canonical threshold of $I$ is
$\min_{\tau\in\mathcal{F}}\left\\{\frac{1}{M_{\tau}}g_{\tau}(\mathbf{w})\right\\}$.

We provide similar descriptions for mixed multiplier ideals, as well as ideal
membership problems on integral closures of powers and symbolic powers in
Section 5. All of these are specializations of the singular combinatorial
geometric principal in last theorem but we work out samples of specific
applications as well as the explicit differential operators at play.

Acknowledgements: We thank Mark Johnson and Paolo Mantero for supportive
conversations and discussions on the material.

##  2\. Preliminaries and notation

Throughout, $x$ will denote a list of variables $x_{1},\ldots,x_{d}$ for
$d\geq 1$. Thus we denote by $\mathbb{C}[x]:=\mathbb{C}[x_{1},\ldots,x_{d}]$
for the polynomial ring with variables $x$. For a vector
$\mathbf{v}\in\mathbb{Z}^{d}$, $\mathbf{v}=(v_{1},\ldots,v_{d})$ we denote by
$x^{\mathbf{v}}$ the monomial $x_{1}^{v_{1}}x_{2}^{v_{2}}\cdots
x_{d}^{v_{d}}$.

We also fix $\sigma=\operatorname{cone}(\mathbf{v}_{1},\ldots,\mathbf{v}_{n})$
a rational polyhedral cone for
$\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\in\mathbb{Z}^{d}$ and set
$R=\mathbb{C}[S]$ where $S=\sigma^{\vee}\cap\mathbb{Z}^{d}$ which identifies
$R=\mathbb{C}[x^{\mathbf{v}_{1}},\ldots,x^{\mathbf{v}_{n}}]$ as a normal
affine semigroup ring. Unless otherwise stated $I\subset R$ is a proper
monomial ideal. We denote the exponent set by
$\operatorname{Exp}I:=\\{\mathbf{a}\in S\colon x^{\mathbf{a}}\in I\\}$ and by
$\operatorname{int}S$ the interior of $S$.

We denote by $\mathcal{D}$ the differential operators of $R$, for which we
utilize the following description. Setting
$\theta_{x_{i}}=x_{i}\partial_{x_{i}}$ and
$\mathbb{C}\langle\theta\rangle:=\mathbb{C}\langle\theta_{x_{1}},\ldots,\theta_{x_{d}}\rangle$,
one may view $\mathcal{D}$ as a subring of $\mathbb{C}[x^{\pm
1}]\langle\theta\rangle$. We also utilize the descending factorial notation
introduced in [BCK+21b, BCK+21a], namely for $g\in\mathbb{C}[x]$,

$(g,n)!=\prod_{i=0}^{n}(g-i)=\begin{cases}g(g-1)(g-2)\cdots(g-n)&\text{if
}n\geq 0\\\ 1&\text{if }n<0.\end{cases}.$

Throughout, for $f\in\mathbb{C}[x]$ we abuse notation, without comment, by
substituting either elements of the lattice $\mathbb{Z}^{d}$ or the tuple of
variables $\theta$. Specifically, for $\mathbf{a}\in\mathbb{Z}^{d}$,
$\mathbf{a}=(a_{1},\ldots,a_{d})$, we denote by
$f(\mathbf{a}):=f(a_{1},\ldots,a_{d})$. Similarly, we can evaluate such a
function on $\theta$ to obtain a differential operator
$f(\theta)=f(\theta_{x_{1}},\ldots,\theta_{x_{d}})$. Note that there is
potential confusion in this abuse, if $\delta$ for example is a differential
operator, then we also denote by $\delta(x^{\mathbf{a}})$ for its action on a
monomial. However, often we consider differential operators
$\delta=x^{\mathbf{d}}f$, so we are forced to consider $f$ both as its
polynomial $\mathbb{C}[x]$ and as the polynomial differential operator
$f(\theta)\in\mathbb{C}\langle\theta\rangle$. For example, the notation
$(f,n)!$ is consistent if we view $f$ as a polynomial or as a differential
operator.

We also have need to consider the vanishing locus of a differential operator
$f$ defined as follows.

######  Definition 2.1.

For $f\in\mathbb{C}[x]$, its monomial vanishing locus is the set

$V_{\textnormal{mon}}(f):=\\{\mathbf{a}\in\mathbb{Z}^{d}\colon
f(\mathbf{a})=\mathbf{0}\\}\subset\mathbb{Z}^{d}.$

The facets $\tau_{1},\ldots,\tau_{m}$ of $\sigma^{\vee}$ determine unique
homogeneous linear forms $h_{i}\in\mathbb{C}[x]$ via their primitive support
functions. For $\mathbf{d}\in\mathbb{Z}^{d}$, the polynomials
$H_{\mathbf{d}}$, where

$H_{\mathbf{d}}=\prod_{i}(h_{i},h_{i}(-\mathbf{d})-1)!,$

play an important role via the following theorem of Saito and Travis, [ST01,
Thm. 3.2.2].

######  Theorem 2.2.

[Saito-Traves] There is a graded decomposition

$\mathcal{D}(R)=\bigoplus_{\mathbf{d}\in\mathbb{Z}^{d}}x^{\mathbf{d}}(H_{\mathbf{d}}(\theta)).$

Where the homogeneous elements are of the form
$\delta=x^{\mathbf{d}}f(\theta)$, where $f\in\mathbb{C}[x]$, $H_{\mathbf{d}}$
divides $f$, and for any monomial $x^{a}\in R$, we have that
$\delta(x^{\mathbf{a}})=f(\mathbf{a})x^{\mathbf{a}+\mathbf{d}}$.

######  Example 2.3.

We take a moment to explore in an explicit example to clarify the introduced
notation. Suppose $S$ is the semigroup generated by $x,xy,xy^{2}$ and
$R=\mathbb{C}[S]\subset\mathbb{C}[x,y]$ so $\mathcal{D}$ is a graded subring
of $\mathbb{C}[x^{\pm 1},y^{\pm 1}]\langle\theta_{x},\theta_{y}\rangle$. The
primitive support functions are $h_{1}=2x-y$ and $h_{2}=y$. This yields
$h_{1}(\theta)=2\theta_{x}-\theta_{y}$ and $h_{2}(\theta)=\theta_{y}$. For
each $\mathbf{d}\in\mathbb{Z}^{2}$, we have a differential operator arising
from $H_{\mathbf{d}}$. For $\mathbf{d}=(-2,-1)$,
$h_{1}(-\mathbf{d})=h_{1}(2,1)=4-1=3$ and $h_{2}(-\mathbf{d})=h_{2}(2,1)=1$.
Therefore,

$\displaystyle H_{\mathbf{d}}(\theta)$ $\displaystyle=$
$\displaystyle\prod_{i}(h_{i}(\theta),h_{i}(-(-2,-1))-1)!$ $\displaystyle=$
$\displaystyle(h_{1}(\theta),2)!(h_{2}(\theta),0)!$ $\displaystyle=$
$\displaystyle(2\theta_{x}-\theta_{y})(2\theta_{x}-\theta_{y}-1)(2\theta_{x}-\theta_{y}-2)(\theta_{y}).$

It is also helpful to see the action of $h_{1}(\theta),h_{2}(\theta)$, and
$H_{\mathbf{d}}(\theta)$ on monomials. For example

$h_{1}(\theta)(x^{a}y^{b})=(2\theta_{x}-\theta_{y})(x^{a}y^{b})=2ax^{a}y^{b}-bx^{a}y^{b}=(2a-b)x^{a}y^{b}=h_{1}(a,b)x^{a}y^{b}.$

Similarly, one may trivially verify

$\displaystyle H_{\mathbf{d}}(\theta)(x^{a}y^{b})$ $\displaystyle=$
$\displaystyle(h_{1},2)!(a,b)(h_{2},0)!(a,b)x^{a}y^{b}$ $\displaystyle=$
$\displaystyle(2a-b)(2a-b-1)(2a-b-2)bx^{a}y^{b}$

Notation: Throughout the paper, unless otherwise stated $R=\mathbb{C}[S]$ is a
normal affine semigroup ring with $S=\sigma^{\vee}\cap\mathbb{Z}^{d}$ for
rational polyhedral cone $\sigma$ and denote by $h_{i}$ the primitive support
functions of $\sigma^{\vee}$. We always denote by $I$ a proper monomial ideal
of $R$ and $\mathbf{d}\in\mathbb{Z}^{d}$. We set $\mathcal{D}:=\mathcal{D}(R)$
and $\delta=x^{\mathbf{d}}f(\theta)$ a homogeneous differential operator of
degree $\mathbf{d}$ with $f\in(H_{\mathbf{d}})\subseteq\mathbb{C}[x]$.

##  3\. Every monomial ideal is $\delta$-fixed

Throughout, the aim of the paper is to study monomial ideals and what
differential operators fix them in the following sense.

######  Definition 3.1.

For $\delta\in\mathcal{D}$, an ideal $I\subset R$ is called $\delta$-fixed if
$\delta(I)=I$ and $\delta$-compatible if $\delta(I)\subset I$.

We characterize $\delta$-compatible ideals for homogeneous $\delta$ in terms
of the exponent set.

######  Theorem 3.2.

The ideal $I$ is $\delta$-compatible if and only if $f(\mathbf{a})=0$ for all

$\mathbf{a}\in\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d}).$

###### Proof.

For any $\mathbf{a}\in\operatorname{Exp}I$, we have that
$\delta(x^{\mathbf{a}})=f(\mathbf{a})x^{\mathbf{d}+\mathbf{a}}$. Therefore,
$\delta(x^{\mathbf{a}})\in I$ if and only if
$\mathbf{d}+\mathbf{a}\in\operatorname{Exp}I$ or $f(\mathbf{a})=0$. This is
equivalent to the statement that if
$\mathbf{a}\notin\operatorname{Exp}I-\mathbf{d}$, then $f(\mathbf{a})=0$. ∎

Next we turn to characterizing when a given ideal is $\delta$-fixed for which
we need the following lemma.

######  Lemma 3.3.

The element $x^{\mathbf{a}}\in R$ is in the image of $\delta$ if and only if
$f(\mathbf{a}-\mathbf{d})\neq 0$.

###### Proof.

Since $\delta$ is homogeneous we may restrict our attention to monomials. We
have that $x^{\mathbf{a}}=\delta(\alpha x^{\mathbf{a}-\mathbf{d}})$ for some
$\alpha\in\mathbb{C}$ if and only if

$x^{\mathbf{a}}=\alpha f(\mathbf{a}-\mathbf{d})x^{\mathbf{a}}.$

This is possible precisely when $f(\mathbf{a}-\mathbf{d})\neq 0$. ∎

######  Theorem 3.4.

The ideal $I$ is $\delta$-fixed if and only if $-\mathbf{d}\in S$ and

$\operatorname{Exp}I\cap
V_{\textnormal{mon}}(f)=\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d}).$

###### Proof.

By Theorem 3.2, $I$ is $\delta$-compatible if and only if

$\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d})\subseteq
V_{\textnormal{mon}}(f).$

To show $I\subseteq\delta(I)$, it suffices to show that for every
$\mathbf{a}\in\operatorname{Exp}I$,
$\mathbf{a}-\mathbf{d}\in\operatorname{Exp}I$ and
$f(\mathbf{a}-\mathbf{d})\neq 0$ by Lemma 3.3. If $-\mathbf{d}\notin S$, then
for some $n\geq 1$, $\mathbf{a}-(n-1)\mathbf{d}\in\operatorname{Exp}I$ but
$\mathbf{a}-n\mathbf{d}\notin\operatorname{Exp}I$. Whence
$x^{\mathbf{a}-(n-1)\mathbf{d}}\notin\delta(I)$. Thus, $-\mathbf{d}\in S$ is
necessary. It is also necessary that $f(\mathbf{a}-\mathbf{d})\neq 0$ for all
$\mathbf{a}\in\operatorname{Exp}I$, or in other words, $f(\mathbf{a})\neq 0$
for every $\mathbf{a}\in\operatorname{Exp}I-\mathbf{d}$. It follows these are
also sufficient again by Lemma 3.3. Therefore $I=\delta(I)$ if and only if
$-\mathbf{d}\in S$ and
$V_{\textnormal{mon}}(f)\cap(\operatorname{Exp}I-\mathbf{d})=\varnothing$,
that is, $\operatorname{Exp}I\cap
V_{\textnormal{mon}}(f)\subseteq\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d})$.
∎

Now we can show for fixed ideal $I$, there are many $\delta$ which fix $I$.

######  Lemma 3.5.

For $\mathbf{a}\in S$, $H_{\mathbf{d}}(\mathbf{a})>0$ if and only if
$\mathbf{a}\in-\mathbf{d}+S$ and $H_{\mathbf{d}}(\mathbf{a})=0$ otherwise.

###### Proof.

Recall $H_{\mathbf{d}}=\prod_{i}(h_{i},h_{i}(-\mathbf{d})-1)!$. As
$\mathbf{a}\in S$, $H_{\mathbf{d}}(\mathbf{a})>0$ if and only if
$h_{i}(\mathbf{a})-k>0$ for all $0\leq k\leq h_{i}(-\mathbf{d})-1$. This
happens if and only if $h_{i}(\mathbf{a})-h_{i}(-\mathbf{d})+1>0$ which by
linearity occurs if and only if $h_{i}(\mathbf{a}+\mathbf{d})\geq 0$.
Equivalently, this occurs when $\mathbf{a}+\mathbf{d}\in S$, or
$\mathbf{a}\in-\mathbf{d}+S$. ∎

######  Theorem 3.6.

Assume $-\mathbf{d}\in S$. There exists a homogeneous element
$\delta\in\mathcal{D}(R)$ of degree $\mathbf{d}$ such that $I$ is
$\delta$-fixed.

###### Proof.

Let $I=(x^{\mathbf{a}_{1}},\ldots,x^{\mathbf{a}_{m}})$, so that
$\operatorname{Exp}I=\bigcup_{i}(\mathbf{a}_{i}+S)$. Let
$f=\sum_{i}H_{\mathbf{d}-\mathbf{a}_{i}}$, and note that
$H_{\mathbf{d}-\mathbf{a}_{i}}(\mathbf{b})\geq 0$ for all $i$ and all
$\mathbf{b}\in S$. Therefore, for any $\mathbf{b}\in\operatorname{Exp}I$, we
have that $f(\mathbf{b})\neq 0$ if and only if there exists $i$ such that
$H_{\mathbf{d}-\mathbf{a}_{i}}(\mathbf{b})>0$, which occurs exactly when
$\mathbf{b}\in\mathbf{a}_{i}-\mathbf{d}+S$ by Lemma 3.5. Thus
$f(\mathbf{b})\neq 0$ if and only if

$\mathbf{b}\in\bigcup_{i}(\mathbf{a}_{i}-\mathbf{d}+S)=-\mathbf{d}+\bigcup_{i}(\mathbf{a}_{i}+S)=-\mathbf{d}+\operatorname{Exp}I.$

Thus, we have that
$V_{\textnormal{mon}}(f)\cap\operatorname{Exp}I=\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d})$.
Furthermore, since $H_{\mathbf{d}}$ divides $H_{\mathbf{d}-\mathbf{a}_{i}}$
for each $i$, we have that $f\in(H_{\mathbf{d}})$, and so
$\delta=x^{\mathbf{d}}f(\theta)\in\mathcal{D}(R)$, and $I$ is $\delta$-fixed.
∎

######  Example 3.7.

Let $R=\mathbb{C}[x,xy,xy^{2},xy^{3}]$. Let

Let

$\displaystyle\tau_{1}$ $\displaystyle=\mathbb{R}_{\geq 0}(1,0)$
$\displaystyle h_{1}(\theta)$ $\displaystyle=\theta_{y}$
$\displaystyle\tau_{2}$ $\displaystyle=\mathbb{R}_{\geq 0}(0,1)$
$\displaystyle h_{2}(\theta)$ $\displaystyle=3\theta_{x}-\theta_{y}$

be the facets of the cone $\sigma^{\vee}$ of $R$. Consider the ideal
$I=(x^{2}y^{4},x^{5}y)$ which is illustrated in Figure 1. We have labeled the
lattice points representing the exponents of monomials which are contained in
the ideal in gray (we have also shaded this region in gray) and those in the
ring which are not members of the ideal in black.

Figure 1. The ideal $I=(x^{2}y^{4},x^{5}y)$

Let $\mathbf{d}=(-2,-1)$. The differential operators in $\mathcal{D}(R)$ of
degree $(-2,-1)$ have the form $x^{-2}y^{-1}H_{(-2,-1)}(\theta)g(\theta)$
where $H_{(-2,-1)}(\theta)=(3\theta_{x}-\theta_{y},4)!\theta_{y}$. In Figure 2
on the left, the lattice points in the solid gray region form the subset of
$\operatorname{Exp}I$ whose images under any differential operator of degree
$(-2,-1)$ also stay in $I$. The lattice points in the vertically lined region
are annihilated by any differential operator of degree $(-2,-1)$. The lattice
points in the crosshatched region represent monomials which lie in $I$, whose
images lie outside of $I$ when acted upon by
$x^{-2}y^{-1}H_{(-2,-1)}(\theta)$. The middle and right images show similar
regions for the homogeneous weights $\mathbf{d}=(-4,-6)$ and
$\mathbf{d}=(-3,-8)$ respectively.

Figure 2. Left: $\mathbf{d}=(-2,-1)$, Middle: $\mathbf{d}=(-4,6)$, Right:
$\mathbf{d}=(-3,-8)$.

######  Example 3.8.

Let $R=\mathbb{C}[x,y,z,xyz^{-1}]$. Let

$\displaystyle\tau_{1}$ $\displaystyle=\mathbb{R}_{\geq
0}(1,0,0)+\mathbb{R}_{\geq 0}(0,0,1)$ $\displaystyle h_{1}(\theta)$
$\displaystyle=\theta_{y}$ $\displaystyle\tau_{2}$
$\displaystyle=\mathbb{R}_{\geq 0}(0,0,1)+\mathbb{R}_{\geq 0}(0,1,0)$
$\displaystyle h_{2}(\theta)$ $\displaystyle=\theta_{x}$
$\displaystyle\tau_{3}$ $\displaystyle=\mathbb{R}_{\geq
0}(0,1,0)+\mathbb{R}_{\geq 0}(1,1,-1)$ $\displaystyle h_{3}(\theta)$
$\displaystyle=\theta_{x}+\theta_{z}$ $\displaystyle\tau_{4}$
$\displaystyle=\mathbb{R}_{\geq 0}(1,0,0)+\mathbb{R}_{\geq 0}(1,1,-1)$
$\displaystyle h_{4}(\theta)$ $\displaystyle=\theta_{y}+\theta_{z}$

be the faces of the cone $\sigma^{\vee}$ of $R$. Now let
$\mathbf{d}=(-2,0,-1)$, and let $I=(x^{2},z,xy^{2}z^{-1})$. We construct a
differential operator of degree $\mathbf{d}$ fixing $I$.

Now, we have that

$\displaystyle H_{\mathbf{d}-(2,0,0)}(\theta)$ $\displaystyle=H_{(-4,0,-1)}$
$\displaystyle=\prod_{i=1}^{4}(h_{i}(\theta),h_{i}(4,0,1)-1)!$
$\displaystyle=(h_{1}(\theta),-1)!\cdot(h_{2}(\theta),3)!\cdot(h_{3}(\theta),4)!\cdot(h_{4}(\theta),0)!$
$\displaystyle=(\theta_{x},3)!(\theta_{x}+\theta_{z},4)!(\theta_{y}+\theta_{z},0)!$
$\displaystyle H_{\mathbf{d}-(0,0,1)}(\theta)$ $\displaystyle=H_{(-2,0,-2)}$
$\displaystyle=\prod_{i=1}^{4}(h_{i}(\theta),h_{i}(2,0,2)-1)!$
$\displaystyle=(h_{1}(\theta),-1)!\cdot(h_{2}(\theta),1)!\cdot(h_{3}(\theta),3)!\cdot(h_{4}(\theta),1)!$
$\displaystyle=(\theta_{x},1)!(\theta_{x}+\theta_{z},3)!(\theta_{y}+\theta_{z},1)!$
$\displaystyle H_{\mathbf{d}-(1,2,-1)}(\theta)$ $\displaystyle=H_{(-3,-2,0)}$
$\displaystyle=\prod_{i=1}^{4}(h_{i}(\theta),h_{i}(3,2,0)-1)!$
$\displaystyle=(h_{1}(\theta),1)!\cdot(h_{2}(\theta),2)!\cdot(h_{3}(\theta),2)!\cdot(h_{4}(\theta),1)!$
$\displaystyle=(\theta_{y},1)!(\theta_{x},2)!(\theta_{x}+\theta_{z},2)!(\theta_{y}+\theta_{z},1)!$

Therefore, a differential operator fixing $I$ is given by

$\delta=x^{-2}z^{-1}\left(H_{(-4,0,-1)}+H_{(-2,0,-2)}+H_{(-3,-2,0)}\right)$

For example, we check that the generator $x^{2}$ of $I$ is in the image of
$\delta$. The preimage of the generator $x^{2}$ should be a multiple of
$x^{4}z$. We have that

$\delta(x^{4}z)=x^{-2}z^{-1}\left(H_{(-4,0,-1)}+H_{(-2,0,-2)}+H_{(-3,-2,0)}\right)(x^{4}z)$
$=x^{2}\left(H_{(-4,0,-1)}(4,0,1)+H_{(-2,0,-2)}(4,0,1)+H_{(-3,-2,0)}(4,0,1)\right)$

Now, we have that

$\displaystyle H_{(-4,0,-1)}(4,0,1)$ $\displaystyle=(4,3)!(5,4)!(1,0)!=4\cdot
3\cdot 2\cdot 1\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 1=2880$ $\displaystyle
H_{(-2,0,-2)}(4,0,1)$ $\displaystyle=(4,1)!(5,3)!(1,1)!=4\cdot 3\cdot 5\cdot
4\cdot 3\cdot 2\cdot 1\cdot 0=0$ $\displaystyle H_{(-3,-2,0)}(4,0,1)$
$\displaystyle=(0,1)!(4,2)!(5,2)!(1,1)!=0\cdot-1\cdot 4\cdot 3\cdot 2\cdot
5\cdot 4\cdot 3\cdot 1\cdot 0=0$

Therefore,

$\delta(x^{4}z)=x^{2}\cdot(2880+0+0)=2880x^{2}$

Hence $x^{2}=\delta\left(\frac{1}{2880}x^{4}z\right)$

##  4\. The Fixed Ideals of a Differential Operator

We turn now to characterizing all ideals fixed by a given differential
operator $\delta=x^{\mathbf{d}}f(\theta)$. To this end, we are interested in
the points in $V_{\textnormal{mon}}(f)$ which are furthest along rays in the
direction of $-\mathbf{d}$.

######  Definition 4.1.

For every lattice point $\mathbf{a}\in\mathbb{Z}^{d}$, set

$\nu(\mathbf{a}):=\inf\\{t\in\mathbb{R}\mid\mathbf{a}+t\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}.$

When $\nu(\mathbf{a})$ is finite, set

$\nu^{\prime}(\mathbf{a}):=\sup\\{t\in[\nu(\mathbf{a}),\nu(\mathbf{a})+1)\mid\mathbf{a}+t\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}$

and $\rm\mathbf{a}:=\mathbf{a}+\nu^{\prime}(\mathbf{a})\mathbf{d}$.

With the above notation, if $\nu(\mathbf{a})$ is finite, then
$\mathbf{a}+\nu(\mathbf{a})\mathbf{d}$ is the point in
$V_{\textnormal{mon}}(f)$ furthest along the ray through $\mathbf{a}$ in the
direction of $-\mathbf{d}$. Also in this case, $\rm\mathbf{a}$ is a point in
$V_{\textnormal{mon}}(f)$ such that $\rm\mathbf{a}-n\mathbf{d}\notin
V_{\textnormal{mon}}(f)$ for any $n\geq 1$. Note that if $\mathbf{d}$ is
primitive, then $\nu(\mathbf{a})=\nu^{\prime}(\mathbf{a})$, since
$\mathbf{a}+t\mathbf{d}$ will not be a lattice point if $t\notin\mathbb{Z}$.

Suppose $\delta=x^{\mathbf{d}}f(\theta)$ is a differential operator on $R$
with $\mathbf{0}\neq\mathbf{d}\in-S$. We have that $\nu(\mathbf{a})<\infty$
for any $\mathbf{a}\in S$. To see this, note if $\nu(\mathbf{a})=\infty$, then
$f(\mathbf{a}+t\mathbf{d})\neq 0$ for all $t\in\mathbb{R}$. If we take $i$
such that $h_{i}(\mathbf{d})<0$, then for large enough $m\in\mathbb{N}$ we
have that
$h_{i}(\mathbf{a}+m\mathbf{d})=h_{i}(\mathbf{a})+mh_{i}(\mathbf{d})<0$.
Therefore $\mathbf{a}+m\mathbf{d}\notin S$. However, we also have that
$x^{\mathbf{a}+m\mathbf{d}}$ is a nonzero multiple of
$\delta^{m}(x^{\mathbf{a}})$ since $f(\mathbf{a}+t\mathbf{d})\neq 0$ for all
$t\in\mathbb{R}$. This is a contradiction since $\delta$ is an operator on $R$
and $x^{\mathbf{a}+m\mathbf{d}}\notin R$.

We note a few more useful points about these functions. If
$\nu(\mathbf{a})>0$, then $f(\mathbf{a})\neq 0$. For any $u\in\mathbb{R}$ we
have that

$\displaystyle\nu(\mathbf{a}+u\mathbf{d})$
$\displaystyle=\inf\\{t\in\mathbb{R}\mid\mathbf{a}+u\mathbf{d}+t\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}$
$\displaystyle=\inf\\{t+u\in\mathbb{R}\mid\mathbf{a}+(t+u)\mathbf{d}\in
V_{\textnormal{mon}}(f)\\}-u$ $\displaystyle=\nu(\mathbf{a})-u.$

Furthermore, notice that if $\mathbf{a}=\mathbf{b}+u\mathbf{d}$ for some
$u\in\mathbb{R}$, then $\rm{\mathbf{a}}=\rm{\mathbf{b}}$. By definition
$\nu(\mathbf{a})\leq\nu^{\prime}(\mathbf{a})<\nu(\mathbf{a})+1$, which means
that for any $\mathbf{a}$, we have that

$\nu(\rm\mathbf{a})=\nu(\mathbf{a}+\nu^{\prime}(\mathbf{a})\mathbf{d})=\nu(\mathbf{a})-\nu^{\prime}(\mathbf{a})$

and so $\nu(\rm\mathbf{a})\in(-1,0]$. If $\mathbf{e}\in\mathbb{Z}^{d}$ is
primitive such that $\mathbf{d}=q\mathbf{e}$, then
$\nu(\mathbf{a}),\nu^{\prime}(\mathbf{a})\in\frac{1}{q}\mathbb{Z}$ for all
$\mathbf{a}\in\mathbb{Z}^{d}$.

In the next theorem, our aim is to “cut out” parts of the semigroup $S$ based
on the values of the primitive support functions of the faces of
$\sigma^{\vee}$. We have that a lattice point $\mathbf{a}\in\mathbb{Z}^{d}$ is
in $S$ if and only if $h_{i}(\mathbf{a})\geq 0$ for all $i$. For any
$\beta=(\beta_{1},\ldots,\beta_{n})\in\mathbb{N}^{n}$, we consider the subset
of the lattice points in $S$ whose values at the functions $h_{i}$ are at
least the values $\beta_{i}$. Let $W_{\beta}=\\{\mathbf{a}\in S\mid\textrm{for
all }i,h_{i}(\mathbf{a})\geq\beta_{i}\\}$. Notice that the set $W_{\beta}$ is
closed under the semigroup action, in other words $W_{\beta}$ is the exponent
set of the ideal $I_{\beta}=(x^{\mathbf{a}}\mid\mathbf{a}\in W_{\beta})$.

We also consider for a collection of vectors
$\mathcal{B}\subseteq\mathbb{N}^{n}$,
$W_{\mathcal{B}}:=\bigcup_{\beta\in\mathcal{B}}W_{\beta}$. Clearly
$W_{\mathcal{B}}$ is also closed under the semigroup action.

######  Definition 4.2.

For a vector $\mathbf{d}\in-S$, we call a tuple $\beta\in\mathbb{N}^{n}$
_compatible with $\mathbf{d}$_ if for all $i$, either $\beta_{i}=0$ or
$h_{i}(\mathbf{d})=0$.

A collection $\mathcal{B}\subseteq\mathbb{N}^{n}$ is called _compatible with
$\mathbf{d}$_ if each $\beta\in\mathcal{B}$ is compatible with $\mathbf{d}$.

The tuples compatible with $\mathbf{d}$ will be part of the data defining the
fixed ideals of a differential operator $\delta$ of degree $\mathbf{d}$. Note
that if $-\mathbf{d}\in\operatorname{int}\sigma^{\vee}$, then the only tuple
compatible with $\mathbf{d}$ is the zero tuple.

######  Lemma 4.3.

If $\delta$ is a homogeneous operator of degree $\mathbf{d}$,
$\mathcal{B}\subseteq\mathbb{N}^{n}$ is compatible with $\mathbf{d}$, and
$I=(x^{\mathbf{a}}\mid\mathbf{a}\in W_{\mathcal{B}})$, then $I$ is
$\delta$-compatible. In particular, if $\mathbf{b}\in W_{\mathcal{B}}$ and
$\nu(\mathbf{b})$ is finite, then $\rm\mathbf{b}\in W_{\mathcal{B}}$.

###### Proof.

If $x^{\mathbf{a}}\in I$, then
$\mathbf{a}\in\operatorname{Exp}I=W_{\mathcal{B}}$. Therefore there exists
$\beta\in\mathcal{B}$ such that for all $i$, $h_{i}(\mathbf{a})\geq\beta_{i}$.
If $\delta(x^{\mathbf{a}})\neq 0$, then $\mathbf{a}+\mathbf{d}\in S$. For
every $i$, if $h_{i}(\mathbf{d})\neq 0$, then $\beta_{i}=0$, and so
$h_{i}(\mathbf{a}+\mathbf{d})\geq\beta_{i}$. If $h_{i}(\mathbf{d})=0$, then
$h_{i}(\mathbf{a}+\mathbf{d})=h_{i}(\mathbf{a})\geq\beta_{i}$. Thus in either
case $\mathbf{a}+\mathbf{d}\in W_{\beta}\subseteq
W_{\mathcal{B}}=\operatorname{Exp}I$. Therefore $\delta(x^{\mathbf{a}})\in I$,
and so $I$ is $\delta$-compatible.

For the last statement, if $\nu^{\prime}(\mathbf{b})\leq 0$, then
$\rm\mathbf{b}=\mathbf{b}+(-\nu^{\prime}(\mathbf{b}))(-\mathbf{d})\in\mathbf{b}+S\subseteq
W_{\mathcal{B}}$. If $\nu^{\prime}(\mathbf{b})>0$, then since
$\nu(\rm\mathbf{b})>-1$, we have that $f(\rm\mathbf{b}-m\mathbf{d})\neq 0$ for
all $m\geq 1$, and so $x^{\rm\mathbf{b}}$ is a nonzero multiple of
$\delta^{m}(x^{\rm\mathbf{b}-m\mathbf{d}})$ for all $m\geq 1$. Taking
$m=\lceil\nu^{\prime}(\mathbf{b})\rceil$, and noting that

$\rm\mathbf{b}-\lceil\nu^{\prime}(\mathbf{b})\rceil\mathbf{d}=\mathbf{b}+\nu^{\prime}(\mathbf{b})\mathbf{d}-\lceil\nu^{\prime}(\mathbf{b})\rceil\mathbf{d}=\mathbf{b}+(\nu^{\prime}(\mathbf{b})-\lceil\nu^{\prime}(\mathbf{b})\rceil)(-\mathbf{d})\in
W_{\mathcal{B}}$

we have that $x^{\rm\mathbf{b}}$ is a nonzero multiple of an element of $I$.
Since $I$ is $\delta$-compatible, $x^{\rm\mathbf{b}}\in I$, and so
$\rm\mathbf{b}\in\operatorname{Exp}I=W_{\mathcal{B}}$. ∎

We now prove necessary and sufficient conditions for a differential operator
$\delta=x^{\mathbf{d}}f(\theta)$ to fix a monomial ideal. This is described
using the following subset of $V_{\textnormal{mon}}(f)$:

$V_{\textnormal{mon}}^{\prime}(f):=\\{\rm\mathbf{a}\mid\mathbf{a}\in\mathbb{Z}^{d},\nu(\mathbf{a})\text{
finite}\\}.$

Notation: We also make the assumption for the rest of the section that
$\mathbf{d}\neq\mathbf{0}$ with $\mathbf{d}=q\mathbf{e}$ with $\mathbf{e}$
primitive.

######  Theorem 4.4.

If there exists a $\mathbf{d}$-compatible subset
$\mathcal{B}\subseteq\mathbb{N}^{n}$ such that

  1. (1)

for all $\mathbf{a}\in W_{\mathcal{B}}$, $\nu(\mathbf{a})>-\infty$

  2. (2)

for all $\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)\cap W_{\mathcal{B}}$
and $i=0,\ldots,q-1$, $\mathbf{a}-i\mathbf{e}\in V_{\textnormal{mon}}(f)$, and

  3. (3)

for all $\mathbf{a},\mathbf{b}\in V_{\textnormal{mon}}^{\prime}(f)\cap
W_{\mathcal{B}}$, $\mathbf{a}-\mathbf{b}\notin S-\mathbf{e}$,

then $\delta$ fixes the ideal $I=(x^{\mathbf{a}}\mid\mathbf{a}\in
V_{\textnormal{mon}}^{\prime}(f)\cap W_{\mathcal{B}})$. Furthermore
$\operatorname{Exp}I=\\{\mathbf{b}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{b})\geq 0\\}$.

###### Proof.

Let $I=(x^{\mathbf{a}}\mid\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)\cap
W_{\mathcal{B}})$. We prove the last statement first. Let
$\mathbf{b}\in\operatorname{Exp}I$ and $\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}(f)\cap W_{\mathcal{B}}$ such that
$\mathbf{b}\in\mathbf{a}+S$. This implies that $\mathbf{b}\in
W_{\mathcal{B}}$. By condition (1), $\rm\mathbf{b}$ exists and therefore
$\rm{\mathbf{b}}\in W_{\mathcal{B}}\cap V^{\prime}_{\textnormal{mon}}(f)$ by
Lemma 4.3. If $\nu^{\prime}(\mathbf{b})<0$, then
$\nu^{\prime}(\mathbf{b})\leq-\frac{1}{q}$, and so
$\nu^{\prime}(\mathbf{b})\mathbf{d}+\mathbf{e}=(q\nu^{\prime}(\mathbf{b})+1)\mathbf{e}\in
S$. Therefore we have that

$\rm\mathbf{b}-\mathbf{a}=\mathbf{b}+\nu^{\prime}(\mathbf{b})\mathbf{d}-\mathbf{a}=(\mathbf{b}-\mathbf{a})+(\nu^{\prime}(\mathbf{b})\mathbf{d}+\mathbf{e})-\mathbf{e}\in
S-\mathbf{e},$

a contradiction to condition (3). Therefore $\nu^{\prime}(\mathbf{b})\geq 0$.
Hence $\operatorname{Exp}I\subseteq\\{\mathbf{b}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{b})\geq 0\\}$.

On the other hand, If $\mathbf{b}\in W_{\mathcal{B}}$ and
$\nu^{\prime}(\mathbf{b})\geq 0$, then $\rm{\mathbf{b}}\in W_{\mathcal{B}}\cap
V^{\prime}_{\textnormal{mon}}(f)\subseteq\operatorname{Exp}I$ and so

$\mathbf{b}=\rm\mathbf{b}+(-\nu^{\prime}(\mathbf{b}))(-\mathbf{d})\in\rm\mathbf{b}+S\subseteq\operatorname{Exp}I.$

Therefore $\\{\mathbf{b}\in W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{b})\geq
0\\}\subseteq\operatorname{Exp}I$, and so we have equality.

Let $\mathbf{b}\in\operatorname{Exp}I$. If $\nu^{\prime}(\mathbf{b})\geq 1$,
then $\nu(\mathbf{b})>0$, and so $f(\mathbf{b})\neq 0$. On the other hand, if
$0\leq\nu^{\prime}(\mathbf{b})<1$, then $\nu^{\prime}(\mathbf{b})=\frac{i}{q}$
for some $i\in\\{0,\ldots,q-1\\}$. Therefore,

$\mathbf{b}=\rm\mathbf{b}-\nu^{\prime}(\mathbf{b})\mathbf{d}=\rm\mathbf{a}-i\mathbf{e}\in
V_{\textnormal{mon}}(f)$

by condition (2), and so $f(\mathbf{b})=0$. Therefore $\mathbf{b}\in
V_{\textnormal{mon}}(f)$ if and only if $\nu^{\prime}(\mathbf{b})<1$, which
occurs exactly when $\nu^{\prime}(\mathbf{b}+\mathbf{d})<0$, that is, when
$\mathbf{b}+\mathbf{d}\notin\operatorname{Exp}I$. Therefore $\delta$ fixes $I$
by Theorem 3.4. ∎

######  Theorem 4.5.

If the operator $\delta$ fixes a nonzero monomial ideal $I$, then there exists
a $\mathbf{d}$-compatible subset $\mathcal{B}\subseteq\mathbb{N}^{n}$ such
that

  1. (1)

for all $\mathbf{a}\in W_{\mathcal{B}}$, $\nu(\mathbf{a})>-\infty$

  2. (2)

for all $\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)\cap W_{\mathcal{B}}$
and $i=0,\ldots,q-1$, $\mathbf{a}-i\mathbf{e}\in V_{\textnormal{mon}}(f)$,

  3. (3)

for all $\mathbf{a},\mathbf{b}\in V_{\textnormal{mon}}^{\prime}(f)\cap
W_{\mathcal{B}}$, $\mathbf{a}-\mathbf{b}\notin S-\mathbf{e}$, and

  4. (4)

$I=(x^{\mathbf{a}}\mid\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)\cap
W_{\mathcal{B}})$ and $\operatorname{Exp}I=\\{\mathbf{a}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$

###### Proof.

Suppose that $\delta$ fixes a monomial ideal
$I=(x^{\mathbf{b}_{1}},\ldots,x^{\mathbf{b}_{m}})$. For $i=1,\ldots,n$ and
$j=1,\ldots,m$, let $\beta_{ij}=h_{i}(\mathbf{b}_{j})$ if
$h_{i}(\mathbf{d})=0$ and $\beta_{ij}=0$ otherwise. For $j=1,\ldots,m$, let
$\beta_{j}=(\beta_{1j},\beta_{2j},\ldots,\beta_{nj})\in\mathbb{N}^{n}$, and
then let $\mathcal{B}=\\{\beta_{1},\ldots,\beta_{m}\\}$. By construction,
$\mathcal{B}$ is $\mathbf{d}$-compatible.

(1) Let $\mathbf{a}\in W_{\mathcal{B}}$, and choose $j$ such that
$\mathbf{a}\in W_{\beta_{j}}$. Fix $i\in\\{1,\ldots,n\\}$. If
$h_{i}(\mathbf{d})<0$, then for sufficiently large $s\in\mathbb{N}$ we have
that
$h_{i}(\mathbf{a}-s\mathbf{d}-\mathbf{b}_{j})=h_{i}(\mathbf{a}-\mathbf{b}_{j})-sh_{i}(\mathbf{d})\geq
0$. If $h_{i}(\mathbf{d})=0$, then for all $s\in\mathbb{N}$ we have that
$h_{i}(\mathbf{a}-s\mathbf{d}-\mathbf{b}_{j})=h_{i}(\mathbf{a})-h_{i}(\mathbf{b}_{j})\geq\beta_{ij}-\beta_{ij}=0$.
So, for some large $s$ we have that $\mathbf{a}-s\mathbf{d}-\mathbf{b}_{j}\in
S$, and so
$\mathbf{a}-s\mathbf{d}\in\mathbf{b}_{j}+S\subseteq\operatorname{Exp}I$.
Therefore, we have that for any $t\in\frac{1}{q}\mathbb{Z}$ with $t\leq-s$,
$\mathbf{a}+t\mathbf{d}=\mathbf{a}-s\mathbf{d}+(t+s)\mathbf{d}\in\operatorname{Exp}I$.
Since $I$ is $\delta$-fixed, this means that for all such $t$,
$\mathbf{a}+(t-1)\mathbf{d}\notin V_{\textnormal{mon}}(f)$. Therefore
$\nu(\mathbf{a})\geq-s-1>-\infty$.

As a consequence to the last paragraph, if we take $\mathbf{a}\in
W_{\mathcal{B}}$ such that $\nu^{\prime}(\mathbf{a})=0$, then the argument
shows that there exists $s\in\mathbb{N}$ such that
$\mathbf{a}-s\mathbf{d}\in\operatorname{Exp}I$, and furthermore
$f(\mathbf{a}-t\mathbf{d})\neq 0$ for all $t\geq 1$. Therefore,
$x^{\mathbf{a}}$ is a nonzero multiple of
$\delta^{s}(x^{\mathbf{a}-s\mathbf{d}})\in\delta^{s}(I)=I$. Therefore
$V^{\prime}_{\textnormal{mon}}(f)\cap
W_{\mathcal{B}}\subseteq\operatorname{Exp}I$.

(2) Let $\mathbf{a}\in V^{\prime}_{\textnormal{mon}}(f)\cap
W_{\mathcal{B}}\subseteq\operatorname{Exp}I$. Suppose that for some
$i\in\\{0,\ldots,q-1\\}$ we have that $\mathbf{a}-i\mathbf{e}\notin
V^{\prime}_{\textnormal{mon}}(f)$. Note this implies that $i\neq 0$. Therefore
$\delta(x^{\mathbf{a}-i\mathbf{e}})\neq 0$, and so
$\mathbf{a}-i\mathbf{e}+\mathbf{d}\in\operatorname{Exp}I$. Therefore, for all
$j=i,i+1,\ldots,q-1$, we have that
$\mathbf{a}-j\mathbf{e}+\mathbf{d}\in\operatorname{Exp}I$, and therefore
$\mathbf{a}-j\mathbf{e}\notin V^{\prime}_{\textnormal{mon}}(f)$ since $I$ is
$\delta$-fixed. Hence $\mathbf{a}-t\mathbf{d}\notin V_{\textnormal{mon}}(f)$
for $t\geq\frac{i}{q}$, and so $\nu(\mathbf{a})\geq-\frac{i-1}{q}$. Therefore,
for any $0<t<1-\frac{i-1}{q}=\frac{q-i+1}{q}$, we must have that
$\mathbf{a}+t\mathbf{d}\notin V_{\textnormal{mon}}(f)$ since otherwise
$\nu^{\prime}(\mathbf{a})$ would be greater than 0. Hence
$\mathbf{a}-i\mathbf{e}+\mathbf{d}=\mathbf{a}+\frac{q-i}{q}\mathbf{d}\notin
V_{\textnormal{mon}}(f)$, and so
$\mathbf{a}-i\mathbf{e}+2\mathbf{d}\in\operatorname{Exp}I$. However, this
implies that $\mathbf{a}+\mathbf{d}\in\operatorname{Exp}I$, a contradiction
since $\mathbf{a}\in V_{\textnormal{mon}}(f)$.

(3) Fix $j\in\\{0,\ldots,m\\}$ and suppose that $\mathbf{a},\mathbf{b}\in
V^{\prime}_{\textnormal{mon}}(f)\cap W_{\mathcal{B}}$, which implies that
$\mathbf{a},\mathbf{b}\in\operatorname{Exp}I$. If $\mathbf{a}-\mathbf{b}\in
S-\mathbf{e}$, then $\mathbf{a}+\mathbf{e}\in\mathbf{b}+S$, and so
$\mathbf{a}+\mathbf{e}\in\operatorname{Exp}I$. However, by condition (2),
$\mathbf{a}-(q-1)\mathbf{e}\in V_{\textnormal{mon}}(f)$, and so
$\mathbf{a}+\mathbf{e}=\mathbf{a}-(q-1)\mathbf{e}+\mathbf{d}\notin\operatorname{Exp}I$,
a contradiction.

(4) If $\mathbf{a}\in\operatorname{Exp}I$, then there exists $\mathbf{b}_{j}$
such that $\mathbf{a}\in\mathbf{b}_{j}+S$, and so $\mathbf{a}\in
W_{\beta_{j}}\subseteq W_{\mathcal{B}}$. Furthermore, if
$\nu^{\prime}(\mathbf{a})<0$, then $\rm\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}\cap W_{\mathcal{B}}$ and
$\mathbf{a}-\lfloor\nu^{\prime}(\mathbf{a})\rfloor\mathbf{d}=\rm\mathbf{a}-i\mathbf{e}$
for some $i\in\\{0,\ldots,q-1\\}$, and so
$\mathbf{a}-\lfloor\nu^{\prime}(\mathbf{a})\rfloor\mathbf{d}\in
V_{\textnormal{mon}}(f)$. Therefore $x^{\mathbf{a}}$ is not in the image of
$\delta$, and so $\mathbf{a}\notin\operatorname{Exp}I$, a contradiction.
Therefore $\operatorname{Exp}I\subseteq\\{\mathbf{a}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$. Hence,

$\\{\mathbf{a}\in W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{a})\geq
0\\}\subseteq(V^{\prime}_{\textnormal{mon}}(f)\cap
W_{\mathcal{B}})+S\subseteq\operatorname{Exp}I\subseteq\\{\mathbf{a}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$

and so we have equality throughout. ∎

######  Corollary 4.6.

Suppose $\mathbf{d}\neq 0$. The operator $\delta$ fixes some nonzero monomial
ideal if and only if there exists a $\mathbf{d}$-compatible subset
$\mathcal{B}\subseteq\mathbb{N}^{n}$ such that

  1. (1)

For all $\mathbf{a}\in W_{\mathcal{B}}$, $\nu(\mathbf{a})>-\infty$

  2. (2)

for all $\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)\cap W_{\mathcal{B}}$
and $i=0,\ldots,q-1$, $\mathbf{a}-i\mathbf{e}\in V_{\textnormal{mon}}(f)$, and

  3. (3)

for all $\mathbf{a},\mathbf{b}\in V_{\textnormal{mon}}^{\prime}(f)\cap
W_{\mathcal{B}}$, $\mathbf{a}-\mathbf{b}\notin S-\mathbf{e}$.

In this case, $\delta$ fixes the ideal $I=(x^{\mathbf{a}}\mid\mathbf{a}\in
V_{\textnormal{mon}}^{\prime}(f)\cap W_{\mathcal{B}})$, and furthermore
$\operatorname{Exp}I=\\{\mathbf{a}\in
W_{\mathcal{B}}\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$.

######  Corollary 4.7.

Suppose $-\mathbf{d}\in\operatorname{int}\sigma^{\vee}$. The operator $\delta$
fixes some nonzero monomial ideal if and only if

  1. (1)

For all $\mathbf{a}\in S$, $\nu(\mathbf{a})>-\infty$

  2. (2)

for all $\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f)$ and $i=0,\ldots,q-1$,
$\mathbf{a}-i\mathbf{e}\in V_{\textnormal{mon}}(f)$, and

  3. (3)

for all $\mathbf{a},\mathbf{b}\in V_{\textnormal{mon}}^{\prime}(f)$,
$\mathbf{a}-\mathbf{b}\notin S-\mathbf{e}$.

In this case, $\delta$ fixes only the ideal
$I=(x^{\mathbf{a}}\mid\mathbf{a}\in V_{\textnormal{mon}}^{\prime}(f))$, and
furthermore $\operatorname{Exp}I=\\{\mathbf{a}\in
S\mid\nu^{\prime}(\mathbf{a})\geq 0\\}$.

###### Proof.

This corollary follows from Corollary 4.6 once we observe that if
$-\mathbf{d}\in\operatorname{int}\sigma^{\vee}$, then $h_{i}(\mathbf{d})<0$
for all $i$. Therefore the only $\mathbf{d}$-compatible vector is
$\mathbf{0}\in\mathbb{N}^{n}$, and $W_{\mathbf{0}}=S$. ∎

######  Corollary 4.8.

If $-\mathbf{d}\in\operatorname{int}\sigma^{\vee}$, then $\delta$ fixes at
most one nonzero monomial ideal.

######  Corollary 4.9.

Let $\mathcal{B}\subseteq\mathbb{N}^{n}$ be a $\mathbf{d}$-compatible subset
such that $\delta$ fixes the ideal
$I_{\mathcal{B}}=(x^{\mathbf{a}}\mid\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}(f)\cap W_{\mathcal{B}})$. Now for any
$\mathbf{d}$-compatible $\mathcal{B}^{\prime}\subseteq\mathbb{N}^{n}$ with
$W_{\mathcal{B}^{\prime}}\subseteq W_{\mathcal{B}}$,
$I_{\mathcal{B}^{\prime}}=(x^{\mathbf{a}}\mid\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}(f)\cap W_{\mathcal{B}^{\prime}})$ is
$\delta$-fixed.

###### Proof.

Clearly, we have that $W_{\mathcal{B}^{\prime}}$ satisfies conditions (1)-(3)
in Corollary 4.6. ∎

######  Corollary 4.10.

If $-\mathbf{d}\in S\setminus\operatorname{int}\sigma^{\vee}$ and $\delta$
fixes at least one monomial ideal, then $\delta$ fixes infinitely many
monomial ideals.

###### Proof.

Let $\mathcal{B}\subseteq\mathbb{N}^{n}$ be a $\mathbf{d}$-compatible subset
such that $\delta$ fixes the ideal
$I_{\mathcal{B}}=(x^{\mathbf{a}}\mid\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}(f)\cap W_{\mathcal{B}})$. Let
$\beta=(\beta_{1},\beta_{2},\ldots,\beta_{n})\in\mathcal{B}$. Without loss of
generality, suppose that $h_{1}(\mathbf{d})=0$. Now for any $m\in\mathbb{N}$,
$\beta_{m}:=(\beta_{1}+m,\beta_{2},\ldots,\beta_{n})$ is
$\mathbf{d}$-compatible and $W_{\beta_{m}}\subseteq W_{\mathcal{B}}$, so
$I_{m}:=(x^{\mathbf{a}}\mid\mathbf{a}\in V^{\prime}_{\textnormal{mon}}(f)\cap
W_{\beta_{m}})$ is $\delta$-fixed.

Each $I_{m}$ is nonzero. Indeed, if we take any $\mathbf{b}\in
S\setminus\operatorname{int}\sigma^{\vee}$, then $h_{i}(\mathbf{b})>0$ for all
$i$, and so for sufficiently large $n\in\mathbb{N}$ we have that
$h_{1}(n\mathbf{b})\geq\beta_{1}+m$ and $h_{i}(n\mathbf{b})\geq\beta_{i}$ for
$i>1$. Thus $n\mathbf{b}\in W_{\beta_{m}}$, and so for any $\mathbf{a}\in
V^{\prime}_{\textnormal{mon}}(f)$, we have that $x^{n\mathbf{b}+\mathbf{a}}\in
I_{m}$. However, we also have that $\bigcap_{m\in\mathbb{N}}I_{m}=0$, and so
there are infinitely many distinct $I_{m}$ in the descending chain
$I_{1}\supseteq I_{2}\supseteq\cdots$, each fixed by $\delta$. ∎

##  5\. Applications

We now review applications, which are all derived from Theorem 5.3 and
elementary combinatorial geometry. This is phrased in suitable abstraction to
find interest beyond the applications we display.

First, we review a bit of the combinatorial geometry, for which one can see
[Grü03] for a deeper introduction. Among other things this will establish
definitions of basic objects which sometimes vary slightly from source to
source. By a polyhedron we mean a region of a Euclidean space of finite
dimension defined as the intersection of finitely many half-spaces. This is
most similar to $H$-polyhedra, as opposed to a $V$-polyehdra where one
utilizes a definition in terms of vertices, in many cases these descriptions
are equivalent. We avoid the term ‘polytope’ which conflicts among standard
sources.

Naturally, the $H$-description arises by preferring the facets, or maximal
faces. We additionally find it convenient to use a related, slightly more
flexible description as the model of computing or specifying a polyhedron $P$.
We motivate this by the following natural set of data. Set $\mathcal{F}$ the
set of faces of $P$. Naturally associated to $\mathcal{F}$ is a sequence
$(g_{\tau},M_{\tau})_{\tau\in\mathcal{F}}$ where
$g_{\tau}\in\mathbb{Z}[x_{1},\ldots,x_{d}]$ is a linear form and
$M_{\tau}\in\mathbb{Q}$ with the property that $\mathbf{v}\in P$ if and only
if $g_{\tau}(\mathbf{v})\geq M_{\tau}$ for all $\tau\in\mathcal{F}$. We have
need to consider a slight generalization of this description to account for
inefficiencies.

######  Definition 5.1.

Fix $P$ a polyhedron. A face data consists of a finite set $\mathcal{S}$ and a
sequence $\\{(g_{\tau},M_{\tau})\\}_{\tau\in\mathcal{S}}$ with
$g_{\tau}\in\mathbb{Z}[x_{1},\ldots,x_{d}]$ is a linear form,
$M_{\tau}\in\mathbb{Q}$ and a point $\mathbf{v}\in P$ if and only if
$g_{\tau}(\mathbf{v})\geq M_{\tau}$ for all $\tau$.

As described there is a canonical face data given the $H$-description of a
polyhedron which motivates the name. However, as we have defined it there is
not a unique face data describing a given polyhedron. In particular, a face
data could have redundant information. Shortly, we will describe an
algorithmic process for computing face data, but this will not be optimal in
the sense that we do not guarantee it returns the canonical face data.
However, in practice these redundancies cause no issue.

By our definition, polyhedra are not necessarily bounded nor convex. However,
at times we will prefer to work with convex polyhedra. Convex or not, a
unifying property we insist holds identifies those polyhedra whose
intersection with the lattice defining the fixed semigroup ring under
consideration in an ideal.

######  Definition 5.2.

A polyhedron $P$ is $\sigma^{\vee}$-closed proved $\mathbf{p}+\mathbf{a}\in P$
for all $\mathbf{p}\in P$ and $\mathbf{a}\in\sigma^{\vee}$.

Notably, $\sigma^{\vee}$-closed polyhedra are never bounded. Any monomial
ideal naturally has associated to it a canonical $\sigma^{\vee}$-closed
polyhedron based on its exponent set. Also, by definition every
$\sigma^{\vee}$-polyhedron defines an ideal. Obviously, this is not however a
bijection as small perturbations in the faces of the polyhedron will often not
change the associated ideal. Nonetheless, we refer to these structures as
associated to each other.

Now we come to the general form of our primary application which articulates
lattice point membership in a union of polyhedra in terms of a differential
operator fixing its associated ideal.

######  Theorem 5.3.

Let $P_{1},\ldots,P_{n}$ be $\sigma^{\vee}$-closed polyhedra with face data
$\\{(g_{\tau},M_{\tau})\\}_{\tau\in\mathcal{S}_{i}}$ for $i=1,\ldots,n$ and
$\mathbf{0}\neq\mathbf{d}\in-S$.

  1. (1)

Let $I=\langle x^{\mathbf{a}}\mid\mathbf{a}\in\bigcup P_{i}\rangle$ and
$f=\sum_{i=1}^{n}\prod_{\tau\in\mathcal{S}_{i}}(g_{\tau}(\theta),\lceil
M_{\tau}-g_{\tau}(\mathbf{d})-1\rceil)!$. For any
$h\in(H_{\mathbf{d}})\cap(f)$, $\delta=x^{\mathbf{d}}h(\theta)$ fixes $I$, and
$x^{\mathbf{a}}\in I$ if and only if $\delta(x^{\mathbf{a}-\mathbf{d}})\neq
0$.

  2. (2)

Let $J=\langle
x^{\mathbf{a}}\mid\mathbf{a}\in\bigcup\operatorname{int}P_{i}\rangle$ and
$f=\sum_{i=1}^{n}\prod_{\tau\in\mathcal{S}_{i}}(g_{\tau}(\theta),\lfloor
M_{\tau}-g_{\tau}(\mathbf{d})\rfloor)!$. For any
$h\in(H_{\mathbf{d}})\cap(f)$, $\delta=x^{\mathbf{d}}h(\theta)$ fixes $J$, and
$x^{\mathbf{a}}\in J$ if and only if $\delta(x^{\mathbf{a}-\mathbf{d}})\neq
0$.

###### Proof.

As the proof of both statements are similar, we only provide the proof of the
first statement. We claim that $g_{\tau}(\mathbf{a})\geq 0$ for all
$\tau\in\mathcal{F}$ and $\mathbf{a}\in S$. Let
$\mathbf{b}\in\operatorname{Exp}I$. For any $m\in\mathbb{N}$,
$\mathbf{b}+m\mathbf{a}\in\operatorname{Exp}I\subseteq P$, and so
$M_{\tau}\leq
g_{\tau}(\mathbf{b}+m\mathbf{a})=g_{\tau}(\mathbf{b})+mg_{\tau}(\mathbf{a})$.
Therefore $g_{\tau}(\mathbf{a})\geq(M_{\tau}-g_{\tau}(\mathbf{b}))/m$ for all
$m\in\mathbb{N}$, and so $g_{\tau}(\mathbf{a})\geq 0$.

Therefore, for any face $\tau$, we have that $(g_{\tau}(\theta),\lceil
M_{\tau}-g_{\tau}(\mathbf{d})\rceil-1)!(\mathbf{a})\geq 0$. Hence
$f(\mathbf{a})\neq 0$ if and only if there exists $1\leq i\leq n$ such that
$(g_{\tau}(\mathbf{a}),\lceil M_{\tau}-g_{\tau}(\mathbf{d})\rceil-1)!>0$ for
all $\tau\in\mathcal{S}_{i}$. This occurs exactly when
$g_{\tau}(\mathbf{a})>\lceil M_{\tau}-g_{\tau}(\mathbf{d})\rceil-1$, which is
equivalent to $g_{\tau}(\mathbf{a}+\mathbf{d})\geq M_{\tau}$. Thus,
$f(\mathbf{a})\neq 0$ if and only if there exists $i$ such that
$g_{\tau}(\mathbf{a}+\mathbf{d})\in P_{i}$, i.e.
$\mathbf{a}+\mathbf{d}\in\bigcup_{i}P_{i}=\operatorname{Exp}I$. Therefore
$V_{\textnormal{mon}}(f)\cap\operatorname{Exp}I=\operatorname{Exp}I\setminus(\operatorname{Exp}I-\mathbf{d})$,
and so $\delta$ fixes $I$ by Theorem 3.4. Also, we have that
$f(\mathbf{a}-\mathbf{d})=0$ for $\mathbf{a}\notin\operatorname{Exp}I$, which
proves the last statement as well. ∎

######  Remark 5.4.

Theorem 3.6 is a special case of Theorem 5.3. This comes from the fact that
every monomial ideal $I$ is of the form in the hypothesis of 5.3. Precisely,
if $I=(x^{\mathbf{a}_{1}},\ldots,x^{\mathbf{a}_{n}})$, we may set
$P_{i}=\mathbf{a}_{i}+\sigma^{\vee}$ for each $i$. Now each $P_{i}$ is a
polyhedron, and $I=(x^{\mathbf{a}}\mid\mathbf{a}\in\bigcup_{i}P_{i})$. The
improvement in Theorem 5.3 is the ability to handle ideals which can be
defined by polyhedra other than shifted cones, e.g. integrally closed ideals,
symbolic powers, and multiplier ideals. See, for instance, Theorem 5.9 below.

The power in the above theorem comes from the ability to compute face data of
polyhedra after fundamental operations. Specifically, for $\mathbf{d}$ a
vector, $P_{1}$ and $P_{2}$ polyhedra, four of the fundamental operations are
the following operations.

Translating | $P_{1}+\mathbf{d}$
---|---
Scaling | $cP_{1}$
Intersection | $P_{1}\cap P_{2}$
Minkowski sum |  $P_{1}+P_{2}$.

Setting $I_{1}$ and $I_{2}$ the ideals associated to $P_{1}$ and $P_{2}$, the
combinatorial geometric operations on polyhedra of course produce a polyhedron
whose associated ideal arises from expected ideal constructions, specifically,
the ideal associated to $P_{1}+P_{2}$ is the ideal product $I_{1}I_{2}$.

######  Remark 5.5.

All fundamental operations described trivially preserve the
$\sigma^{\vee}$-closed conditions.

One should consider unions as a fundamental operation, but as defined the
union of polyhedra may fail to be a polyhedra and taking the convex hull is a
lossy process in that working this way would not allow us to distinguish
between ideals and their integral closures. As such, the statement of Theorem
5.3 assumes polyhedral input but works to describe the ideal associated to the
union so this operation is not lost in our application.

Note, in total generality, computing face data of the output of some of these
fundamental operations from the face data of its inputs can be computationally
difficult, notably NP-hard even in more restrictive settings than our general
polyhedra [Tiw08]. However, we quickly review some easy and rather explicit
computations in some specialized settings sufficient for our applications.

######  Lemma 5.6.

Fix $P_{1}$ and $P_{2}$ $\sigma^{\vee}$-closed polyhedra with face data
$\\{(g_{\tau},M_{\tau})\\}_{\tau\in\mathcal{S}_{i}}$ for $i=1,2$, $\mathbf{d}$
a vector, and $c>0$.

  1. (1)

$P_{1}+\mathbf{d}$ is defined by face data
$\\{(g_{\tau},M_{\tau}+g_{\tau}(\mathbf{d}))\\}_{\tau\in\mathcal{S}_{1}}$

  2. (2)

$cP_{1}$ is defined by face data
$\\{(g_{\tau},cM_{\tau})\\}_{\tau\in\mathcal{S}_{1}}$

  3. (3)

$P_{1}\cap P_{2}$ is defined by face data
$\\{(g_{\tau},M_{\tau})\\}_{\tau\in\mathcal{S}_{1}\cup\mathcal{S}_{2}}$

  4. (4)

$P_{1}+P_{2}$ is defined by face data
$\\{(g_{\tau},N_{\tau})\\}_{\tau\in\mathcal{S}_{1}\cup\mathcal{S}_{2}}$ for
some $N_{\tau}\in\mathbb{R}$. In the two-dimensional case, for each
$\tau\in\mathcal{S}_{i}$, we have that
$N_{\tau}=M_{\tau}+g_{\tau}(\mathbf{a}_{\tau})$, where $\mathbf{a}_{\tau}$ is
a vertex of $P_{3-i}$ such that line through $a_{\tau}$ parallel to $\tau$
does not intersect the interior of $P_{3-i}$.

###### Proof.

(1) We have that $\mathbf{v}\in P_{1}+\mathbf{d}$ if and only if
$\mathbf{v}-\mathbf{d}\in P_{1}$, which occurs exactly when
$g_{\tau}(\mathbf{v}-\mathbf{d})\geq M_{\tau}$ for all
$\tau\in\mathcal{S}_{1}$. Therefore $\mathbf{v}\in P_{1}+\mathbf{d}$ if and
only if $g_{\tau}(\mathbf{v})\geq M_{\tau}+g_{\tau}(\mathbf{d})$ for all
$\tau\in\mathcal{S}_{1}$.

(2) We have that $\mathbf{v}\in cP_{1}$ if and only if $(1/c)\mathbf{v}\in
P_{1}$, which occurs exactly when $g_{\tau}((1/c)\mathbf{v})\geq M_{\tau}$ for
all $\tau\in\mathcal{S}_{1}$. Therefore $\mathbf{v}\in cP_{1}$ if and only if
$g_{\tau}(\mathbf{v})\geq cM_{\tau}$ for all $\tau\in\mathcal{S}_{1}$.

(3) We have that $\mathbf{v}\in P_{1}\cap P_{2}$ if and only if $\mathbf{v}$
is in both $P_{1}$ and $P_{2}$, i.e. if $g_{\tau}(\mathbf{v})\geq M_{\tau}$
for all $\tau\in\mathcal{S}_{1}\cup\mathcal{S}_{2}$.

(4) The result is well-known, and algorithms for computing the numbers
$N_{\tau}$ can be found in [dBCvKO08, Ch. 13]. However, there are some
technical adaptations to derive the stated claim. Notably, we are working with
$\sigma^{\vee}$-closed regions which are not compact and also any two
polyhedra have parallel faces.

We note the selection criteria for which $\mathbf{a}_{\tau}$ is chosen is
simply done by linear programming. Thus, for each face
$\tau\in\mathcal{S}_{1}$, there is a face $\tau^{\prime}$ of $P_{1}+P_{2}$
which is defined by $\tau^{\prime}=\tau+\mathbf{a}_{\tau}$, where
$\mathbf{a}_{\tau}$ is a vertex of $P_{2}$ which minimizes $g_{\tau}$, and
similarly for every $\tau\in\mathcal{S}_{2}$, there is a face
$\tau^{\prime}=\tau+\mathbf{a}_{\tau}$, where $\mathbf{a}_{\tau}$ is a vertex
of $P_{1}$. A vector $\mathbf{v}$ is in the half-plane defined by
$\tau^{\prime}$ if and only if $\mathbf{v}-\mathbf{a}_{\tau}$ is in the half-
plane defined by $\tau$, which occurs exactly when
$g_{\tau}(\mathbf{v}-\mathbf{a}_{\tau})\geq M_{\tau}$, implying
$g_{\tau}(\mathbf{v})\geq M_{\tau}+g_{\tau}(\mathbf{a}_{\tau})$. This proves
the last statement of (4). ∎

######  Remark 5.7.

The primary consequence of Lemma 5.6 is that combined with Theorem 5.3 gives
an algorithm for computing face data of many common polyhedra and the
differential operators fixing the associated monomial ideals.

###  5.1. Symbolic powers and Integral closure of powers

We now turn out attention to sharper results in the case of ideal membership
problems for the integral closure of powers

$\overline{I^{n}}=\\{x\in R\mid\text{there exists }n\in\mathbb{N}\text{ and
}a_{i}\in I^{ni}\text{ for }1\leq i\leq n\text{ such that
}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}=0\\}$

and the symbolic power $I^{(n)}=\bigcap\limits_{\mathfrak{p}\in{\rm
Ass}(I)}I^{n}R_{\mathfrak{p}}\cap R$. For monomial ideals $I$, there are
important descriptions of these in terms of a polyhedra.

######  Definition 5.8.

Let $R=\mathbb{C}[S]$ be an affine semigroup ring defined by a cone
$\sigma^{\vee}$ and $I\subseteq R$ a proper monomial ideal with primary
decomposition $I=Q_{1}\cap\cdots Q_{r}$. For a prime $P\in{{\rm maxAss}(I)}$
set $Q_{\subseteq P}=IR_{P}\cap R$ (the intersection of the $Q_{i}\subseteq
P$).

  1. (1)

The Newton polyhedron of $I$ is $\operatorname{Newt}{(I)}={\rm convex\
hull}\langle\operatorname{Exp}{(I)}\rangle$.

  2. (2)

The symbolic polyhedron of $I$ is ${\rm
SP}(I)=\bigcap\limits_{\mathfrak{p}\in{\rm
maxAss}(I)}\operatorname{Newt}{(Q_{\subseteq P})}$.

######  Theorem 5.9.

Let $R=\mathbb{C}[S]$ be an affine semigroup ring defined by a cone
$\sigma^{\vee}$ and $I\subseteq R$ a proper monomial ideal.

  1. (1)

[DFMS19, Proposition 2.7] $x^{\mathbf{d}}\in\overline{I^{n}}$ if and only if
$\displaystyle\frac{\mathbf{d}}{n}\in\operatorname{Newt}{(I)}$.

  2. (2)

[CEHH17, Theorem 5.4] If $x^{\mathbf{d}}\in I^{(n)}$, then
$\displaystyle\frac{\mathbf{d}}{n}\in{\rm SP}{(I)}$.

  3. (3)

[DFMS19, Proposition 2.10] If we assume $S=\mathbb{N}^{d}$ for some $d$, i.e.
$R$ is a polynomial ring and $I$ is a squarefree monomial ideal, then
$x^{\mathbf{d}}\in I^{(n)}$ if and only if
$\displaystyle\frac{\mathbf{d}}{n}\in{\rm SP}{(I)}$.

######  Remark 5.10.

Teissier notes in the exercises of [Tei02, Section 3] that the proof of (1)
above also holds for any normal affine semigroup ring. For (2), the proof
provided in [CEHH17, Theorem 5.4] goes through for normal affine semigroup
rings. For (3), we believe that the generalization from $S=\mathbb{N}^{d}$ to
any strongly convex semigroup $S$ is probably true for ideals $I$ which are
intersections of monomial primes. While we don’t pursue this in full detail
here, Example 5.12 illustrates a normal affine semigroup ring and an ideal $I$
which is an intersection of all height 2 primes and it is clear that $n{\rm
SP}(I)=\operatorname{Exp}{(I^{(n)})}.$

######  Remark 5.11.

Theorem 5.9 (3) also holds for a wider class of ideals as noted in [HN21,
Remark 2.6], namely when $I$ is of linear-power type, specifically for those
ideals $I=\bigcap\limits_{\mathfrak{p}\in{\rm
Min}(I)}{\mathfrak{p}}^{\omega_{\mathfrak{p}}}$ for some
$\omega_{\mathfrak{p}}\in\mathbb{N}$ and all minimal primes are monomial
primes.

######  Example 5.12.

Let $R=\mathbb{C}[x,y,z,xyz^{-1}]$ and

$I=(xy,xz,yz,x^{2}yz^{-1},xy^{2}z^{-1})=(x,y,z)\cap(y,z,xyz^{-1})\cap(x,y,xyz^{-1})\cap(x,z,xyz^{-1}).$

(The intersection of the height 2 primes of $R$; the equivalent of a square-
free monomial ideal in a polynomial ring.) The facets of $R$ are

$\sigma_{1}=\\{(x,y,z)\mid y=0\\},\sigma_{2}=\\{(x,y,z)\mid
y+z=0\\},\sigma_{3}=\\{(x,y,z)\mid x+z=0\\},\text{ and
}\sigma_{4}=\\{(x,y,z)\mid x=0\\},$

and the polyhedra defining the primes are

$\displaystyle P_{1}$ $\displaystyle=\\{(x,y,z)\mid x+y\geq 1,y\geq 0,y+z\geq
0,x+z\geq 0,x\geq 0\\},$ $\displaystyle P_{2}$ $\displaystyle=\\{(x,y,z)\mid
2y+z\geq 1,y\geq 0,y+z\geq 0,x+z\geq 0,x\geq 0\\},$ $\displaystyle P_{3}$
$\displaystyle=\\{(x,y,z)\mid x+y+2z\geq 1,y\geq 0,y+z\geq 0,x+z\geq 0,x\geq
0\\},$ $\displaystyle P_{4}$ $\displaystyle=\\{(x,y,z)\mid 2x+z\geq 1,y\geq
0,y+z\geq 0,x+z\geq 0,x\geq 0\\}.$

The symbolic polyhedron

$\displaystyle{\rm SP}(I)$ $\displaystyle=P_{1}\cap P_{2}\cap P_{3}\cap P_{4}$
$\displaystyle=\\{(x,y,z)\mid x\geq 0,y\geq 0,x+z\geq 0,y+z\geq 0,x+y\geq
1,2y+z\geq 1,x+y+2z\geq 1\text{ and }2x+z\geq 1\\}$

is a polyhedron with vertex $(1/2,1/2,0)$. It is easy to see that
$I^{(2)}=(xy,x^{2}z^{2},y^{2}z^{2},x^{4}y^{2}z^{-2},x^{2}y^{4},z^{-2})$ is the
ideal generated by $2S(P)$ giving credence to the fact that Theorem 5.9 (3)
also holds for ideals which are intersections of homogeneous primes defined by
the faces of a normal affine semigroup ring. For an example of a differential
operator fixing $2{\rm SP}(I)=\operatorname{Exp}{(I^{(2)})}$, consider
$\mathbf{d}=(-1,-1,0)$; then
$H_{\mathbf{d}}=\theta_{x}\theta_{y}(\theta_{x}+\theta_{z})(\theta_{y}+\theta_{z})$
and

$f=\theta_{x}\theta_{y}(\theta_{x}+\theta_{z})(\theta_{y}+\theta_{z})(\theta_{x}+\theta_{y},3)!(2\theta_{x}+\theta_{z},3)!(2\theta_{y}+\theta_{z},3)!(\theta_{x}+\theta_{y}+2\theta_{z},3)!$

and $\delta=x^{-1}y^{-1}f$ fixes $I^{(2)}$.

$x$$z$$y$$xyz^{-1}$

$x^{2}$$z^{2}$$y^{2}$$x^{2}y^{2}z^{-2}$

$x^{3}$$z^{3}$$y^{3}$$x^{3}y^{3}z^{-3}$

$x^{4}$$z^{4}$$y^{4}$$x^{4}y^{4}z^{-4}$

$x^{5}$$z^{5}$$y^{5}$$x^{5}y^{5}z^{-5}$

Figure 3. Cross-sections of multiples of the symbolic polyhedra of $I$

In Figure 3, we illustrate cross-sections of the lattice points of the cone
defining $R$. The lattice points in grey represent the monomials which are in
$I\setminus I^{(2)}$; those in blue represent the monomials which are in
$I^{(2)}\setminus I^{(3)}$; those in teal represent the monomials which are in
$I^{(3)}\setminus I^{(4)}$; and those in violet represent the the monomials
which are in $I^{(4)}$. We have included the vertices of the symbolic
polyhedra ${\rm SP}(I)$ and $3{\rm SP}(I)$ in gray and teal respectively
although they do not correspond to monomials.

######  Example 5.13.

Let $R=\mathbb{C}[x,y,z]$ and
$I=(x^{2}z,xy^{2},yz^{2},xyz)=(x,z^{2})\cap(x^{2},y)\cap(y^{2},z)$. The
minimal (and maximal) primes are $P_{1}=(x,z),P_{2}=(x,y)$ and $P_{3}=(y,z)$.
Set $Q_{1}=(x,z^{2}),Q_{2}=(x^{2},y)$ and $Q_{3}=(y^{2},z)$. Then
$Q_{\subseteq P_{i}}=Q_{i}$ and the faces of the Newton Polyhedra of $Q_{i}$
are the following:

$P_{1}=\\{(x,y,z)\mid 2x+z\geq 2,x\geq 0,y\geq 0\text{ and }z\geq 0\\},$
$P_{2}=\\{(x,y,z)\mid 2y+x\geq 2,x\geq 0,y\geq 0\text{ and }z\geq 0\\},$
$P_{3}=\\{(x,y,z)\mid 2y+z\geq 2,x\geq 0,y\geq 0\text{ and }z\geq 0\\}.$

Thus,

${\rm SP}(I)=P_{1}\cap P_{2}\cap P_{3}$

which is a polyhedral cone with vertex $(2/3,2/3,2/3)$ with facets

$\tau_{1}=\\{(x,y,z)\mid 2x+z=2\\},\tau_{2}=\\{(x,y,z)\mid 2y+x=2\\},\text{
and }\tau_{3}=\\{(x,y,z)\mid 2z+y=2\\}.$

In [CEHH17, Example 4.6], the authors note that $x^{2}y^{2}z^{2}\in I^{(3)}$
and Theorem 5.9 (2) in fact verifies that the vertex of the cone above lies in
${\rm SP}(I)$. In fact, it is also easy to verity that $x^{2n}y^{2n}z^{2n}\in
P^{(3n)}$.

Applying Theorem 5.3, we will exhibit a differential operator which fixes the
ideal generated by ${\rm SP}(I)$. Note that the facet data for ${\rm SP}(I)$
is $\\{(2x+z,2),(2y+x,2),(2z+y,2)\\}$. Set $\mathbf{d}=(-1,-1,-1)$ and note
that $H_{\mathbf{d}}=\theta_{x}\theta_{y}\theta_{z}$ and
$f=(2\theta_{x}+\theta_{z},4)!(2\theta_{y}+\theta_{x},4)!(2\theta_{z}+\theta_{y},4)!$.
Since $H_{\mathbf{d}}f\in(H_{\mathbf{d}})\cap(f)$, then
$\delta=x^{\mathbf{d}}H_{\mathbf{d}}f$ fixes the ideal generated by ${\rm
SP}(I)$.

######  Remark 5.14.

In the above example, we can replace 2 by any natural number $n$ and the facet
data for ${\rm SP}(I_{n})$ for
$I_{n}=(x^{n}z,xy^{n},yz^{n},xyz)=(x,z^{n})\cap(x^{n},y)\cap(y^{n},z)$ will
instead be $\\{(nx+z,n),(ny+x,n),(nz+y,n)$ and ${\rm SP}(I)$ will be a cone
with vertex $(\frac{n}{n+1},\frac{n}{n+1},\frac{n}{n+1})\\}$. It is
interesting to note that as $n$ gets large the vertices of these cones are
approaching $(1,1,1)$. Of course, the ideals $I_{n}$ are just nilpotent
thickenings of $I_{1}=(x,y)\cap(x,z)\cap(y,z)=(xy,xz,yz)$. As $n$ gets large,
the symbolic polyhedron of these thickenings is approaching the cone of the
ideal $J=(xyz)=(x)\cap(y)\cap(z)$, the intersection of height 1 primes.

Similarly, when $a<b$ and
$I=(x^{a}z^{b},x^{b}y^{a},y^{b}z^{a},x^{a}y^{a}z^{a})=(x^{b},z^{a})\cap(x^{a},y^{b})\cap(y^{a},z^{b})$,
the facet data for ${\rm SP}(I)$ will instead be
$\\{(ax+bz,ab),(ay+bx,ab),(az+by,ab)$ and ${\rm SP}(I)$ will be a cone with
vertex $(\frac{ab}{a+b},\frac{ab}{a+b},\frac{ab}{a+b})\\}$. Again these ideals
are thickenings of $J=(x^{a}y^{a},x^{a}z^{a},y^{a},z^{a})$ and their symbolic
polyhedra again are converging toward the cone of the ideal
$(x^{a}y^{a}z^{a})$.

######  Example 5.15.

The vertex of the cone need not be a multiple of $(1,1,1)$. Consider

$I=(x^{a}y^{b},x^{a}z^{c},y^{b}z^{c})=(x^{a},z^{c})\cap(x^{a},y^{b})\cap(y^{b},z^{c}).$

The polyhedra defining the primary components are

$P_{1}=\\{(x,y,z)\mid cx+az\geq ac,x\geq 0,y\geq 0,\text{ and }z\geq 0\\},$
$P_{2}=\\{(x,y,z)\mid bx+ay\geq ab,x\geq 0,y\geq 0,\text{ and }z\geq 0\\},$
$P_{3}=\\{(x,y,z)\mid cy+bz\geq bc,x\geq 0,y\geq 0,\text{ and }z\geq 0\\}.$

Thus,

${\rm SP}(I)=P_{1}\cap P_{2}\cap P_{3}$

which is a polyhedral cone with vertex $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$
with facets

$\tau_{1}=\\{(x,y,z)\mid cx+az=ac\\},\tau_{2}=\\{(x,y,z)\mid
bx+ay=ab\\},\text{ and }\tau_{3}=\\{(x,y,z)\mid cy+bz=bc\\}.$

It is easy to check that $x^{a}y^{b}z^{c}\in I^{(2)}\setminus I^{2}$ and
$\frac{1}{2}(a,b,c)\in{\rm SP}(I)$ as in Theorem 5.9(b).

Applying Theorem 5.3, we will exhibit a differential operator which fixes the
ideal generated by $2{\rm SP}(I)$. Note that this ideal contains $I^{(2)}$ but
is not $I^{(2)}$ since $x^{2a-1}y^{b}z$ is one example of an element in $2{\rm
SP}(I)\setminus\operatorname{Exp}{(I^{(2)})}$. Note that the facet data for
$2{\rm SP}(I)$ is $\\{(cx+az,2ac),(bx+ay,2ab),(cy+bz,2bc)\\}$. Set
$\mathbf{d}=(-1,-1,-1)$ and note that
$H_{\mathbf{d}}=\theta_{x}\theta_{y}\theta_{z}$ and
$f=(c\theta_{x}+a\theta_{z},2ac+a+c-1)!(b\theta_{x}+a\theta_{y},2ab+a+b-1)!(c\theta_{y}+b\theta_{z},2bc+b+c-1)!$.
Since $H_{\mathbf{d}}f\in(H_{\mathbf{d}})\cap(f)$, then
$\delta=x^{\mathbf{d}}H_{\mathbf{d}}f$ fixes the ideal generated by $2{\rm
SP}(I)$.

###  5.2. Multiplier ideals and jumping numbers

We now turn to sharper applications of Theorem 5.3 towards multiplier ideals
and their variants. In the general framework, if $X$ is a normal variety,
$\Delta$ a $\mathbb{Q}$-divisor so that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier
where $K_{X}$ is a canonical divisor, $\mathcal{I}$ an ideal sheaf on $X$ and
$c\geq 0$ a real number, the multiplier ideals $J(X,\Delta,\mathcal{I}^{c})$
are defined in terms of a log resolution of this data. Famously, on toric
varieties, so $X=\operatorname{Spec}\mathbb{C}[S]$, there is a canonical torus
invariant choice for $K_{X}$. Moreover, choosing $\Delta$ torus invariant then
there is a multiple $r$ and an exponent $\mathbf{u}$ so that $\textrm{div
}x^{\mathbf{u}}=r(K_{X}+\Delta)$, whence to each $\Delta$ one can associate a
vector $\mathbf{w}=\mathbf{u}/r$. The exponent associated to $\Delta=0$ is
$\mathbf{w}=(1,1,\ldots,1)$. Finally, for $\mathcal{I}$ defined by a monomial
ideal $I$, each $J(X,\Delta,\mathcal{I}^{c})$ is a monomial ideal and we have
the following explicit description of the multiplier ideals [Bli04, How01].

######  Theorem 5.16 (Blickle-Howald).

The monomial ideal

$J(X,\Delta,\mathcal{I}^{c})=\langle x^{\mathbf{a}}\in
R\colon\mathbf{a}+\mathbf{w}\in
c\cdot\operatorname{int}\operatorname{Newt}I\rangle$

where $\operatorname{int}\operatorname{Newt}I$ denotes the relative interior.

From here on, we’ll use the notation $J(X,\mathbf{w},I^{c})$ instead of
$J(X,\Delta,\mathcal{I}^{c})$ for this ideal or even just $J(\mathbf{w},c)$
when $X$ and $I$ are clear. As is well-known, as the value of $c$ gets larger,
the multiplier ideals $J(X,\mathbf{w},I^{c})$ form a decreasing family of
ideals and values where $J(X,\mathbf{w},I^{c-\epsilon})\neq
J(X,\mathbf{w},I^{c})$ for all $\epsilon>0$ are called jumping numbers for the
triple $(X,\Delta,\mathfrak{a})$. The smallest jumping number is the log
canonical threshold which plays a most critical role in the study of
singularities. Detecting when a value $c$ is the log canonical threshold
therefore comes down to determining if the monomial ideal
$J(X,\mathbf{w},I^{c})$ is proper or not. Our applications will characterize
the jumping numbers $c$ of a monomial ideal $I$ in terms of the differential
operators that fix $I$. The following simple example demonstrates the idea.

######  Example 5.17.

Let $R=\mathbb{C}[x,y]$ and $I=(x^{2},y^{3})$. Let $\mathbf{w}=(1,1)$. Define

$J(\mathbf{w},c)=\\{x^{\mathbf{v}}\mid\mathbf{v}+\mathbf{w}\in
c\cdot\operatorname{int}{(\operatorname{Newt}(I))}\\}.$

Solving for $t$ and $u$, we see that $t=\frac{1}{2}$ and $u=\frac{1}{3}$,
making $c<\frac{5}{6}$.

(2c,0)(0,3c)(2c-1,-1)(1,3c-1) Figure 4. Log canonical threshold

We want to visualize this threshold $c$ through differential operators.

(2c,0)(0,3c) Figure 5. LCT/Differential operator

To determine which values $c$ satisfy
$\mathbf{0}\in\operatorname{int}{(\operatorname{Exp}{J(\mathbf{w},c)})}$, note
that

$(1,1)=t(2,0)+u(0,3)$

for $t,u\geq 0$, $t+u>c$. See Figure 5. Let
$f_{c}(\theta)=H_{(-1,-1)}(\theta)\prod\limits_{i=0}^{\lfloor
6c\rfloor}(3\theta_{x}-2\theta_{y}-i)$. $\delta=x^{-1}y^{-1}f_{c}(\theta)$
fixes $J_{c}=R$ if and only if $(1,1)\notin V(f_{c})$. Note that
$f_{c}(1,1)=5\ldots(5-\lfloor 6c\rfloor)\neq 0$ if $c<5/6$.

Thus a direct application of Theorem 5.3 for the shifted Newton polygon gives
the following explicit manner of computing multiplier ideals and their jumping
numbers from differential operators.

######  Corollary 5.18.

Let $\mathbf{w}\in S$, $c\geq 0$ a real number, and
$J(X,\mathbf{w},I^{c})=\langle x^{\mathbf{a}}\mid\mathbf{a}+\mathbf{w}\in
c\cdot\operatorname{int}\operatorname{Newt}I\rangle$. Set
$(g_{\tau},M_{\tau})_{\tau\in\mathcal{F}}$ a face data for
$\operatorname{Newt}I$.

  1. (1)

The differential operator
$\delta_{c}=x^{-\mathbf{w}}H_{-\mathbf{w}}(\theta)\prod_{\tau\in\mathcal{F}}(g_{\tau}(\theta),\lfloor
cM_{\tau}\rfloor)!$ fixes $J_{c}$.

  2. (2)

The jumping numbers of $I$ are exactly
$\left\\{\min_{\tau\in\mathcal{F}}\left\\{\frac{1}{M_{\tau}}g_{\tau}(\mathbf{a}+\mathbf{w})\right\\}\mid\mathbf{a}\in
S\right\\}$.

  3. (3)

The log canonical threshold of $I$ is
$\min_{\tau\in\mathcal{F}}\left\\{\frac{1}{M_{\tau}}g_{\tau}(\mathbf{w})\right\\}$.

###### Proof.

The first claim holds directly by Theorem 5.3. A monomial $x^{\mathbf{a}}$ is
in $J_{c}$ if and only if $\delta_{c}(x^{\mathbf{a}+\mathbf{w}})\neq 0$, which
occurs if and only if $f_{c}(\mathbf{a}+\mathbf{w})\neq 0$. This happens
exactly when for all $\tau\in\mathcal{F}$,
$g_{\tau}(\mathbf{a}+\mathbf{w})>cM_{\tau}$, i.e. when
$c<\frac{1}{M_{\tau}}g_{\tau}(\mathbf{a}+\mathbf{w})$. In particular, this
means that
$\min\left\\{\frac{1}{M_{\tau}}g_{\tau}(\mathbf{a}+\mathbf{w})\mid\tau\in\mathcal{F}\right\\}$
is a jumping number for $I$. ∎

######  Example 5.19.

Let $R=\mathbb{C}[x,xy,xy^{2}]$ and $I=(x^{4},x^{2}y,x^{3}y^{6})$. For
$\mathbf{w}\in S$, set

$J_{c}=\\{x^{\mathbf{v}}\in R\mid\mathbf{v}+\mathbf{w}\in
c\operatorname{int}(\operatorname{Newt}I)\\}.$

Let us assume that $\mathbf{w}=(1,1)$. The linear forms associated to the two
faces $\tau_{1}$ and $\tau_{2}$ that bound the convex hull of $I$ are
$g_{\tau_{1}}:=x+2y\text{ and }g_{\tau_{2}}:=5x-y$ and it is easy to see that
$M_{\tau_{1}}=4$ and $M_{\tau_{2}}=9$. We define

$\delta_{c}=x^{-1}y^{-1}H_{(-1,-1)}(\theta)(g_{\tau_{1}}(\theta),\lfloor
4c\rfloor)!(g_{\tau_{2}}(\theta),\lfloor 9c\rfloor)!.$

By Corollary 5.18(2), the jumping numbers of $I$ with respect to the divisor
$0$ are

$\left\\{\text{min}\left\\{\frac{g_{\tau_{1}}(\mathbf{a}+(1,1))}{4},\frac{g_{\tau_{2}}(\mathbf{a}+(1,1))}{9}\right\\}\mid\mathbf{a}\in
S\right\\}.$

The table in Figure 6 illustrates the jumping numbers in ascending order in
blue. Note that the jumping numbers can be determined by $\tau_{1}$ or
$\tau_{2}$. The illustration in Figure 6 illustrates the boundaries of
$c\operatorname{int}{(\operatorname{Newt}{I})}$ for the various jumping
numbers in the table.

$\mathbf{a}$$\displaystyle\frac{g_{\tau_{1}}(\mathbf{a}+(1,1))}{4}$$\displaystyle\frac{g_{\tau_{2}}(\mathbf{a}+(1,1))}{9}$(0,0)3/44/9(1,2)27/9(1,1)3/28/9(1,0)11(2,4)13/410/9(2,3)11/411/9(2,0)5/414/9(2,2)9/412/9(3,6)15/413/9(3,5)13/414/9

Figure 6. Jumping numbers

Looking at Figure 6, we see that the ideals $J_{c}$ for the jumping numbers
given are:

$J_{c}=\begin{cases}R&\text{ if }c<4/9\\\ \mathfrak{m}&\text{ if }4/9\leq
c<7/9\\\ (x,xy,x^{2}y^{4})&\text{ if }7/9\leq c<8/9\\\
(x,x^{2}y^{3},x^{2}y^{4})&\text{ if }8/9\leq c<1\\\ \mathfrak{m}^{2}&\text{ if
}1\leq c<10/9\\\ (x^{2},x^{2}y,x^{2}y^{2},x^{2}y^{3},x^{3}y^{6})&\text{ if
}10/9\leq c<11/9\\\ (x^{2},x^{2}y,x^{2}y^{2},x^{3}y^{5},x^{3}y^{6})&\text{ if
}11/9\leq c<5/4\\\ (x^{3},x^{2}y,x^{2}y^{2},x^{3}y^{5},x^{3}y^{6})&\text{ if
}5/4\leq c<12/9\\\ (x^{3},x^{2}y,x^{3}y^{4},x^{3}y^{5},x^{3}y^{6})&\text{ if
}12/9\leq c<13/9\\\
(x^{3},x^{3}y,x^{3}y^{2},x^{3}y^{3},x^{3}y^{4},x^{3}y^{5},x^{4}y^{8})&\text{
if }13/9\leq c<14/9\\\ \end{cases}$

Often in the study of multiplier ideals, natural useful generalizations emerge
and our methods apply in much more generally to these settings. Indeed, many
variants of multiplier ideals, such a mixed multiplier ideals [Laz04, Gen.
9.2.8], Fujino’s non-LC ideals, and many of the intermediate adjoint ideals,
as well as mixed versions, introduced in [HSZ14, Sec. 4,5] all enjoy variants
of the Blickle-Howald theorem. As such, we articulate this more general class
as follows.

######  Definition 5.20.

Let $I_{1},\ldots,I_{n}$ be monomial ideals, $\mathbf{c}\in\mathbb{R}_{\geq
0}^{n}$, $\mathbf{w}\in\mathbb{Z}^{d}$. A Blickle-Howald ideal is an ideal of
the form

$\displaystyle J_{I_{1},\ldots,I_{n}}(\mathbf{w},\mathbf{c})$
$\displaystyle:=$ $\displaystyle\left\langle
x^{\mathbf{a}}\mid\mathbf{a}+\mathbf{w}\in\sum_{j=1}^{n}c_{j}\operatorname{Newt}I_{j}\right\rangle\textrm{
or }$ $\displaystyle J^{\circ}_{I_{1},\ldots,I_{n}}(\mathbf{w},\mathbf{c})$
$\displaystyle:=$ $\displaystyle\left\langle
x^{\mathbf{a}}\mid\mathbf{a}+\mathbf{w}\in\sum_{j={1}}^{n}c_{j}\operatorname{int}\operatorname{Newt}I_{j}\right\rangle.$

######  Example 5.21.

A few natural examples are the mixed multiplier ideals already mentioned
$J(X,\Delta,I_{1}^{a_{1}}\cdots I_{n}^{a_{n}})$ which by definition is the
ideal $J_{I_{1},\ldots,I_{n}}^{\circ}(\mathbf{w},\mathbf{a})$ where
$\mathbf{w}$ corresponds to $\Delta$ and $\mathbf{a}=(a_{1},\ldots,a_{n})$.

######  Example 5.22.

For fixed ideals $I_{1}$ and $I_{2}$, the ideal
$J_{I_{1},I_{2}}(\mathbf{w},(c_{1},c_{2}))=\left\langle
x^{\mathbf{a}}\mid\mathbf{a}+\mathbf{w}\in
c_{1}\operatorname{Newt}I_{1}+c_{2}\operatorname{Newt}I_{2}\right\rangle$ is
utilized in [HSZ14, Prop. 5.3].

We refer to BH ideals for Blickle-Howald ideals and when the ideals
$I_{1},\ldots,I_{n}$ and the vector $\mathbf{w}$ they can be suppressed in the
notation. As the input data for BH-ideals is more complicated than that for
multiplier ideals, instead of having simple jumping numbers, BH ideals enjoy
various constancy regions similar to the mixed multiplier ideals, see [LM11]
and [ACAMDC18, ACAMDCGA20] for more in that latter case. We summarize the
relevant terminology here however the basic results on BH-ideals follow
immediately from the methods of [ACAMDC18].

######  Remark 5.23.

It is immediate to verify the following from the definition. Set
$I_{1},\ldots,I_{n}$ be monomial ideals, $\mathbf{c}\in\mathbb{R}_{\geq
0}^{n}$, $\mathbf{w}\in\mathbb{Z}^{d}$. For any
$\mathbf{c}^{\prime}\in\mathbf{c}+\mathbb{R}_{\geq 0}^{n}$,
$J^{\circ}(\mathbf{w},\mathbf{c})\supseteq
J^{\circ}(\mathbf{w},\mathbf{c}^{\prime})$ and
$J^{\circ}(\mathbf{w},\mathbf{c})=J^{\circ}(\mathbf{w},\mathbf{c}^{\prime})$
for $\mathbf{c}^{\prime}\in(\mathbf{c}+\mathbb{R}_{\geq 0}^{n})\cap
B_{\epsilon}(\mathbf{c})$ for $0<\epsilon\ll 1$.

Fix $I_{1},\ldots,I_{n}$ and $\mathbf{w}$. Again, adapting terminology from
the mixed multiplier ideal setting, we say for $\mathbf{c}$ the locus
$C_{\mathbf{c}}:=\\{\mathbf{c}^{\prime}\in\mathbb{R}_{\geq 0}^{n}\colon
J^{\circ}(\mathbf{w},\mathbf{c}^{\prime})=J^{\circ}(\mathbf{w},\mathbf{c})\\}$
is its consistency region. A jumping point is a value $\mathbf{c}$ so that
$J^{\circ}(\mathbf{w},\mathbf{c}^{\prime})\supsetneq
J^{\circ}(\mathbf{w},\mathbf{c})$ for all
$\mathbf{c}^{\prime}\in\\{\mathbf{c}-\mathbb{R}_{\geq 0}^{n}\\}\cap
B_{\epsilon}(\mathbf{c})$ and $0<\epsilon\ll 1$. A jumping wall is the
boundary of $C_{\mathbf{c}}$.

We are interested in the jumping wall of consistency region where
$J^{\circ}(\mathbf{w},\mathbf{c})=R$ which in the mixed multiplier ideal is
the log-canonical wall. Clearly, the methods combining Theorem 5.3 and Lemma
5.6 combine to extend Corollary 5.18 to many more BH-ideals. We characterize
this region in terms of differential operators in the following special case
which is an immediate consequence.

######  Example 5.24.

Let $R=\mathbb{C}[x,xy,xy^{2}]$, $I_{1}=(x^{8}y,x^{3}y^{3})$, and
$I_{2}=(x^{7}y^{2},x^{4}y^{6})$. We have that $\operatorname{Newt}I_{1}$ is
defined by the face data $\\{(y,1),(2x+5y,21),(2x-y,3)\\}$, and
$\operatorname{Newt}I_{2}$ is defined by the face data
$\\{(y,2),(4x+3y,34),(2x-y,2)\\}$. For any
$\mathbf{c}=(c_{1},c_{2})\in\mathbb{R}_{>0}^{2}$, we have that
$c_{1}\operatorname{Newt}I_{1}$ has face data
$\\{(y,c_{1}),(2x+5y,21c_{1}),(2x-y,3c_{1})\\}$, and
$\operatorname{Newt}I_{2}$ is defined by the face data
$\\{(y,2c_{2}),(4x+3y,34c_{2}),(2x-y,2c_{2})\\}$. In the first graph in Figure
7, we illustrate the Newton Polyhedra of $I_{1}$ and $I_{2}$ in gray and blue
respectively.

Figure 7. $\operatorname{Newt}{(I_{1})}$ with $\operatorname{Newt}{(I_{2})}$
and $c_{1}\operatorname{Newt}{(I_{1})}+c_{2}\operatorname{Newt}{(I_{2})}$ for
$(c_{1},c_{2})\in\\{(\frac{2}{7},\frac{1}{34}),(\frac{1}{5},\frac{1}{5}),(\frac{1}{2},\frac{1}{6}),(\frac{1}{5},\frac{4}{5})\\}$.

Now, by part (4) of Lemma 5.6, we can compute face data of
$Q_{\mathbf{c}}:=c_{1}\operatorname{Newt}I_{1}+c_{2}\operatorname{Newt}I_{2}$.

$\begin{array}[]{ccc}\text{Face of }I_{1}&\text{Corr.\ vertex of
}c_{2}\operatorname{Newt}I_{2}&\text{Face datum of }Q_{\mathbf{c}}\\\
(y,c_{1})&(7c_{2},2c_{2})&(y,c_{1}+2c_{2})\\\
(2x+5y,21c_{1})&(7c_{2},2c_{2})&(2x+5y,21c_{1}+24c_{2})\\\
(2x-y,3c_{1})&(4c_{2},6c_{2})&(2x-y,c_{1}+2c_{2})\\\ \\\ \text{Face of
}I_{2}&\text{Corr.\ vertex of }c_{1}\operatorname{Newt}I_{1}&\text{Face datum
of }Q_{\mathbf{c}}\\\ (y,2c_{2})&(8c_{1},c_{1})&(y,c_{1}+2c_{2})\\\
(4x+3y,34c_{2})&(3c_{1},3c_{1})&(4x+3y,21c_{1}+34c_{2})\\\
(2x-y,2c_{2})&(3c_{1},3c_{1})&(2x-y,3c_{1}+2c_{2})\\\ \end{array}$

Removing the redundant conditions, we have that $Q_{\mathbf{c}}$ is defined by
face data

$\\{(y,c_{1}+2c_{2}),(2x+5y,21c_{1}+24c_{2}),(4x+3y,21c_{1}+34c_{2}),(2x-y,3c_{1}+2c_{2})\\}.$

In second graph in Figure 7, we illustrate the Minkowski sum
$c_{1}\operatorname{Newt}{(I_{1})}+c_{2}\operatorname{Newt}{(I_{2})}$ for
three different pairs $(c_{1},c_{2})$: $(\frac{2}{7},\frac{1}{34})$ in violet,
$(\frac{1}{5},\frac{1}{5})$ in gray, $(\frac{1}{2},\frac{1}{6})$ in blue, and
$(\frac{1}{5},\frac{4}{5})$ in green.

Set $\mathbf{w}=(1,1)$. We consider examples BH-ideals which arise as a mixed
multiplier ideal and Fujino’s non-LC ideal respectively. In particular, set
$J:=J_{I_{1},I_{2}}(\mathbf{w},\mathbf{c})=\left\langle
x^{\mathbf{a}}\mid\mathbf{a}\in P_{\mathbf{c}}\right\rangle$, where
$P_{\mathbf{c}}=Q_{\mathbf{c}}-(1,1)$. Now $P_{\mathbf{c}}$ is defined by face
data

$\\{(y,c_{1}+2c_{2}-1),(2x+5y,21c_{1}+24c_{2}-7),(4x+3y,21c_{1}+34c_{2}-7),(2x-y,3c_{1}+2c_{2}-1)\\}.$

Let $\mathbf{d}=(-2,-1).$ By Theorem 5.3, $J$ is fixed by the differential
operator
$\delta_{\mathbf{c}}=x^{-2}y^{-1}H_{\mathbf{d}}(\theta)f_{\mathbf{c}}(\theta)$,
where

$f_{\mathbf{c}}=(y,\lceil c_{1}+2c_{2}-1\rceil)!\cdot(2x+5y,\lceil
21c_{1}+24c_{2}+1\rceil)!\cdot(4x+3y,\lceil
21c_{1}+34c_{2}+3\rceil)!\cdot(2x-y,\lceil 3c_{1}+2c_{2}+1\rceil)!.$

Also $J^{\circ}:=J^{\circ}(\mathbf{w},\mathbf{c})=\left\langle
x^{\mathbf{a}}\mid\mathbf{a}\in\operatorname{int}P_{\mathbf{c}}\right\rangle$
is fixed by
$\delta^{\circ}_{\mathbf{c}}=x^{-2}y^{-1}H_{\mathbf{d}}(\theta)g_{\mathbf{c}}(\theta)$,
where

$g_{\mathbf{c}}=(y,\lfloor c_{1}+2c_{2}\rfloor)!\cdot(2x+5y,\lfloor
21c_{1}+24c_{2}+2\rfloor)!\cdot(4x+3y,\lfloor
21c_{1}+34c_{2}+4\rfloor)!\cdot(2x-y,\lfloor 3c_{1}+2c_{2}+2\rfloor)!$

Now $J^{\circ}$ is the unit ideal if $1$ is in the image of
$\delta^{\circ}_{\mathbf{c}}$, which occurs exactly when
$\delta^{\circ}_{\mathbf{c}}(x^{2}y)\neq 0$. This happens when

$0\neq g(2,1)=(1,\lfloor c_{1}+2c_{2}\rfloor)!\cdot(9,\lfloor
21c_{1}+24c_{2}+2\rfloor)!\cdot(11,\lfloor
21c_{1}+34c_{2}+4\rfloor)!\cdot(3,\lfloor 3c_{1}+2c_{2}+2\rfloor)!,$

which is equivalent to the system of inequalities

$\displaystyle c_{1}+2c_{2}$ $\displaystyle<1$ $\displaystyle 21c_{1}+24c_{2}$
$\displaystyle<7$ $\displaystyle 21c_{1}+34c_{2}$ $\displaystyle<7$
$\displaystyle 3c_{1}+2c_{2}$ $\displaystyle<1$

The third inequality above implies the other three, so $J^{\circ}$ is the unit
ideal for all $\mathbf{c}=(c_{1},c_{2})$ with $21c_{1}+34c_{2}<7$. A similar
computation shows that $J$ is the unit ideal for all
$\mathbf{c}=(c_{1},c_{2})$ with $21c_{1}+34c_{2}\leq 7$.

More generally, for any monomial $x^{a}y^{b}\in
J^{\circ}(\mathbf{w},\mathbf{c})$ if and only if

$0\neq g(a+2,b+1)=(b+1,\lfloor c_{1}+2c_{2}\rfloor)!\cdot(2a+5b+9,\lfloor
21c_{1}+24c_{2}+2\rfloor)!\cdot(4a+3b+11,\lfloor
21c_{1}+34c_{2}+4\rfloor)!\cdot(2a-b+3,\lfloor 3c_{1}+2c_{2}+2\rfloor)!,$

equivalently,

$\displaystyle c_{1}+2c_{2}$ $\displaystyle<b+1$ $\displaystyle
21c_{1}+24c_{2}$ $\displaystyle<2a+5b+7$ $\displaystyle 21c_{1}+34c_{2}$
$\displaystyle<4a+3b+7$ $\displaystyle 3c_{1}+2c_{2}$ $\displaystyle<2a-b+1.$

Solving the above system of inequalities, we can determine regions in
$\mathbb{R}_{\geq 0}^{2}$ where the various monomials of $R$, must be in
$J^{o}(\mathbf{w},\mathbf{c})$. Looking at the intersections of these regions,
we can determine the consistency regions for $J^{o}(\mathbf{w},\mathbf{c})$
and we plot some of them in Figure 8, for

$\\{(c_{1},c_{2})\mid c_{1}\geq 0,c_{2}\geq
0,c_{1}\leq\frac{2}{3},c_{2}\leq\frac{2}{3}\\}.$

123456912107811$(\frac{1}{3},0)$$(0,\frac{1}{3})$$(\frac{2}{3},0)$$(0,\frac{2}{3})$

RegionIdeal1$R$2$\mathfrak{m}$3$(x^{2},xy,xy^{2})$4$(x^{2},x^{2}y,xy^{2}$)5$(x^{3},x^{2}y,xy^{2})$6$(x,xy)$7$(x^{2},xy)$8$(x^{3},xy)$9$(x^{4},xy)$10$(x^{2}y,x^{2}y^{2},x^{2}y^{3})$11$(x^{3},x^{2}y,x^{2}y^{2},x^{2}y^{3})$12$(x^{4},x^{2}y,x^{2}y^{2},x^{2}y^{3})$

Figure 8. Some consistency regions for $J^{o}(\mathbf{w},\mathbf{c})$ for
$(c_{1},c_{2})\in[0,\frac{2}{3}]\times[0,\frac{2}{3}]$

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