# Rees algebras of ideals of star configurations

A. Costantini , B. Drabkin , L. Guerrieri University of California, Riverside,
Riverside CA 92521, USA. _e-mail_<EMAIL_ADDRESS>University of
Technology and Design, Singapore, Singapore _e-mail_ :
<EMAIL_ADDRESS>University, Instytut Matematyki,
30-348 Kraków, Poland. _e-mail_<EMAIL_ADDRESS>
( )

###### Abstract

In this article we study the defining ideal of Rees algebras of ideals of star
configurations. We characterize when these ideals are of linear type and
provide sufficient conditions for them to be of fiber type. In the case of
star configurations of height two, we give a full description of the defining
ideal of the Rees algebra, by explicitly identifying a minimal generating set.

##  1 Introduction

Ideals of star configurations arise in algebraic geometry, in connection with
the study of intersections of subvarieties or subschemes in a projective
space. Given a family of hypersurfaces meeting properly in $\mathbb{P}^{n}$, a
_star configuration of codimension $c$_ is the union of all the codimension
$c$ complete intersection subschemes obtained by intersecting $c$ of the
hypersurfaces (see [15]). The terminology derives from the special case of ten
points located at pairwise intersections of five lines in $\mathbb{P}^{2}$,
with the lines positioned in the shape of a star.

From a commutative algebra perspective, ideals defining star configurations
represent an interesting class, since a great amount of information is known
about their free resolutions, Hilbert functions and symbolic powers (see for
instance [14, 15, 12, 23, 25, 2, 3, 29, 24]). In this article we study their
Rees algebras, about which little is currently known (see for instance [19,
13, 27, 5]).

If $I=(g_{1},\ldots,g_{\mu})$ is an ideal in a Noetherian ring $R$, the _Rees
algebra_ of $I$ is the subalgebra

$\mathcal{R}(I)\coloneq R[It]=R[g_{1}t,\dots,g_{\mu}t]\subseteq R[t]$

of the polynomial ring $R[t]$. In particular, $\,g_{1}t,\dots,g_{\mu}t\,$ are
$R$-algebra generators of $\mathcal{R}(I)$, and the algebraic structure of
$\mathcal{R}(I)$ is classically understood by determining the ideal of
relations among these generators. The latter is called the _defining ideal_ of
the Rees algebra, and its generators are called the _defining equations_ of
$\mathcal{R}(I)$. Geometrically, $\mathrm{Proj}(\mathcal{R}(I))$ is the blow-
up of the affine scheme $X=\mbox{\rm Spec}(R)$ along the subscheme $V(I)$.

Determining the defining ideal of Rees algebras is usually difficult. Indeed,
although the defining equations of degree one can be easily determined from a
presentation matrix of the given ideal, a full understanding of the defining
ideal of $\mathcal{R}(I)$ often requires prior knowledge of a free resolution
of $I$ and of its powers $I^{m}$. On the other hand, only a few classes of
ideals have well-understood free resolutions, and usually a free resolution
for $I$ does not provide information on the free resolutions of its powers
$I^{m}$. Nevertheless, the problem becomes manageable if one imposes algebraic
conditions on a presentation matrix of $I$ (see for instance [30, 26, 22]),
especially when one can exploit methods from algebraic combinatorics (see for
instance [32, 19, 11, 8, 13, 16, 1]).

The rich combinatorial structure of ideals of star configurations sometimes
allows to deduce information on their Rees algebra. For instance, _monomial
star configurations_ , which are constructed choosing the hypersurfaces to be
coordinate hyperplanes in $\mathbb{P}^{n}$, are monomial ideals associated
with discrete polymatroids, and hence are of fiber type by work of Herzog,
Hibi and Vladoiu (see [19, 3.3]). Recall that an ideal $\,I\subseteq
R=K[x_{1},\ldots,x_{n}]\,$ is said to be of _fiber type_ if the non-linear
equations of the Rees algebra $\mathcal{R}(I)$ are given by the defining
equations of the _fiber cone_ $\,F(I)\coloneq\mathcal{R}(I)\otimes_{R}K$.

In order to study _linear star configurations_ , which are constructed
choosing arbitrary hyperplanes in $\mathbb{P}^{n}$, one can instead exploit
the combinatorial properties of hyperplane arrangements. In particular,
Garrousian, Simis and Tohăneanu proved that ideals of this kind of height two
are of fiber type, and provided a (non-minimal) generating set for the
defining ideal of their Rees algebra (see [13, 4.2 and 3.5]). Similar results
were obtained for linear star configurations of height three in
$\mathbb{P}^{2}$ by Burity, Tohăneanu and Xie (see [5, 3.4 and 3.5]), who also
conjectured that ideals of linear star configurations of any height are of
fiber type.

In this context, it is then natural to ask the following question.

######  Question 1.1.

Let $\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$ be a family of homogeneous
polynomials in $K[x_{1},\ldots,x_{n}]$ and let $I_{c,\mathcal{F}}$ be the
ideal of the star configuration of height $c$ obtained from the hypersurfaces
defined by the $F_{i}$’s. Under what conditions on $\mathcal{F}$ is
$I_{c,\mathcal{F}}$ of fiber type?

Although in general ideals of star configurations may not be of fiber type, we
show that this is always the case when the elements of $\mathcal{F}$ form a
regular sequence. More precisely, our first main result (see 3.3 and 3.4) is
the following.

######  Theorem 1.2.

Let $\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$ be a homogeneous regular sequence
in $K[x_{1},\ldots,x_{n}]$. Let $I$ be the ideal of a star configuration of
height $c\geq 2$ constructed on the hypersurfaces defined by the elements of
$\mathcal{F}$. Then for any $m\geq 1$, $I^{m}$ is of fiber type. Moreover, the
defining equations of the fiber cone $F(I^{m})$ have degree at most two.

Our key observation is that, under these assumptions, $I$ and its powers
$I^{m}$ are generated by _monomials in the $F_{i}$’s_, i.e. elements of the
form $\,F_{1}^{i_{1}}\cdots F_{t}^{i_{t}}$. Hence, the defining ideal of the
Rees algebra of $I^{m}$ can be deduced from its Taylor resolution (see [28,
Chapter IV]). This method was previously used by several authors to study Rees
algebras of squarefree monomial ideals (see for instance [32, 11, 21, 16]). We
remark that the content of 1.2 was already known in the case when
$\mathcal{F}=\\{x_{1},\ldots,x_{n}\\}$ (see [19, 3.3] and [18, 5.3(b)]).
However, our proof is substantially different, since we only perform algebraic
manipulations on the generators of $I^{m}$, while the proof of [18, 5.3(b)]
heavily relies on the use of Gröbner bases and monomial orders (via results of
De Negri [7, 2.5 and 2.6], see [18, 5.2 and 5.3]).

In the light of 1.2, it is natural to explore how moving away from the
assumption that the elements of $\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}\,$
form a regular sequence affects the structure of the defining ideal of the
Rees algebra $\mathcal{R}(I_{c,\mathcal{F}})$. Our second main result (see 4.4
and 4.5) shows that the length of a maximal regular sequence in the family
$\mathcal{F}$ characterizes the linear-type property of ideals of linear star
configurations. Recall that an ideal is said to be of _linear type_ if the
defining ideal of its Rees algebra only consists of linear equations.

More deeply, when the $F_{i}$’s are all linear, in 4.10 we characterize the
regular sequences contained in $\mathcal{F}$ in terms of the matrix whose
entries are the coefficients of the $F_{i}$’s. This turns out to be
particularly useful in the case of linear star configurations of height two.
Indeed, an ideal $I$ of this kind is perfect, hence it is classically known
that the maximal minors of the _Jacobian dual_ of a Hilbert-Burch matrix of
$I$ are defining equations of the Rees algebra $\mathcal{R}(I)$ (see Section 2
for the definition of Jacobian dual matrices). Thanks to 4.10, we can
interpret the defining ideal described by Garrousian, Simis and Tohăneanu in
[13, 4.2 and 3.5] in terms of the associated primes of the ideal of maximal
minors of the Jacobian dual. More precisely, our main result (see 6.5 and
6.14) is the following.

######  Theorem 1.3.

Let $I$ be the ideal of a linear star configuration of height two. Then, the
defining ideal of the Rees algebra $\mathcal{R}(I)$ is
$\mathcal{L}+\mathcal{P}$, where $\mathcal{L}$ consists of linear equations
and (under mild assumptions) $\mathcal{P}$ is the only associated prime of the
ideal of maximal minors of a Jacobian dual for $I$ that is not generated by
monomials.

The proof of 1.3 proceeds in three steps. First, we identify a minimal
generating set for the ideal of maximal minors of a Jacobian dual for $I$ (see
5.4). Next, we prove that suitable irreducible factors of these minimal
generators span all the non-linear equations described in [13] (see 6.5),
hence they are the minimal non-linear equations of the Rees algebra. Finally,
using 4.10 and under mild assumptions on the matrix of coefficients of the
$F_{i}$’s, we provide a primary decomposition of the ideal of maximal minors
of the Jacobian dual and show that $\mathcal{P}$ satisfies the required
property (see 6.14). Our approach is entirely algebraic, combining linear
algebra with divisibility arguments. As a biproduct, it allows us to identify
the degrees of each non-linear generator of the defining ideal of
$\mathcal{R}(I)$ (see Remark 6.6), which was not obvious from the
combinatorial arguments appearing in the proof of [13, 3.5].

We remark that our methods can be potentially extended to ideals of linear
star configurations of height greater than two, since the notion of Jacobian
dual matrix is defined with no restriction on the height of the ideal under
examination (see [31, p. 191]). Moreover, it seems reasonable to believe that
the defining ideal of the fiber cone can be determined from the associated
primes of the ideal of maximal minors of the Jacobian dual matrix in other
cases as well. In fact, the defining ideal of the fiber cone is sometimes
known to coincide with the radical of the ideal of maximal minors of a
Jacobian dual. This is, for instance, the case for certain equigenerated
ideals with a linear presentation (including perfect ideals of height three
that are linearly presented, see [22, 7.1, 7.2 and 7.4]), or certain
equigenerated homogeneous ideals of arbitrary height (see [19, 3.5]).

We now describe how this article is structured.

In Section 2 we collect the necessary background on ideals of star
configurations and Rees algebras that we will use throughout. In Section 3 we
study the Rees algebras of (powers of) ideals of star configurations defined
with respect to a regular sequence $\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$.
The main results of this section are 3.3 and 3.4.

Section 4 is devoted to the linear type property of ideals of star
configurations. In particular, 4.5 provides a criterion for such an ideal to
be of linear type. In the case of linear star configurations, we also fully
characterize when these ideals are of linear type locally up to a certain
height (see 4.4 and 4.5).

In the remaining sections we focus on linear star configurations of height
two. In Section 5 we construct a minimal generating set for the ideal of
maximal minors of their Jacobian dual, which we exploit in Section 6 in order
to characterize the defining ideal of the Rees algebra of linear star
configurations of height two (see 6.5 and 6.14).

##  2 Background

In this section we collect the necessary background information on ideals of
star configurations and Rees algebras.

###  2.1 Star configurations

Throughout this article $R=K[x_{1},\ldots,x_{n}]\,$ denotes a standard graded
polynomial ring over a field $K$ and
$\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}\,$ denotes a set of homogeneous
elements in $R$. For any integer $c\,$ with $\,1\leq c\leq t$, let

$I_{c,\mathcal{F}}=\bigcap_{1\leq i_{1}<\ldots<i_{c}\leq
t}(F_{i_{1}},\ldots,F_{i_{c}}),$

######  Definition 2.1.

If $\,1\leq c\leq\mathrm{min}\\{n,t\\}$ and any subset of $\mathcal{F}$ of
cardinality $c+1$ is a regular sequence, then $\,I_{c,\mathcal{F}}\,$ is
called the _ideal of the star configuration of height $c$ on the set
$\mathcal{F}$_.

We say that $\,I_{c,\mathcal{F}}\,$ defines a _linear star configuration_ if
in addition all the $F_{i}$ are homogeneous of degree one, and a _monomial
star configuration_ if $\,\mathcal{F}=\\{x_{1},\ldots,x_{n}\\}$.

The following proposition summarizes useful results about ideals of star
configurations.

######  Proposition 2.2.

Let $I_{c,\mathcal{F}}$ be the ideal of the star configuration of height $c$
on the set $\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$. For $s\leq t$, denote
$\,\displaystyle{J_{s,\mathcal{F}}=\sum_{1\leq i_{1}<\ldots<i_{s}\leq
t}(F_{i_{1}}\cdots F_{i_{s}})}$. Then:

  1. (1)

([15, 2.3]) $\,\displaystyle{\,I_{c,\mathcal{F}}=J_{t-c+1,\mathcal{F}}}$. In
particular, the minimal number of generators of $I_{c,\mathcal{F}}$ is
$\,\mu(I_{c,\mathcal{F}})=\binom{t}{t-c+1}\geq t,\,$ and
$\,\mu(I_{c,\mathcal{F}})=t\,$ if $\,c=2$.

  2. (2)

([15, 3.3 and 3.5]) $\,I_{c,\mathcal{F}}\,$ is a perfect ideal and has a
linear resolution.

  3. (3)

([5, 2.2 and 3.1] and [6, 3.1, 5.2 and 5.3]) If the $F_{i}$’s are all linear
or $\,\mathcal{F}=\\{x_{1},\ldots,x_{n}\\}$, then $\,I_{c,\mathcal{F}}\,$ has
linear powers (i.e. for every $m$, $\,I^{m}_{c,\mathcal{F}}\,$ has a linear
resolution).

###  2.2 Rees algebras

Although some of the definitions included in this subsection make sense over
any Noetherian ring and for any ideal, we assume throughout that $R\coloneq
K[x_{1},\ldots,x_{n}]$ is a polynomial ring over a field $K$ and that
$\,I=(g_{1},\ldots,g_{\mu})\,$ is a graded ideal of $R$. The _Rees algebra_ of
$I$ is the subalgebra

$\mathcal{R}(I)\coloneq\bigoplus_{i\geq 0}I^{i}t^{i}\subseteq R[t]$

of the polynomial ring $R[t]$, where $I^{0}\coloneq R$. Notice that
$\,\mathcal{R}(I)=R[g_{1}t,\dots,g_{\mu}t]\,$ is a graded ring (with grading
inherited from that of $R[t]$) and there is a natural graded epimorphism

$\varphi\colon S\coloneq
R[T_{1},\ldots,T_{\mu}]\twoheadrightarrow\mathcal{R}(I),$

defined by $\,\varphi(T_{i})=g_{i}t\,$ for all $\,1\leq i\leq{\mu}$. The ideal
$\mathcal{J}\coloneq\mathrm{ker}(\varphi)$ is called the _defining ideal_ of
the Rees algebra $\mathcal{R}(I)$, and the generators of $\mathcal{J}$ are
called the _defining equations_ of $\mathcal{R}(I)$. Notice that
$\,\displaystyle{\mathcal{J}=\bigoplus_{s\geq 0}\mathcal{J}_{s}}\,$ is a
graded ideal (in the $T_{i}$ variables) of the polynomial ring $S$.

The linear equations of $\mathcal{R}(I)$ can be easily determined from a
presentation matrix of $I$. Specifically, if

$R^{s}\stackrel{{\scriptstyle M}}{{\longrightarrow}}R^{\mu}\longrightarrow
I\to 0$

is a presentation of $I$, then
$\mathcal{J}_{1}=(\lambda_{1},\ldots,\lambda_{s})$, where the $\lambda_{i}$’s
are homogeneous linear polynomials in $S$ satisfying the matrix equation

$[\lambda_{1},\ldots,\lambda_{s}]=[T_{1},\ldots,T_{\mu}]\cdot M.$

We denote $\mathcal{J}_{1}$ by $\mathcal{L}$. The ideal $I$ is said to be _of
linear type_ if $\mathcal{J}=\mathcal{L}$.

Given an arbitrary ideal $I$, one should expect that the defining ideal of the
Rees algebra $\mathcal{R}(I)$ also contains non-linear equations. The latter
are usually difficult to determine, however if the $g_{i}$ all have the same
degree, the non-linear equations can sometimes be identified by analyzing the
_fiber cone_ of $I$, which is defined as

$F(I)\coloneq\mathcal{R}(I)\otimes_{R}K=K[g_{1},\dots,g_{\mu}]\cong\frac{K[T_{1},\ldots,T_{\mu}]}{\mathcal{I}}.$

Indeed, by construction one always has
$\,\mathcal{L}+\mathcal{I}S\subseteq\mathcal{J},\,$ and the ideal $I$ is
called of _fiber type_ when the latter inclusion is an equality. In fact, the
rings $S$ and $\mathcal{R}(I)$ can be given natural structures of bigraded
$K$-algebras by setting $\mbox{\rm deg}(T_{i})=(0,1)$ and $\mbox{\rm
deg}(x_{i})=(d_{1},0)$, where $d_{i}$ is the degree of $x_{i}$ in $R$. Then,
$\,\displaystyle{\mathcal{J}=\bigoplus_{i,j\geq 0}\mathcal{J}_{(i,j)}}\,$ is a
bigraded ideal and $I$ is of fiber type if and only if the defining ideal of
$\mathcal{R}(I)$ is

$\mathcal{J}=[\mathcal{J}]_{(\ast,1)}+[\mathcal{J}]_{(0,\ast)}.$

When $I$ is a perfect ideal of height two, one can best exploit the
information contained in a Hilbert-Burch presentation matrix $M$ of $I$
through the notion of a Jacobian dual matrix, which was first introduced in
[31]. Recall that the _Jacobian dual_ $B(M)$ of $M$ is an $n\times(\mu-1)$
matrix with coefficients in $S$, satisfying the equation

$[x_{1},\ldots,x_{n}]\cdot B(M)=[T_{1},\ldots,T_{\mu}]\cdot M.$

A priori one could find several possible Jacobian duals associated with a
given presentation $M$, however $B(M)$ is uniquely determined in the case when
the entries of $M$ are linear polynomials in $R=K[x_{1},\ldots,x_{n}]$.
Moreover, if $\,s=\mbox{\rm max}\\{n,\mu-1\\}\,$ and $I_{s}(B(M))$ denotes the
ideal of maximal minors of $B(M)$, then the defining ideal $\mathcal{J}$ of
$\mathcal{R}(I)$ always satisfies the inclusion

$\hypertarget{eqJacdual}{}\mathcal{L}+I_{s}(B(M))\subseteq\mathcal{J}$ (2.1)

Notice that, if $M$ has linear entries, then the entries of $B(M)$ are in
$K[T_{1},\ldots,T_{\mu}]$. Hence, the images of the generators of
$I_{s}(B(M))$ in the fiber cone $F(I)$ are in the defining ideal $\mathcal{I}$
of $F(I)$. Although the inclusion in Eq. 2.1 is usually strict, 2.5 below
provides an instance when equality holds. Before we state the theorem we need
to recall the following definition, which we will use often throughout this
article.

######  Definition 2.3.

An ideal $I$ satisfies the _$G_{s}$ condition_ if
$\mu(I_{\mathfrak{p}})\leq\dim R_{\mathfrak{p}}$ for every $\mathfrak{p}\in
V(I)$ of height at most $s-1$. Equivalently, if $M$ is any presentation matrix
of $I$, then $I$ satisfies the $G_{s}$ condition if and only if for every
$\,1\leq i\leq s-1$, $\,\mbox{\rm ht}(I_{\mu(I)-i}(M))\geq i+1$.

Moreover, $I$ is said to satisfy $G_{\infty}$ if it satisfies the $G_{s}$
condition for every integer $s$.

######  Remark 2.4.

Ideals of linear type always satisfy $G_{\infty}$.

Under the assumption that $I$ satisfies $G_{n}$, the Rees algebra of $I$ is
described by the following result of Morey and Ulrich. Recall that the Krull
dimension of the fiber cone $F(I)$ is called the _analytic spread_ of $I$ and
is denoted with $\ell(I)$.

######  Theorem 2.5 ([26, 1.3]).

Let $R=K[x_{1},\ldots,x_{n}]$ and assume that $K$ is infinite. Let $I\subseteq
R$ be a linearly presented perfect ideal of height 2 with $\mu(I)\geq n+1$ and
assume that $I$ satisfies $G_{n}$. Let $M$ be a Hilbert-Burch matrix resolving
$I$, then the defining ideal of $\mathcal{R}(I)$ is

$\mathcal{J}=\mathcal{L}+I_{n}(B(M))$

where $B(M)$ is the Jacobian dual of $M$. Moreover, $\mathcal{R}(I)$ and
$F(I)$ are Cohen-Macaulay, $I$ is of fiber type and $\ell(I)=n$.

In Section 6 we will also use some results on Rees algebras of ideals
generated by $a$-fold products of linear forms from [13], which we briefly
recall here. Let $\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$ be a set of
homogeneous linear polynomials in $R=K[x_{1},\ldots,x_{n}]$ and suppose that
$t\geq n$. Notice that each $F_{i}$ defines a line through the origin in
$\mathbb{P}^{n-1}$, hence the set $\mathcal{F}$ defines a _central hyperplane
arrangement_ in $\mathbb{P}^{n-1}$. A _linear dependency_ among $s$ of the
given linear forms is a relation of the form

$\hypertarget{dependency}{}D\colon
c_{i_{1}}F_{i_{1}}+\ldots+c_{i_{s}}F_{i_{s}}=0.$ (2.2)

Given a linear dependency $D$, one can define the following homogeneous
polynomial in $S=R[T_{1},\ldots,T_{\mu}]$

$\hypertarget{deltadependency}{}\partial
D\colon\sum_{j=1}^{s}c_{i_{j}}\prod_{j\neq k=1}^{s}T_{i_{k}}.$ (2.3)

The following result of Garrousian, Simis and Tohăneanu relates the $\partial
D$’s to the fiber cone and Rees algebra of ideals generated by $(t-1)$-fold
products of linear forms.

######  Theorem 2.6.

With the notation above, let $\,\displaystyle{I=\sum_{1\leq
i_{1}<\ldots<i_{t-1}\leq t}(F_{i_{1}}\cdots F_{i_{t-1}})}$. Then,

  1. (1)

([13, 4.2]) $I$ is of fiber type.

  2. (2)

([13, 3.5]) The defining ideal of the fiber cone $F(I)$ is generated (possibly
not minimally) by all elements of the form $\partial D$ as in Eq. 2.3, where
$D$ varies within the set of linear dependencies among any $\,t-1$ of the
$F_{i}$’s.

  3. (3)

([13, 4.9 and 4.10]) $\mathcal{R}(I)$ and $F(I)$ are Cohen-Macaulay.

##  3 Star configurations on a regular sequence

In this section we assume the following setting.

######  Setting 3.1.

Let $R=K[X_{1},\ldots,X_{n}]\,$ be a polynomial ring over a field $K$ and let
$\,\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}\,$ be a homogeneous regular sequence
in $R$. For $\,1\leq c\leq\mathrm{min}\\{n,t\\}$, let

$I=I_{c,\mathcal{F}}=\bigcap_{1\leq i_{1}<\ldots<i_{c}\leq
t}(F_{i_{1}},\ldots,F_{i_{c}})$

be the ideal of the star configuration of height $c$ on the set $\mathcal{F}$.
For a fixed $m\geq 1$, let $\,g_{1},\ldots,g_{\mu}$ be a minimal generating
set for the $m$-th power $I^{m}$ of $I$, and write
$S=R[T_{1},\ldots,T_{\mu}]$.

Since the elements of $\mathcal{F}$ form a regular sequence, from 2.2 it
follows that the $I$ is minimally generated by _monomials in the $F_{i}$_,
i.e. products $\,F_{1}^{i_{1}}\ldots F_{n}^{i_{n}}\,$ for some integers
$i_{j}\geq 0$. Notice that the exponents $i_{j}$ are uniquely determined
because the $F_{i}$’s form a regular sequence. Moreover, since every $m$-th
power $I^{m}$ is minimally generated by $m$-fold products of minimal
generators of $I$, with the assumptions and notations of 3.1 we may also
assume that $\,g_{1},\ldots,g_{\mu\,}$ are monomials in the $F_{i}$’s.

The defining ideal of $\,\mathcal{R}(I^{m})$ can then be determined from the
Taylor resolution of $I^{m}$ (see [28, Chapter IV]). More precisely, for
$\,1\leq s\leq t\,$ let

$\mathcal{I}_{s}=\\{\alpha=(i_{1},\ldots,i_{s})\,|\,1\leq i_{1}\leq\ldots\leq
i_{s}\leq t\\}.$

For each $\,\alpha\in\mathcal{I}_{s}\,$ denote $\,g_{\alpha}=g_{i_{1}}\cdots
g_{i_{s}},\,$ $\,T_{\alpha}=T_{i_{1}}\cdots T_{i_{s}}\,$ and

$\hypertarget{Talphabeta}{}T_{\alpha,\beta}=\frac{g_{\beta}}{\gcd(g_{\alpha},g_{\beta})}\,T_{\alpha}-\frac{g_{\alpha}}{\gcd(g_{\alpha},g_{\beta})}\,T_{\beta}.$
(3.1)

Then, the defining ideal of $\,\mathcal{R}(I^{m\,})$ is

$\mathcal{J}=\mathcal{J}_{1}+\bigcup_{s=2}^{\infty}\mathcal{J}_{s},$

where
$\,\displaystyle{\mathcal{J}_{s}=\\{T_{\alpha,\beta}\,|\,\alpha,\beta\in\mathcal{I}_{s}\,\\}}\,$
for every $s\geq 2\,$.

Since $\mathcal{R}(I^{m\,})$ is Noetherian, there exists an $N\geq 2$ so that
$\,\displaystyle{\mathcal{J}_{s}\subseteq\sum_{i=1}^{N}\mathcal{J}_{i}}\,$ for
all $s\geq N$. In order to estimate such an $N$, the first step is to identify
redundant relations, which we do in the following lemma. To simplify the
notation, for multi-indices $\alpha=(i_{1},\cdots,i_{s})$ and
$\beta=(j_{1},\cdots,j_{s})$, denote
$\,\theta=\displaystyle{\frac{g_{\beta}}{\gcd(g_{\alpha},g_{\beta})}}\,$ and
$\,\delta=\displaystyle{\frac{g_{\alpha}}{\gcd(g_{\beta},g_{\alpha})}}$. Then,
Eq. 3.1 can be rewritten as

$\hypertarget{relation}{}T_{\alpha,\beta}=\theta\,T_{i_{1}}\cdots
T_{i_{s}}-\delta\,T_{j_{1}}\cdots T_{j_{s}}$ (3.2)

######  Lemma 3.2.

With the notation above, let
$\,\displaystyle{\theta_{1}\coloneq\frac{g_{j_{1}}}{\gcd(g_{i_{1}},g_{j_{1}})}}$.
Assume that, up to reordering the indices $i_{k}$ and $j_{k}$ in Eq. 3.2,
$\theta_{1}$ divides $\theta$. Then, for $s\geq 2$ the relation in Eq. 3.2 can
be expressed as an $S$-linear combination of relations in $\mathcal{J}$ of
degree at most $s-1$.

###### Proof.

Write $\theta=\theta_{1}\theta^{\prime}$ and define
$\,\displaystyle{\delta_{1}\coloneq\frac{g_{i_{1}}}{\gcd(g_{i_{1}},g_{j_{1}})}}$.
Notice that $\theta_{1}T_{i_{1}}-\delta_{1}T_{j_{1}}\in\mathcal{J}_{1}$. Using
this linear relation we can rewrite

$\theta\,T_{i_{1}}\cdots T_{i_{s}}-\delta\,T_{j_{1}}\cdots
T_{j_{s}}=\theta^{\prime}\,T_{i_{2}}\cdots
T_{i_{s}}(\theta_{1}T_{i_{1}}-\delta_{1}T_{j_{1}})-T_{j_{1}}(\delta\,T_{j_{2}}\cdots
T_{j_{s}}-\delta_{1}\theta^{\prime}\,T_{i_{2}}\cdots T_{i_{s}}).$

Now, the term $\,\theta_{1}T_{i_{1}}-\delta_{1}T_{j_{1}}\,$ is clearly in
$\mathcal{J}_{1}$ and the term $\,\delta\,T_{j_{2}}\cdots
T_{j_{s}}-\delta_{1}\theta^{\prime}\,T_{i_{2}}\cdots T_{i_{s}}\,$ is in
$\mathcal{J}_{s-1}$. Indeed, observe that

$\theta^{\prime}g_{i_{2}}\cdots g_{i_{s}}=\frac{\theta
g_{\alpha}}{\theta_{1}g_{i_{1}}}=\frac{\delta
g_{\beta}}{\delta_{1}g_{j_{1}}}=\frac{\delta g_{j_{2}}\cdots
g_{j_{s}}}{\delta_{1}},$

and therefore $\,\delta_{1}\theta^{\prime}g_{i_{2}}\cdots g_{i_{s}}=\delta
g_{j_{2}}\cdots g_{j_{s}}.$ ∎

Notice that, whenever there exists a grading so that the $F_{i}\in\mathcal{F}$
have the same degree, then $I$ is generated by elements of the same degree
$\alpha(I)$. Then, each power $I^{m}$ is generated by all the monomials in the
$F_{i}$’s of degree $\,\alpha(I)m\,$ that are not divisible by $F^{m+1}$ for
any $F\in\mathcal{F}$. Moreover, the fiber cone $F(I^{m})$ is isomorphic to
the _toric ring_ of $\,\\{g_{1},\ldots,g_{\mu}\\}$. In particular, its
defining ideal is generated by all binomials of the form
$\,T_{\alpha}-T_{\beta}\,$ so that $\,g_{\alpha}=g_{\beta}$ (see [17,
Proposition 10.1.1] for a proof).

######  Theorem 3.3.

With the assumptions and notations of 3.1, assume also that $I$ is generated
by elements of the same degree with respect to some $\mathbb{Z}_{>0}$-grading.
Then, the ideal $I^{m}$ is of fiber type.

###### Proof.

We need to prove that all the relations of the form described in Eq. 3.2 can
be expressed as an $S$-linear combination of linear relations and fiber-type
relations. Working by induction on $s$, it suffices to show that any relation
$\,\displaystyle{T_{\alpha,\beta}=\theta\,T_{i_{1}}\cdots
T_{i_{s}}-\delta\,T_{j_{1}}\cdots T_{j_{s}}}\,$ of degree $s\geq 2$ that is
not of fiber type can be expressed as an $S$-linear combination of relations
of smaller degree in the $T_{i}$ variables.

Since $T_{\alpha,\beta}$ is not of fiber type, there exists a form
$F_{1}\in\mathcal{F}$ so that $F_{1}$ divides $\theta$. Moreover, since for
any $F\in\mathcal{F}$ we know that $F^{m+1}$ cannot divide any generator of
$I^{m}$, by possibly relabeling the indices, we may assume that $F_{1}$
divides $g_{j_{1}}$ and $F_{1}^{m}$ does not divide $g_{i_{1}}$. As in 3.2,
let
$\,\displaystyle{\theta_{1}\coloneq\frac{g_{j_{1}}}{\gcd(g_{i_{1}},g_{j_{1}})}}$.
If $\theta_{1}$ divides $\theta$, we are done by 3.2. Hence we assume the
opposite condition. Write

$\theta_{1}=F_{1}^{\,p_{1}}\cdots F_{a}^{\,p_{a}}G_{1}^{\,q_{1}}\cdots
G_{b}^{\,q_{b}}$

where the $F_{k}$’s and $G_{k}$’s are elements of $\mathcal{F}$ such that, for
all $k$, $F_{k}^{\,p_{k}}$ divides $\theta$, while no $G_{k}^{\,q_{k}}$
divides $\theta$. Notice that $p_{1}\geq 1$ by what was said above about
$F_{1}$. Moreover, since $T_{\alpha,\beta}$ does not satisfy the assumption of
3.2, necessarily we must have that $b\geq 1$, so there exists at least one
such form $G_{1}$. In particular, notice that $G_{1}$ divides $g_{j_{1}}$ but
$G_{1}^{\,m}$ does not divide $g_{i_{1}}$.

Similarly, let
$\,\displaystyle{\delta_{1}\coloneq\frac{g_{i_{1}}}{\gcd(g_{i_{1}},g_{j_{1}})}}\,$.
If $\delta_{1}$ divides $\delta$, the thesis follows by applying 3.2 to
$\delta$ and $\delta_{1}$. So, assume that $\delta_{1}$ does not divide
$\delta$. Then, there exist a form $H_{1}\in\mathcal{F}$ and an integer
$r_{1}\geq 1$ such that $H_{1}^{\,r_{1}}$ divides $\delta_{1}$ but does not
divide $\delta$. In particular, $H_{1}$ divides $g_{i_{1}}$ and $H_{1}^{m}$
does not divide $g_{j_{1}}$. Moreover, since $\gcd(\theta_{1},\delta_{1})=1$,
we also know that $H_{1}$ does not divide $\theta_{1}$ and $G_{1}$ does not
divide $\delta_{1}$.

Now, rewriting
$\,\displaystyle{g_{i_{1}}=\gcd(g_{i_{1}},g_{j_{1}})\delta_{1}}$ and
$\,\displaystyle{g_{j_{1}}=\gcd(g_{i_{1}},g_{j_{1}})\theta_{1}}$, we get that

$\hypertarget{gik-gkkequation}{}\theta\,\delta_{1\,}g_{i_{2}}\cdots
g_{i_{s}}=\delta\,\theta_{1\,}g_{j_{2}}\cdots g_{j_{s}}$ (3.3)

We claim that either there exists a $g_{j_{k}}$ with $k\geq 2$ such that
$H_{1}$ divides $g_{j_{k}}$ and $G_{1}^{m}$ does not divide $g_{j_{k}}$, or
there exists a $g_{i_{k}}$ such that $G_{1}$ divides $g_{i_{k}}$ and
$H_{1}^{m}$ does not divide $g_{i_{k}}$. Indeed, suppose that for all $k\geq
2$, $H_{1}$ divides $g_{j_{k}}$ if and only if $G_{1}^{m}$ divides $g_{j_{k}}$
and $H_{1}^{m}$ divides $g_{i_{k}}$ if and only if $G_{1}$ divides
$g_{i_{k}}$. Assume that $H_{1}$ divides exactly $c$ of the $g_{j_{k}}$’s for
$k\geq 2$ and that $G_{1}$ divides exactly $d$ of the $g_{i_{k}}$’s for $k\geq
2$. Then, the degree of $G_{1}$ on each side of Eq. 3.3 is at least $q_{1}+cm$
and at most $q_{1}-1+dm$. Similarly, the degree of $H_{1}$ on each side of Eq.
3.3 is at least $r_{1}+dm$ and at most $r_{1}-1+cm$. Hence, we must
simultaneously have that $cm\leq dm-1$ and $dm\leq cm-1$, which is impossible.
This proves our claim. Up to relabeling the $j_{k}$’s or $i_{k}$’s, we may
assume that $k=2$.

From the discussion above it follows that there exist generators
$g_{h_{1}},g_{h_{2}}$ of $I^{m}$ such that either
$H_{1}g_{j_{1}}=G_{1}g_{h_{1}}$ and $H_{1}g_{h_{2}}=G_{1}g_{j_{2}}$, or
$G_{1}g_{i_{1}}=H_{1}g_{h_{1}}$ and $G_{1}g_{h_{2}}=H_{1}g_{i_{2}}$. Let us
consider the first case (the second case is equivalent). We can write

$\displaystyle T_{\alpha,\beta}$ $\displaystyle=$ $\displaystyle\theta
T_{i_{1}}\cdots T_{i_{s}}-\delta T_{j_{1}}\cdots T_{j_{s}}$ $\displaystyle=$
$\displaystyle\theta T_{i_{1}}\cdots T_{i_{s}}-\delta T_{h_{1}}T_{h_{2}}\cdots
T_{j_{s}}+\delta T_{j_{3}}\cdots
T_{j_{s}}(T_{h_{1}}T_{h_{2}}-T_{j_{1}}T_{j_{2}}).$

The last term is a multiple of a fiber-type relation by a monomial in $S$.
Moreover, from our choice of $h_{1}$ and $h_{2}$ it follows that
$\,\displaystyle{\delta=\frac{g_{\alpha}}{\gcd(g_{\alpha},g_{\gamma})}},\,$
where $\gamma=(h_{1},h_{2},j_{3},\ldots,j_{s})$. Hence,

$T_{\alpha,\beta}^{(1)}=\theta T_{i_{1}}\cdots T_{i_{s}}-\delta
T_{h_{1}}T_{h_{2}}\cdots T_{j_{s}}\in\mathcal{J}_{s}$

and we only need to prove our claim for this new relation. Set
$\,\displaystyle{\theta_{1}^{(1)}\coloneq\frac{g_{h_{1}}}{\gcd(g_{i_{1}},g_{h_{1}})}=\frac{\theta_{1}}{G_{1}}}$
(where the latter equality holds because
$\,\displaystyle{\gcd(g_{i_{1}},g_{h_{1}})=H_{1}\gcd(g_{i_{1}},g_{j_{1}})}$).
Then there are two possibilities: either this new relation
$T_{\alpha,\beta}^{(1)}$ satisfies the assumption of 3.2 and the proof is
complete, or we iterate the process considering another form
$G_{k}\in\\{G_{1},\ldots,G_{b}\\}$. As before, by subtracting a multiple of a
fiber-type relation from $T_{\alpha,\beta}^{(1)}$, we reduce to a new relation
$T_{\alpha,\beta}^{(2)}$ such that the term corresponding to
$\theta_{1}^{(1)}$ is
$\,\displaystyle{\theta_{1}^{(2)}\coloneq\frac{\theta_{1}^{(1)}}{G_{k}}}$.
Since the monomial $F_{1}^{p_{1}}\cdots F_{a}^{p_{a}}$ divides $\theta$,
iterating this argument finitely many times, we reduce to prove our claim for
a relation satisfying the assumption of 3.2. ∎

######  Theorem 3.4.

With the assumptions and notations of 3.1, assume also that $I$ is generated
by elements of the same degree with respect to some $\mathbb{Z}_{>0}$-grading.
Then for all $m\geq 1$, the defining ideal of the Rees algebra
$\mathcal{R}(I^{m})$ is generated in degree at most 2 in the $T$-variables.

###### Proof.

It is sufficient to show that any fiber-type relation of the form

$\hypertarget{eqfiber}{}T_{i_{1}}\cdots T_{i_{s}}-T_{j_{1}}\cdots
T_{j_{s}}\in\mathcal{J}$ (3.4)

with $s\geq 3$ can be expressed as linear combination of fiber-type relations
of degree at most $s-1$. First, assume that the following condition holds.
$(\ast)$: Up to relabeling the indices, there exists a generator $g_{h}\in I$
such that $g_{i_{1}}g_{i_{2}}=g_{j_{1}}g_{h}$.
In this case, we get

$T_{i_{1}}\cdots T_{i_{s}}-T_{j_{1}}\cdots T_{j_{s}}=T_{i_{3}}\cdots
T_{i_{s}}(T_{i_{1}}T_{i_{2}}-T_{j_{1}}T_{h})-T_{j_{1}}(T_{j_{2}}\cdots
T_{j_{s}}-T_{h}T_{i_{3}}\cdots T_{i_{s}}),$

and observe that the last summand corresponds to a relation of degree $s-1$,
since

$g_{j_{2}}\cdots g_{j_{s}}=\frac{g_{i_{1}}g_{i_{2}}\cdots
g_{i_{s}}}{g_{j_{1}}}=g_{h}g_{i_{3}}\cdots g_{i_{s}}.$

Hence, this case is concluded and we may assume that condition $(\ast)$ does
not hold for the relation (3.4). Write
$g_{i_{1}}=\gcd(g_{i_{1}},g_{j_{1}})F_{1}^{p_{1}}\cdots F_{a}^{p_{a}}$ and
$g_{j_{1}}=\gcd(g_{i_{1}},g_{j_{1}})G_{1}^{q_{1}}\cdots G_{b}^{q_{b}}$, where
$F_{1},\dots,F_{a},G_{1},\dots G_{b}\in\mathcal{F}$. Observe that since the
ideal $I^{m}$ is equigenerated, then $\mbox{\rm deg}(F_{1}^{p_{1}}\cdots
F_{a}^{p_{a}})=\mbox{\rm deg}(G_{1}^{q_{1}}\cdots G_{b}^{q_{b}})$. Moreover,

$\hypertarget{Fl-Glequation}{}F_{1}^{p_{1}}\cdots F_{a}^{p_{a}}g_{i_{2}}\cdots
g_{i_{s}}=G_{1}^{q_{1}}\cdots G_{b}^{q_{b}}g_{j_{2}}\cdots g_{j_{s}}.$ (3.5)

Now, necessarily, we can find a $k\geq 2$ so that either $g_{i_{k}}$ is
divisible by some $G_{l}$ and not divisible by at least one power $F_{l}^{m}$
or $g_{j_{k}}$ is divisible by some $F_{l}$ and not divisible by at least one
power $G_{l}^{m}$. Indeed, if exactly $c$ elements among
$g_{i_{2}},\ldots,g_{i_{s}}$ are divisible by some of the $G_{1},\ldots,G_{b}$
and they are all divisible also by $F_{1}^{m}\cdots F_{a}^{m}$, then the
degree of each $F_{l}$ in each side of Eq. 3.5 is at least $cm+p_{l}$ while
the degree of each $G_{l}$ is at most $cm$. Similarly, if exactly $d$ elements
among $g_{j_{2}},\ldots,g_{j_{s}}$ are divisible by some of the
$F_{1},\ldots,F_{a}$ and they are all divisible also by $G_{1}^{m}\cdots
G_{b}^{m}$, then the degree of each $G_{l}$ in each side of Eq. 3.5 is at
least $dm+q_{l}$ while the degree of each $F_{l}$ is at most $dm$. These
conditions cannot be satisfied simultaneously, hence for some $k\geq 2$ there
must exist a generator $g_{j_{k}}$ divisible by some $F_{l}$ and not divisible
by at least one power $G_{l}^{m}$. In particular, this argument implies that
if $\mbox{\rm deg}(F_{1}^{p_{1}}\cdots F_{a}^{p_{a}})=1$, then
$g_{i_{1}}=\gcd(g_{i_{1}},g_{j_{1}})F_{1}$,
$g_{j_{1}}=\gcd(g_{i_{1}},g_{j_{1}})G_{1}$ and
$g_{j_{1}}g_{j_{k}}=g_{i_{1}}g_{h}$ where $g_{h}=\frac{G_{1}}{F_{1}}g_{j_{k}}$
is a generator of $I$. Therefore, condition $(\ast)$ is satisfied.

If $\mbox{\rm deg}(F_{1}^{p_{1}}\cdots F_{a}^{p_{a}})\geq 2$, without loss of
generality, possibly relabelling appropriately we may assume that $g_{i_{2}}$
is divisible by $G_{1}$ and not divisible by $F_{1}^{m}$ and consider the
generators of $I^{m}$, $g_{h_{1}}:=g_{i_{1}}\frac{G_{1}}{H_{1}}$ and
$g_{h_{2}}:=g_{i_{2}}\frac{F_{1}}{G_{1}}$. It then follows that

$T_{i_{1}}\cdots T_{i_{s}}-T_{j_{1}}\cdots T_{j_{s}}=T_{i_{3}}\cdots
T_{i_{s}}(T_{i_{1}}T_{i_{2}}-T_{h_{1}}T_{h_{2}})+T_{h_{1}}T_{h_{2}}T_{i_{3}}\cdots
T_{i_{s}}-T_{j_{1}}\cdots T_{j_{s}}.$

Since the first summand is a multiple of a fiber-type relation of degree 2, we
only need to prove the theorem for the relation
$T_{h_{1}}T_{h_{2}}T_{i_{3}}\cdots T_{i_{s}}-T_{j_{1}}\cdots T_{j_{s}}$.
Observe that now
$g_{h_{1}}=\gcd(g_{h_{1}},g_{j_{1}})F_{1}^{p_{1}-1}F_{2}^{p_{2}}\cdots
F_{a}^{p_{a}}$. If this new relation does not satisfy condition $(\ast)$, by
iterating the process, in a finite number of steps we get a relation in which
the total degree of the monomial in the forms $F_{1},\ldots,F_{a}$ is one, and
hence condition $(\ast)$ must finally be satisfied. This concludes the proof.
∎

######  Remark 3.5.

As observed in the introduction, when $\mathcal{F}=\\{x_{1},\ldots,x_{n}\\}$,
the powers $I_{c,\mathcal{F}}^{m}$ are polymatroidal ideals satisfying the
strong exchange property. Hence, by [19, 3.3] and [18, 5.3(b)] it was already
known that these ideals are of fiber type and their fiber-type relations have
degree at most two The result [18, 5.3] is proved using a sorting technique,
relying on Gröbner bases and monomial orders. Also, mapping variables $y_{i}$
to the $F_{i}\in\mathcal{F}$ defines a flat map
$\,\displaystyle{K[y_{1},\ldots,y_{t}]\to K[F_{1},\ldots,F_{t}]}$. Since
formation of Rees algebras commutes with flat base change (see [10, 1.3]),
then 3.3 and 3.4 follow from the case of a monomial regular sequence. However,
our direct proof recovers the known results while also giving a new proof in
the monomial case that requires less technical machinery.

##  4 The linear type property of ideals of star configurations

In this section we study under what conditions the ideal of a star
configuration is of linear type. Moreover, we give a criterion to determine
how this property may fail in the case of linear star configurations. Our
first result characterizes the linear type property of star configurations of
hypersurfaces.

######  Theorem 4.1.

Suppose that $I_{c,\mathcal{F}}\subseteq R=K[x_{1},\ldots,x_{n}]$ is a star
configuration on $\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}$. Let $n=\dim R$ and
assume that $\mathcal{F}$ contains a regular sequence of length $n$. The
following are equivalent:

  1. (1)

$I_{c,\mathcal{F}}$ is of linear type.

  2. (2)

$\mu(I_{c,\mathcal{F}})\leq n$.

  3. (3)

$c=2$ and $n=t$, i.e. $\mathcal{F}$ is a regular sequence.

###### Proof.

Assume that $I_{c,\mathcal{F}}$ is of linear type. Then, $I_{c,\mathcal{F}}$
satisfies the $G_{\infty}$ condition and in particular
$\mu(I_{c,\mathcal{F}})\leq n$. In general, by 2.2
$\,\mu(I_{c,\mathcal{F}})=\binom{t}{t-c+1}\geq t\geq n$. Hence, since $2\leq
c\leq n-1$, $\mu(I_{c,\mathcal{F}})\leq n$ if and only if $c=2$ and $n=t$. In
particular, in this case $\mu(I_{c,\mathcal{F}})=n$.

Finally, if $c=2$ and $n=t$, we know by 3.3 and 3.4 that $I_{c,\mathcal{F}}$
is of fiber type and the non-linear relations are generated by binomials of
degree 2. But, if there was a nonzero relation of degree two of the form
$T_{i_{1}}T_{i_{2}}-T_{i_{3}}T_{i_{4}}$ for distinct indices
$i_{1},i_{2},i_{3},i_{4}$, setting $I_{c,\mathcal{F}}=(g_{1},\ldots,g_{n})$,
we would have $g_{i_{1}}g_{i_{2}}-g_{i_{3}}g_{i_{4}}=0$. Since $c=2$, we have
that $g_{i}=\prod_{j\neq i}F_{j}$ and this implies
$F_{i_{1}}F_{i_{2}}=F_{i_{3}}F_{i_{4}}$ that is a contradiction since $n=t$
and $F_{1},\ldots,F_{t}$ form a regular sequence. Therefore there are no
relations of degree two and $I_{c,\mathcal{F}}$ is of linear type. ∎

######  Remark 4.2.

From the proof of (3) implies (1) it follows that if $c=2$ and all the
elements of $\mathcal{F}$ form a regular sequence then $I=I_{2,\mathcal{F}}$
is of linear type. In this case, since $I$ is a perfect ideal of height 2,
then the Rees algebra of $I$ is Cohen-Macaulay by [20, 2.6].

It is clear from the definition that localizing an ideal of linear type at any
prime ideal produces an ideal of linear type. In the rest of this section we
aim to measure how an ideal of star configurations may fail to be of linear
type by examining its linear type property locally. For this purpose, we first
need to understand how ideals of star configurations behave under
localization.

######  Lemma 4.3.

Let $I_{c,\mathcal{F}}\subseteq R$ be a star configuration on $\mathcal{F}$
and let $\mathfrak{p}$ be a prime ideal of $R$. Set
$\mathcal{F}^{\prime}:=\mathcal{F}\cap\mathfrak{p}$. Then

$(I_{c,\mathcal{F}})_{\mathfrak{p}}=\left\\{\begin{array}[]{ccc}I_{c,\mathcal{F}^{\prime}}R_{\mathfrak{p}}&\mbox{if
}|\mathcal{F}^{\prime}|>c\\\ \mbox{ complete intersection of height
}c&\mbox{if }|\mathcal{F}^{\prime}|=c\\\ R_{\mathfrak{p}}&\mbox{if
}|\mathcal{F}^{\prime}|<c.\\\ \end{array}\right.$

###### Proof.

Finite intersection of ideals commutes with localization, hence

$(I_{c,\mathcal{F}})_{\mathfrak{p}}=\bigcap_{i_{1},\ldots,i_{c}}(F_{i_{1}},\ldots,F_{i_{c}})_{\mathfrak{p}}.$

If some $F_{i_{j}}\not\in\mathfrak{p}$, the ideal
$(F_{i_{1}},\ldots,F_{i_{c}})_{\mathfrak{p}}=R_{\mathfrak{p}}$. If
$|\mathcal{F}^{\prime}|<c$, this necessarily happens for all the ideals in the
intersection and $(I_{c,\mathcal{F}})_{\mathfrak{p}}=R_{\mathfrak{p}}$.

If some $F_{i_{1}},\ldots,F_{i_{s}}\in\mathfrak{p}$ and they form a regular
sequence in $R$, then they form a regular sequence also in the ring
$R_{\mathfrak{p}}$. Hence, in the case $|\mathcal{F}^{\prime}|=c$, then
$(I_{c,\mathcal{F}})_{\mathfrak{p}}=(F_{i_{1}},\ldots,F_{i_{c}})R_{\mathfrak{p}}$
where $\mathcal{F}^{\prime}=\\{F_{i_{1}},\ldots,F_{i_{c}}\\}$ and it is a
complete intersection of height $c$. Instead, if $|\mathcal{F}^{\prime}|>c$,
the ideal
$(I_{c,\mathcal{F}})_{\mathfrak{p}}=I_{c,\mathcal{F^{\prime}}}R_{\mathfrak{p}}$
is a star configuration. ∎

Although the proofs of 4.1 and 4.3 work for any star configuration of
hypersurfaces, in the rest of this section we restrict to the case when the
$F_{i}$’s are all linear.

######  Proposition 4.4.

Let $R=K[x_{1},\ldots,x_{n}]$ and let $I_{c,\mathcal{F}}$ be a linear star
configuration of height $c$. The following are equivalent:

  1. (1)

$(I_{c,\mathcal{F}})_{\mathfrak{p}}$ is of linear type for every prime
$\mathfrak{p}\in\mbox{\rm Spec}(R)$ with $\mbox{\rm ht}{\mathfrak{p}}\leq
s-1$.

  2. (2)

$I_{c,\mathcal{F}}$ satisfies the $G_{s}$ condition.

###### Proof.

By Remark 2.4, condition (1) always implies condition (2). Thus assume that
$I_{c,\mathcal{F}}$ satisfies the $G_{s}$ condition. By way of contradiction,
say that there exists a prime ideal $\mathfrak{p}$ of height $\leq s-1$ such
that $(I_{c,\mathcal{F}})_{\mathfrak{p}}$ is not of linear type. By 4.3, this
implies $|\mathcal{F}\cap\mathfrak{p}|>c.$ Call $\mathfrak{q}$ the ideal
generated by the elements of $\mathcal{F}\cap\mathfrak{p}$. Clearly
$\mathfrak{q}$ is prime since the elements of $\mathcal{F}$ are linear forms
and $\mathfrak{q}\subseteq\mathfrak{p}$. Furthermore
$\mathcal{F}\cap\mathfrak{q}=\mathcal{F}\cap\mathfrak{p}$ and thus by 4.3,
$(I_{c,\mathcal{F}})_{\mathfrak{q}}$ is a star configuration of height $c$. We
want to show that $\,\mu((I_{c,\mathcal{F}})_{\mathfrak{q}})>\mbox{\rm
ht}\mathfrak{q}$; since $\mbox{\rm ht}\mathfrak{q}\leq s-1,\,$ this would
contradict the assumption that $I_{c,\mathcal{F}}$ satisfies the $G_{s}$
condition.

Notice that the height of $\mathfrak{q}$ is equal to the length of a maximal
regular sequence contained in $\mathcal{F}\cap\mathfrak{p}$ and for this
reason $(I_{c,\mathcal{F}})_{\mathfrak{q}}$ satisfies the assumptions of 4.1.
Hence, from 4.1 it follows immediately that
$\,\mu((I_{c,\mathcal{F}})_{\mathfrak{q}})>\mbox{\rm ht}\mathfrak{q}$ whenever
$c\geq 3$. If $c=2$, observe that
$\,\mu((I_{c,\mathcal{F}})_{\mathfrak{q}})=|\mathcal{F}\cap\mathfrak{q}|=|\mathcal{F}\cap\mathfrak{p}|$
and it suffices to show that $\mathcal{F}\cap\mathfrak{p}$ is not a regular
sequence. But if this were a regular sequence,
$(I_{c,\mathcal{F}})_{\mathfrak{p}}$ would be a star configuration of height 2
generated over a regular sequence, hence it would be of linear type by Remark
4.2. ∎

We now describe the set of primes at which $I_{c,\mathcal{F}}$ fails to be of
linear type. We call this set of primes the _non-linear type locus_ of
$I_{c,\mathcal{F}}$ and denote it by

$NLT(I_{c,\mathcal{F}})=\\{\mathfrak{p}\in\mbox{\rm Spec}(R)\mbox{ :
}(I_{c,\mathcal{F}})_{\mathfrak{p}}\mbox{ is not of linear type}\\}.$

######  Proposition 4.5.

Let $R=K[x_{1},\ldots,x_{n}]$ and let $I_{c,\mathcal{F}}$ be a linear star
configuration of height $c$. Then, any set
$\,\mathcal{H}\subseteq\mathcal{F}\,$ of cardinality $s\leq n$ that is not a
regular sequence generates a prime ideal $\mathfrak{q}\in
NLT(I_{c,\mathcal{F}})\setminus\\{(x_{1},\ldots,x_{n})\\}$.

Moreover, if $c=2$ the minimal elements of $NLT(I_{2,\mathcal{F}})$ that are
not maximal ideals of $R$ are all of this form. If instead $c\geq 3$, then
$\,NLT(I_{c,\mathcal{F}})=\\{\mathfrak{p}\in\mbox{\rm Spec}(R)\mbox{ :
}|\mathcal{F}\cap\mathfrak{p}|>c\\}$.

###### Proof.

First consider a set $\mathcal{H}\subseteq\mathcal{F}$ of cardinality $s\leq
n$ that is not a regular sequence. Since $I_{c,\mathcal{F}}$ is a star
configuration, clearly $|\mathcal{H}|>c+1$. The elements of $\mathcal{H}$ are
linear forms, thus they generate a prime ideal $\mathfrak{q}$ of $R$ which is
clearly non-maximal since it has depth strictly smaller than $n$. By 4.3,

$\mbox{\rm
ht}(\mathfrak{q})<|\mathcal{H}|\leq|\mathcal{F}\cap\mathfrak{q}|\leq\mu((I_{c,\mathcal{F}})_{\mathfrak{q}}).$

Hence, $(I_{c,\mathcal{F}})_{\mathfrak{q}}$ is not of linear type.

In the case $c=2$, to show that all minimal primes in $NLT(I_{2,\mathcal{F}})$
arise in this way, let $\mathfrak{p}$ be a non-maximal prime ideal of $R$ such
that $(I_{2,\mathcal{F}})_{\mathfrak{p}}$ is not of linear type. By 4.3,
$(I_{2,\mathcal{F}})_{\mathfrak{p}}$ is the star configuration of height 2
over the set $\mathcal{F}\cap\mathfrak{p}$ in the ring $R_{\mathfrak{p}}$. If
$|\mathcal{F}\cap\mathfrak{p}|>n$, consider any subset
$\mathcal{H}\subseteq(\mathcal{F}\cap\mathfrak{p})$ of cardinality $n$. Since
$\mathfrak{p}$ is non-maximal, necessarily $\mathcal{H}$ is not a regular
sequence. Then the prime ideal $\mathfrak{q}$ generated by the elements of
$\mathcal{H}$ is contained in $\mathfrak{p}$ and is such that
$(I_{2,\mathcal{F}})_{\mathfrak{q}}$ is not of linear type by the first part
of this proof.

If, instead, $|\mathcal{F}\cap\mathfrak{p}|\leq n$, call $\mathfrak{q}$ the
prime ideal generated by all the elements of $\mathcal{F}\cap\mathfrak{p}$.
Clearly $\mathfrak{q}\subseteq\mathfrak{p}$ and
$\mathcal{F}\cap\mathfrak{p}=\mathcal{F}\cap\mathfrak{q}$. Notice that
$\mathcal{F}\cap\mathfrak{p}$ is not a regular sequence, since by Remark 4.2 a
star configuration of height 2 generated over a regular sequence is always of
linear type. Therefore,

$\mbox{\rm
ht}(\mathfrak{q})<|\mathcal{F}\cap\mathfrak{p}|=|\mathcal{F}\cap\mathfrak{q}|=\mu((I_{2,\mathcal{F}})_{\mathfrak{q}}).$

Hence $(I_{2,\mathcal{F}})_{\mathfrak{q}}$ is not of linear type.

When $c\geq 3$, choose a non-maximal prime ideal $\mathfrak{p}.$ By 4.3, if
$|\mathcal{F}\cap\mathfrak{p}|\leq c$, then
$(I_{c,\mathcal{F}})_{\mathfrak{p}}$ is of linear type. Otherwise, if
$|\mathcal{F}\cap\mathfrak{p}|>c$, let $\mathfrak{q}$ be the prime ideal
generated by the elements of $\mathcal{F}\cap\mathfrak{p}$. Proceeding as in
the proof of 4.4, we can apply 4.1 to deduce that
$(I_{c,\mathcal{F}})_{\mathfrak{q}}$ is not of linear type. Since
$\mathfrak{q}\subseteq\mathfrak{p}$, then also
$(I_{c,\mathcal{F}})_{\mathfrak{p}}$ is not of linear type. ∎

We next apply the previous result to characterize when $I_{c,\mathcal{F}}$
satisfies the $G_{n}$ condition.

######  Theorem 4.6.

Let $R=K[x_{1},\ldots,x_{n}]$ and let $I_{c,\mathcal{F}}$ be a linear star
configuration of height $c$.

  1. (1)

The ideal $I_{c,\mathcal{F}}$ satisfies the $G_{n}$ condition.

  2. (2)

Every subset of $\mathcal{F}$ of cardinality $n$ is a regular sequence and
$c\in\\{2,n-1\\}.$

###### Proof.

Recall that by 4.4, $I_{c,\mathcal{F}}$ satisfies the $G_{n}$ condition if and
only if $NLT(I_{c,\mathcal{F}})=\\{(x_{1},\ldots,x_{n})\\}$.

First assume that there exists one subset of cardinality $n$ of $\mathcal{F}$
that is not a regular sequence. By 4.5, it then follows that
$NLT(I_{c,\mathcal{F}})$ contains some non-maximal prime ideal of $R$. Thus
$I_{c,\mathcal{F}}$ does not satisfy the $G_{n}$ condition.

Hence, we can assume that every subset of $\mathcal{F}$ of cardinality $n$ is
a regular sequence. If $c=2$, the conclusion follows by 4.5. If $c=n-1$, let
$\mathfrak{p}$ be a non-maximal prime ideal of $R$. We show that in this case
$|\mathcal{F}\cap\mathfrak{p}|\leq c$ and conclude that
$(I_{c,\mathcal{F}})_{\mathfrak{p}}$ is of linear type by 4.3. Indeed,
$|\mathcal{F}\cap\mathfrak{p}|>c=n-1$ if and only if at least $n$ distinct
elements of $\mathcal{F}$ are in $\mathfrak{p}$. But, since
$I_{n-1,\mathcal{F}}$ is a star configuration, any $n$ distinct elements of
$\mathcal{F}$ form a regular sequence and hence they cannot be contained in a
non-maximal prime ideal of $R$.

Finally, if $2<c<n-1$, let $\mathcal{H}$ be a subset of $\mathcal{F}$ of
cardinality $n-1$, and let $\mathfrak{p}$ be the prime ideal generated by the
elements of $\mathcal{H}$. Since every subset of $\mathcal{F}$ of cardinality
$n$ is a regular sequence, it follows that any
$F_{i}\in\mathcal{F}\setminus\mathcal{H}$ is regular modulo $\mathfrak{p}$,
hence is not in $\mathfrak{p}$. Therefore,
$\mathcal{F}\cap\mathfrak{p}=\mathcal{H}$ has cardinality $n-1>c$, hence the
conclusion follows from 4.5. ∎

######  Corollary 4.7.

Let $R=K[x_{1},\ldots,x_{n}]$ and assume that any subset of $\mathcal{F}$ of
cardinality $n$ is a regular sequence. Let $I_{2,\mathcal{F}}$ be a linear
star configuration of height $2$. Then $I$ is of fiber type and the defining
ideal of $\mathcal{R}(I)$ is given by

$\mathcal{J}=\mathcal{L}+I_{n}(B(M)),$

where $B(M)$ is the ideal of maximal minors of the Jacobian dual of $M$.
Moreover, $\mathcal{R}(I)$ and $F(I)$ are Cohen-Macaulay.

###### Proof.

From 4.6 it follows that $I$ satisfies the $G_{n}$ condition. Then the thesis
follows from 2.5. ∎

In the next section we give a more accurate description of the Rees algebra of
linear star configurations of height two, providing an explicit generating set
for the ideal of maximal minors of the Jacobian dual. This also allows us to
determine the defining ideal in the case when the $G_{n}$ condition is not
satisfied. For this purpose, it will be convenient to better control the
subsets of regular sequences contained in $\mathcal{F}$. The following lemma
will be useful.

######  Lemma 4.8.

Let $K$ be an infinite field and let $I_{c,\mathcal{F}}$ be a linear star
configuration of height $c$ defined over the set
$\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}\subseteq K[y_{1},\ldots,y_{d}]$. Assume
that the maximal regular sequence contained in $\mathcal{F}$ has length $n$.
Then, after renaming the variables of the ring, we can always assume

$\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}=\\{x_{1},\ldots,x_{n},L_{1},\ldots,L_{r}\\}\subseteq
K[x_{1},\ldots,x_{n}]$

where $L_{1},\ldots,L_{r}\in(x_{1},\ldots,x_{n})$ are linear forms.

###### Proof.

By relabeling the indices assume that $F_{1},\ldots,F_{n}$ is a regular
sequence of maximal length contained in $\mathcal{F}$. After a linear change
of variables, this regular sequence of linear forms can be always expressed as
$x_{1},\ldots,x_{n}$. If some linear form $F_{j}$ with $j>n$ has a monomial of
the form $y_{k}$, with $y_{k}$ distinct from $x_{1},\ldots,x_{n}$, then
clearly $F_{j},x_{1},\ldots,x_{n}$ is a regular sequence of length $n+1$
contradicting the hypothesis. It follows that $F_{n+1},\ldots,F_{t}$ must be
contained in $(x_{1},\ldots,x_{n})$. ∎

Thanks to 4.8, whenever the $F_{i}$’s are linear forms we can always reduce to
the following setting.

######  Setting 4.9.

Let $K$ be an infinite field and let $R=K[x_{1},\ldots,x_{n}]$. Assume that

$\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}=\\{x_{1},\ldots,x_{n},L_{1},\ldots,L_{r}\\}\subseteq
R$

where $L_{i}\coloneq\sum_{j=1}^{n}u_{ij}x_{j}$ with $u_{ij}\in K$ and $t=n+r$.
Let $U$ denote the $n\times r$ matrix on the elements $u_{ij}$ and let
$I_{c,\mathcal{F}}\subseteq R$ be a star configuration on $\mathcal{F}$.

######  Proposition 4.10.

Fix $2\leq s\leq n$. With the assumptions and notations of 4.9, the following
conditions are equivalent:

  1. (1)

Any subset of $\mathcal{F}$ of cardinality $s$ is a regular sequence.

  2. (2)

For every $1\leq h\leq\mbox{\rm min}\\{r,s\\}$, all the submatrices of the
matrix $U$ of size $(h+n-s)\times h$ have maximal rank.

###### Proof.

Recall that a finite set of linear forms in $R$ is a regular sequence if and
only if those forms are linearly independent over the base field $K$. This is
equivalent to have that the matrix expressing their coefficients as a function
of the variables $x_{1},\ldots,x_{n}$ has maximal rank.

After fixing $s$, assume one submatrix $V$ of $U$ of size $(h+n-s)\times h$
has not maximal rank. For simplicity, up to permuting rows and columns, we may
assume this to be the matrix obtained considering the first $h+n-s$ rows and
the first $h$ columns of $U$ for some $h\leq\mbox{\rm min}\\{r,s\\}$. If
$h=s$, this implies that the linear forms $L_{1},\ldots,L_{h}$ are not
linearly independent over $K$, thus not a regular sequence. Otherwise, if
$h<s$, consider the set $\\{L_{1},\ldots,L_{h},x_{n+h-s+1},\ldots,x_{n}\\}$.
Going modulo the regular sequence $\\{x_{n+h-s+1},\ldots,x_{n}\\}$, this set
reduces to the set of linear forms
$\\{\overline{L_{1}},\ldots,\overline{L_{h}}\\}$. The matrix of coefficients
of these linear forms with respect to $x_{1},\ldots,x_{n+h-s}$ is exactly the
matrix $V$. It follows that $\overline{L_{1}},\ldots,\overline{L_{h}}$ are not
linearly independent, hence not a regular sequence. Therefore
$\\{L_{1},\ldots,L_{h},x_{n+h-s+1},\ldots,x_{n}\\}$ is not a regular sequence
and has cardinality $s$.

Conversely, assume that condition 2 is satisfied. Consider a subset
$\mathcal{H}$ of $\mathcal{F}$ of cardinality $s$. Possibly going modulo the
forms of type $x_{1},\ldots,x_{n}$ belonging to $\mathcal{H}$, we reduce to
consider a set of linear forms
$\\{\overline{L_{i_{1}}},\ldots,\overline{L_{i_{h}}}\\}$ with $h\leq s$ and
whose coefficients with respect to $\overline{x_{1}},\ldots,\overline{x_{n}}$
form a submatrix of $U$ of size $(h+n-s)\times h$. It follows that those forms
are linearly independent and hence a regular sequence. Thus also the elements
of $\mathcal{H}$ form a regular sequence. ∎

######  Remark 4.11.

The proof of the 4.10 shows that there is a one-to-one correspondence between
the subsets of $\mathcal{F}$ of cardinality $s\leq n$ and the submatrices of
$U$ of size $(h+n-s)\times h$. One of such subsets of $\mathcal{F}$ is a
regular sequence if and only if the corresponding matrix has maximal rank.

##  5 The maximal minors of the Jacobian dual of linear star configurations of
height two

The main goal of this section is to determine a minimal generating set for the
ideal of maximal minors of the Jacobian dual of a presentation of
$I_{2,\mathcal{F}}$ (see 5.4). In the case when $I_{2,\mathcal{F}}$ satisfies
$G_{n}$, by 2.5 these are the non-linear equations defining the Rees algebra
of $I_{2,\mathcal{F}}$. In the next section, we will use 5.4 to identify the
non-linear equations of the Rees algebra also when the $G_{n}$ condition is
not satisfied (see 6.5).

######  Proposition 5.1.

With the assumptions and notations of 4.9, let $c=2$. There exists a
presentation matrix $M$ of $I_{2,\mathcal{F}}$ whose Jacobian dual can be
expressed in the following form:

$\hypertarget{jacobiandual}{}\footnotesize
B(M)=\begin{bmatrix}T_{1}&0&\ldots&0&u_{11}T_{n+1}&u_{21}T_{n+2}&\ldots&u_{r1}T_{n+r}\\\
-T_{2}&T_{2}&\ldots&0&u_{12}T_{n+1}&u_{22}T_{n+2}&\ldots&u_{r2}T_{n+r}\\\
0&-T_{3}&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
\vdots&0&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
\vdots&\vdots&\ldots&0&\vdots&\vdots&\ldots&\vdots\\\
0&0&\ldots&T_{n-1}&u_{1,n-1}T_{n+1}&u_{2,n-1}T_{n+2}&\ldots&u_{r,n-1}T_{n+r}\\\
0&0&\ldots&-T_{n}&(u_{1n}T_{n+1}-T_{n})&(u_{2n}T_{n+2}-T_{n})&\ldots&(u_{rn}T_{n+r}-T_{n})\\\
\end{bmatrix}.$ (5.1)

###### Proof.

Since $I_{2,\mathcal{F}}$ is perfect of height two, by the Hilbert-Burch
Theorem a presentation matrix of $I_{2,\mathcal{F}}$ is

$\hypertarget{presentationmatrix}{}\footnotesize
M=\begin{bmatrix}F_{1}&0&\ldots&0\\\ -F_{2}&F_{2}&\ldots&\vdots\\\
0&-F_{3}&\ldots&\vdots\\\ \vdots&0&\ldots&0\\\ \vdots&\vdots&\ldots&F_{t-1}\\\
\vdots&\vdots&\ldots&-F_{t}\\\ \end{bmatrix}.$ (5.2)

Recall that
$\mathcal{F}=\\{F_{1},\ldots,F_{t}\\}=\\{x_{1},\ldots,x_{n},L_{1},\ldots,L_{r}\\}$
where $L_{i}=\sum_{j=1}^{n}u_{ij}x_{j}$. Therefore, the Jacobian dual $B(M)$
is equal to

$\footnotesize\begin{bmatrix}T_{1}&0&\ldots&0&-u_{11}T_{n+1}&(u_{11}T_{n+1}-u_{21}T_{n+2})&\ldots&(u_{r-1,1}T_{n+r-1}-u_{r1}T_{n+r})\\\
-T_{2}&T_{2}&\ldots&0&-u_{12}T_{n+1}&(u_{12}T_{n+1}-u_{22}T_{n+2})&\ldots&(u_{r-1,2}T_{n+r-1}-u_{r2}T_{n+r})\\\
0&-T_{3}&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
\vdots&0&\ldots&\vdots&\vdots&\vdots&\ldots&\vdots\\\
\vdots&\vdots&\ldots&0&\vdots&\vdots&\ldots&\vdots\\\
0&0&\ldots&T_{n-1}&-u_{1,n-1}T_{n+1}&(u_{1,n-1}T_{n+1}-u_{2,n-1}T_{n+2})&\ldots&(u_{r-1,n-1}T_{n+r-1}-u_{r,n-1}T_{n+r})\\\
0&0&\ldots&-T_{n}&(T_{n}-u_{1n}T_{n+1})&(u_{1n}T_{n+1}-u_{2n}T_{n+2})&\ldots&(u_{r-1,n}T_{n+r-1}-u_{rn}T_{n+r})\\\
\end{bmatrix}.$

By column operations this matrix can be reduced to the form given in Eq. 5.1.
∎

Notice that the Jacobian dual of $I_{2,\mathcal{F}}$ is of size
$n\times(t-1)$. To compute the ideal of maximal minors of the Jacobian dual we
introduce the following notations.

######  Definition 5.2.

Assume $r=t-n\geq 1$. As in 4.9, for $i=1,\ldots,r$ consider the linear forms
$L_{i}\coloneq\sum_{k=1}^{n}u_{ik}x_{k}$ with $u_{ik}\in K$ and let $U$ be the
$n\times r$ matrix on the elements $u_{ik}$.

Consider a set of indexes $\chi\subseteq\\{1,\ldots,t\\}$ such that
$|\chi|=r$. Write

$\chi=\\{i_{1},\ldots,i_{h},j_{1},\ldots,j_{r-h}\\}$

with $i_{1},\ldots,i_{h}\leq n$ and $j_{1},\ldots,j_{r-h}\geq n+1$. Denote by
$U_{\chi}$ the $h\times h$ minor of $U$ defined by taking the rows
$i_{1},\ldots,i_{h}$ and removing the columns $j_{1}-n,\ldots,j_{r-h}-n$. In
the case $h=0$, we set $U_{\chi}\coloneq 1$.

######  Definition 5.3.

Consider the polynomial ring $K[T_{1},\ldots,T_{t}]$. Suppose $r=t-n\geq 1$
and define the following polynomials. Given $\Theta\subseteq\\{1,\ldots,t\\}$
such that $|\Theta|=r-1$, set
$\\{1,\ldots,t\\}\setminus\Theta=\\{k_{1},\ldots,k_{n+1}\\},$ where
$k_{i}<k_{i+1}$ for every $i=1,\ldots,n$. Define

$m_{\Theta}:=\sum_{i=1}^{n+1}(-1)^{\alpha(\Theta,k_{i})}\left(\dfrac{T_{k_{1}}\cdots
T_{k_{n+1}}}{T_{k_{i}}}\right)U_{\Theta\cup\\{k_{i}\\}}\in
K[T_{1},\ldots,T_{t}],$

where $U_{\Theta\cup\\{k_{i}\\}}$ is defined as in Definition 5.2. The
exponent $\alpha(\Theta,k_{i})$ is obtained as follows. Let $h<n+1$ be the
integer such that $k_{h}\leq n$ and $k_{h+1}>n$. Then

$\alpha(\Theta,k_{i})=\left\\{\begin{array}[]{cc}n-h+i-k_{i}&\mbox{ if }i\leq
h\\\ n+i&\mbox{ if }i>h.\end{array}\right.$

Notice that none of the monomials of $m_{\Theta}$ is divisible by any variable
$T_{j}$ for $j\in\Theta$ and that $m_{\Theta}\in(T_{k_{i}},T_{k_{l}})$ for any
$k_{i},k_{l}$.

We now state our main theorem about the ideal of maximal minors of the
Jacobian dual matrix of $I_{2,c}(\mathcal{F})$.

######  Theorem 5.4.

With the assumptions of 4.9, let $c=2$, and let $B$ be the Jacobian dual
matrix for $I_{2,\mathcal{F}}$ described by Eq. 5.1. Moreover, assume that
$r=t-n\geq 1$. The ideal $I_{n}(B)$ of the maximal minors of $B$ is minimally
generated by all the polynomials $m_{\Theta}$ defined in Definition 5.3 such
that $\,n\not\in\Theta$.

Before proving the theorem, we need a technical definition and a lemma that
give us control on the minors of the submatrices of $B$ containing the last
$r$ columns.

######  Definition 5.5.

Adopt the same notation of Definition 5.3. Let $B$ be the Jacobian dual matrix
for $I_{2,\mathcal{F}}$ described by Eq. 5.1 and denote its columns by
$A_{1},\ldots,A_{n-1},C_{1},\ldots,C_{r}$. Consider a set of indices
$\Theta\subseteq\\{1,\ldots,n-1\\}$ such that $|\Theta|=r-1.$
If $1\not\in\Theta$ write $\Theta=\Theta_{1}\cup\ldots\cup\Theta_{s}\,$ such
that for every $\,i=1,\ldots,s$:

  * •

$\Theta_{i}=\\{k_{i},k_{i}+1,\ldots,k_{i}+l_{i}-1\\}$ contains $l_{i}$
consecutive indexes.

  * •

$k_{i}+l_{i}-1<k_{i+1}-1$.

In particular, for every $i$, $k_{i}-1\not\in\Theta$. If $\,1\in\Theta$, write
in the same way $\,\Theta=\Theta_{0}\cup\Theta_{1}\cup\ldots\cup\Theta_{s}$
with $1\in\Theta_{0}$.

For $s=0$ (i.e. $\Theta=\\{1,\ldots,r-1\\}$), let $p_{\Theta}^{(1,1)}$ denote
the maximal minor of the submatrix of $B$ computed by only removing the first
$r-1$ columns. If $s\geq 1$, for $i=1,\ldots,s$ and $j=1,\ldots,l_{i}$ denote
by $p_{\Theta}^{(i,j)}$ the maximal minor of the matrix obtained from $B$ by
removing all the columns $A_{k}$ for $k\in\Theta$ and by doing the following
column operations:

  * •

for $1\leq h<i$ replace the column $A_{k_{h}-1}$ with the sum of consecutive
columns $A_{k_{h}-1}+A_{k_{h}}+\ldots+A_{k_{h}+l_{h}-1}$.

  * •

replace the column $A_{k_{i}-1}$ with the sum of consecutive columns
$A_{k_{i}-1}+A_{k_{i}}+\ldots+A_{k_{i}+j-2}$.

Only for $i=s$, we define in an analogous way another minor
$p_{\Theta}^{(s,l_{s}+1)}$.

In the proof of 5.4 we show that the minor $p_{\Theta}^{(s,l_{s}+1)}$
coincides with the polynomial $m_{\Theta}$, and that the ideal generated by
all the polynomials $m_{\Theta}$ coincides with the ideal generated by all the
minors $p_{\Theta}^{(1,1)}$. A key fact is that the minors
$p_{\Theta}^{(i,j)}$ define a sequence, where every element is obtained from
the previous one by replacing one column of the corresponding submatrix with
its sum with a column of $B$ excluded from such submatrix. We rename this
sequence of minors as

$p_{\Theta}^{(1)},p_{\Theta}^{(2)},\ldots,p_{\Theta}^{(e)}:=p_{\Theta}^{(1,1)},\ldots,p_{\Theta}^{(1,l_{1})},p_{\Theta}^{(2,1)},\ldots,p_{\Theta}^{(2,l_{2})},\ldots,p_{\Theta}^{(s,l_{s})},p_{\Theta}^{(s,1)},\ldots,p_{\Theta}^{(s,l_{s}+1)}.$

The following lemma provides a formula relating different elements of this
sequence of minors, which allows us to prove 5.4 by induction. To help the
reader dealing with the technicality of our argument, in Example 5.7 we show
how to apply the lemma for small values of $r$.

######  Lemma 5.6.

Adopt the same notations as in Definition 5.3 and Definition 5.5. For
$i=1,\ldots,s$ and $j=1,\ldots,l_{i}$, set
$\,\Theta(i,j)\coloneq\Theta\setminus\\{k_{i}+j-1\\}\cup\\{k_{i}-1\\}$. For
$h=1,\ldots,e-1$ we have

$\hypertarget{minorsformula}{}p_{\Theta}^{(h)}\coloneq
p_{\Theta}^{(i,j)}=p_{\Theta}^{(h+1)}-p_{\Theta(i,j)}^{(h-j+1)}.$ (5.3)

###### Proof.

Recalling that the matrix $B$ can be expressed as in Eq. 5.1, up to permuting
columns, by definition $p_{\Theta}^{(i,j)}$ is the determinant of a matrix of
the form

$\footnotesize\begin{bmatrix}0&&\\\ \vdots&&\\\ T_{k_{i}-1}&&\\\ 0&&\\\
\vdots&B^{\prime}\\\ -T_{k_{i}-1+j}&&\\\ 0&&\\\ \vdots&&\\\ 0&&\end{bmatrix}.$

Thus we can write
$p_{\Theta}^{(h)}=p_{\Theta}^{(i,j)}=(-1)^{k_{i}}T_{k_{i}-1}M_{1}-(-1)^{k_{i}+j}T_{k_{i}-1+j}M_{2}$,
where $M_{1}$ and $M_{2}$ are minors of the matrix $B^{\prime}$. Similarly, by
definition $p_{\Theta}^{(h+1)}$ is the determinant of the same matrix, where
the first column is replaced by its sum with the column of $B$ containing the
two variables $T_{k_{i}-1+j},-T_{k_{i}+j}$. Hence it has the form
$p_{\Theta}^{(h+1)}=(-1)^{k_{i}}T_{k_{i}-1}M_{1}-(-1)^{k_{i}+j+1}T_{k_{i}+j}M_{3}$
for some minor $M_{3}$ of $B^{\prime}$. It follows that

$p_{\Theta}^{(h)}=p_{\Theta}^{(h+1)}-(-1)^{k_{i}+j}(T_{k_{i}+j}M_{3}+T_{k_{i}-1+j}M_{2}).$

We have to show that
$p_{\Theta(i,j)}^{(h-j+1)}=(-1)^{k_{i}+j}(T_{k_{i}-1+j}M_{2}+T_{k_{i}+j}M_{3})$.
The second term of the equality is clearly the determinant of the matrix

$\footnotesize\begin{bmatrix}0&&\\\ \vdots&&\\\ T_{k_{i}-1+j}&&\\\
-T_{k_{i}+j}&B^{\prime}\\\ 0&&\\\ \vdots&&\\\ 0&&\end{bmatrix}.$

The first column of this matrix is equal to the column $A_{k_{i}-1+j}$ of $B$.
The remaining columns are obtained by performing the operations described by
Definition 5.5 on the set $\Theta$ to obtain the minor $p_{\Theta}^{(i,j)}$.
Now, notice that all the indexes smaller than $k_{i}-1$ are in $\Theta$ if and
only if they are in the set
$\Theta(i,j)=\Theta\setminus\\{k_{i}+j-1\\}\cup\\{k_{i}-1\\}$. Then, by
performing $h-j$ of the operations described in Definition 5.5 on the set
$\Theta(i,j)$ we obtain the same matrix as above. Hence, the determinant of
the matrix above coincides with $p_{\Theta(i,j)}^{(h-j+1)}$. ∎

######  Example 5.7.

Adopt the same notation as in Definition 5.5 and 5.6.

  * •

In the case when $r=1$, $B$ has only one maximal minor. In the proof of 5.4,
this minor will be shown to be equal to $m_{\Theta}$ with $\Theta=\emptyset$.

  * •

In the case when $r=2$, following the notation of Definition 5.5 and 5.6, we
deal with sets $\Theta=\\{i\\}$ with $1\leq i\leq n-1$. If $i=1$, we have
$s=0$ and $p_{\Theta}^{(1,1)}=m_{\Theta}$. For $i\geq 2$, $s=1$ and the
sequence corresponding to $\Theta$ is $p_{\Theta}^{(1,1)},p_{\Theta}^{(1,2)},$
where $p_{\Theta}^{(1,1)}$ is the minor of $B$ obtained removing the column
$A_{i}$ and $p_{\Theta}^{(1,2)}$ is the minor obtained by removing the column
$A_{i}$ and replacing $A_{i-1}$ by $A_{i-1}+A_{i}$. This second minor is equal
to $m_{\Theta}$. 5.6 gives

$p_{\Theta}^{(1,1)}=p_{\Theta}^{(1,2)}-p_{\Theta^{\prime}}^{(1,1)},$

with $\Theta^{\prime}=\\{i-1\\}$. Inductively this shows that
$p_{\Theta}^{(1,1)}$ is in the ideal generated by the minors of the form
$m_{\\{j\\}}$ for $j\leq i$.

  * •

Consider also the case when $r=3$. Here the sequences correspond to sets
$\Theta=\\{i,j\\}$ with $1\leq i<j\leq n-1$. Again we have
$p_{\\{1,2\\}}=m_{\\{1,2\\}}$. Then we have to describe three possible cases:
$\\{1,j\\}$ with $j\geq 3$, $\\{i,i+1\\}$, and $\\{i,j\\}$ with $i-j\geq 2$.
In the first case $s=1,l_{1}=1$, and similarly to the case $r=2$ we get

$p_{\\{1,j\\}}^{(1,1)}=p_{\\{1,j\\}}^{(1,2)}-p_{\\{1,j-1\\}}^{(1,1)},$

and $p_{\\{1,j\\}}^{(1,2)}=m_{\\{1,j\\}}$. For $\Theta=\\{i,i+1\\}$ we find
$s=1,l_{1}=2$. The minor $p_{\Theta}^{(1,1)}$ is obtained by removing columns
$A_{i},A_{i+1}$, $p_{\Theta}^{(1,2)}$ is obtained by also replacing the column
$A_{i-1}$ by $A_{i-1}+A_{i}$, and $p_{\Theta}^{(1,3)}=m_{\\{i,i+1\\}}$ is
obtained replacing the column $A_{i-1}$ by $A_{i-1}+A_{i}+A_{i+1}$. 5.6 gives

$p_{\\{i,i+1\\}}^{(1,1)}=p_{\\{i,i+1\\}}^{(1,2)}-p_{\\{i-1,i+1\\}}^{(1,1)}=(p_{\\{i,i+1\\}}^{(1,3)}-p_{\\{i-1,i\\}}^{(1,1)})-p_{\\{i-1,i+1\\}}^{(1,1)}.$

In the case $\Theta=\\{i,j\\}$ with $i>1$, $i-j\geq 2$, we have
$s=2,l_{1}=1,l_{2}=1$. Here we have $p_{\Theta}^{(1,1)}$,
$p_{\Theta}^{(2,1)}$, $p_{\Theta}^{(2,2)}=m_{\Theta}$ that are obtained
subsequently by first removing columns $A_{i},A_{j}$, then replacing $A_{i-1}$
by $A_{i-1}+A_{i}$, and finally replacing also $A_{j-1}$ by $A_{j-1}+A_{j}$.
By 5.6

$p_{\\{i,j\\}}^{(1,1)}=p_{\\{i,j\\}}^{(2,1)}-p_{\\{i-1,j\\}}^{(1,1)}=(p_{\\{i,j\\}}^{(2,2)}-p_{\\{i,j-1\\}}^{(2)})-p_{\\{i-1,j\\}}^{(1,1)},$

where the notation $(2)$ stands for $(1,2)$ if $j-1=i+1$ and for $(2,1)$ if
$j-1>i+1$. Also in this case it follows that each $p_{\Theta}^{(1,1)}$ is in
the ideal generated by the polynomials $m_{\Theta}$. Indeed one can combine
all the previous formulas and use inductively the fact that the second term of
each new equality corresponds to a set $\Theta^{\prime}$ containing smaller
indexes.

We are now ready to prove 5.4.

###### Proof.

(of 5.4). As in Definition 5.5 denote the columns of $B$ by
$A_{1},\ldots,A_{n-1},C_{1},\ldots$, $C_{r}$. Given
$\Psi\subseteq\\{A_{1},\ldots,A_{n-1},C_{1},\ldots,C_{r}\\}$ such that
$|\Psi|=r-1$, denote by $p_{\Psi}$ the $n\times n$ minor of $B$ obtained by
removing all the columns contained in $\Psi$. Observe that we need to prove
that the ideal

$I_{n}(B)=(p_{\Psi}\mbox{ :
}\Psi\subseteq\\{A_{1},\ldots,A_{n-1},C_{1},\ldots,C_{r}\\},\mbox{
}|\Psi|=r-1)$

is equal to the ideal

$(m_{\Theta}\mbox{ : }\Theta\subseteq\\{1,\ldots,n-1,n+1,\ldots,t\\},\mbox{
}|\Theta|=r-1).$

We fix $n$ and work by induction on $r$. If $r=1$, the matrix $B$ described in
Eq. 5.1 reduces to the form

$\footnotesize\begin{bmatrix}T_{1}&0&0&\ldots&0&u_{11}T_{n+1}\\\
-T_{2}&T_{2}&0&\ldots&0&u_{12}T_{n+1}\\\
0&-T_{3}&T_{3}&\ldots&\vdots&\vdots\\\ \vdots&0&-T_{4}&\ldots&\vdots&\vdots\\\
\vdots&\vdots&0&\ldots&0&\vdots\\\
\vdots&\vdots&\ldots&\vdots&T_{n-1}&u_{1,n-1}T_{n+1}\\\
0&0&0&\ldots&-T_{n}&(u_{1n}T_{n+1}-T_{n})\\\ \end{bmatrix}.$

A quick computation by induction on $n$ shows that for every $n$ the
determinant of this matrix is equal to

$m_{\Theta}=-T_{1}\cdots T_{n}+\sum_{i=1}^{n}\left(\dfrac{T_{1}\cdots
T_{n}}{T_{i}}\right)u_{1i}$

which, according to Definition 5.3, corresponds to the set $\Theta=\emptyset$.

Hence, we assume that the result is true for $r-1$ and we prove it for some
$1<r<n$. The case $r\geq n$ will be considered later.

Consider the matrix in Eq. 5.1 and take the submatrix $B^{\prime}$ obtained by
eliminating one column $C_{j}$ with $j\in\\{1,\ldots,r\\}$. The ideal of
maximal minors of $B^{\prime}$ is contained in $I_{n}(B)$ and its generators
are also generators of $I_{n}(B)$. By the inductive hypothesis we have

$I_{n}(B^{\prime})=(m_{\Theta^{\prime}}\mbox{ :
}\Theta^{\prime}\subseteq\\{1,\ldots,n-1,n+1,\ldots,t\\}\setminus\\{n+j\\},\mbox{
}|\Theta^{\prime}|=r-2)$

as ideal of the polynomial ring
$K[T_{1},\ldots,\widehat{T_{n+j}},\ldots,T_{t}]$. Following the notation of
Definition 5.3 and working back in the polynomial ring $K[T_{1},\ldots,T_{t}]$
we observe that each of such $m_{\Theta^{\prime}}$ coincides with $m_{\Theta}$
with $\Theta\coloneq\Theta^{\prime}\cup\\{n+j\\}$. Since the same argument can
be applied to any $j\in\\{1,\ldots,r\\}$, we reduce to considering only the
minors of $B$ for submatrices containing all the last $r$ columns
$C_{1},\ldots,C_{r}$. In particular we have to show that

$(p_{\Psi}\mbox{ : }\Psi\subseteq\\{A_{1},\ldots,A_{n-1}\\},\mbox{
}|\Psi|=r-1)=(m_{\Theta}\mbox{ : }\Theta\subseteq\\{1,\ldots,n-1\\},\mbox{
}|\Theta|=r-1).$

Consider first the submatrix obtained from $B$ by deleting the first $r-1$
columns. This matrix is equal to

$\footnotesize
B^{\star}:=\begin{bmatrix}0&0&\ldots&0&u_{11}T_{n+1}&\ldots&u_{r1}T_{t}\\\
\vdots&\vdots&\ldots&\vdots&\vdots&\ldots&\vdots\\\
T_{r}&0&\ldots&\vdots&\vdots&\ldots&\vdots\\\
-T_{r+1}&T_{r+1}&\ldots&\vdots&\vdots&\ldots&\vdots\\\
0&-T_{r+2}&\ldots&\vdots&\vdots&\ldots&\vdots\\\
\vdots&\vdots&\ldots&0&\vdots&\ldots&\vdots\\\
0&0&\ldots&T_{n-1}&u_{1,n-1}T_{n+1}&\ldots&u_{r,n-1}T_{t}\\\
0&0&\ldots&-T_{n}&(u_{1n}T_{n+1}-T_{n})&\ldots&(u_{rn}T_{t}-T_{n})\\\
\end{bmatrix}.$

One can check by induction on $n$ that its determinant is equal to

$p_{\\{A_{1},\ldots,A_{r-1}\\}}=(-1)^{r}\left[\sum_{i=r}^{t}(-1)^{\alpha_{i}}\left(\dfrac{T_{r}\cdots
T_{t}}{T_{i}}\right)U_{\Theta\cup\\{i\\}}\right]=m_{\Theta^{\star}},$

where $\Theta^{\star}:=\\{1,\ldots,r-1\\}$ and $\alpha_{i}=\mbox{\rm
max}\\{i-n-1,0\\}=\alpha(\Theta^{\star},i)$ as in Definition 5.3.

Consider now an arbitrary set of indices $\Theta\subseteq\\{1,\ldots,n-1\\}$
such that $|\Theta|=r-1,$ and $\Theta\neq\Theta^{\star}$. Using the notation
of Definition 5.5, we want to show that $m_{\Theta}=p_{\Theta}^{(s,l_{s}+1)}$
and therefore is in the ideal $I_{n}(B)$. Write
$\\{1,\ldots,t\\}\setminus\Theta=\\{k_{1},\ldots,k_{n+1}\\}$ such that
$k_{i}<k_{i+1}$ for every $i=1,\ldots,n$. By construction,
$p_{\Theta}^{(s,l_{s}+1)}$ is the minor of a matrix in which all the variables
$T_{j}$ for $j\in\Theta$ do not appear. In particular, after permuting the
rows and replacing the variables $T_{r},\ldots,T_{n-1}$ by
$T_{k_{1}},\ldots,T_{k_{n-r}}$ keeping the same order, this matrix is equal to
the matrix $B^{\star}$. This implies that, up to a sign,

$p_{\Theta}^{(s,l_{s}+1)}=\sum_{i=1}^{n+1}(-1)^{\beta_{i}}\left(\dfrac{T_{k_{1}}\cdots
T_{k_{n+1}}}{T_{k_{i}}}\right)U_{\Theta\cup\\{k_{i}\\}}=m_{\Theta},$

where each $\beta_{i}$ is determined by the permutations of the rows performed
in the process, and equals $\alpha(\Theta,k_{i})$.

This proves that each $m_{\Theta}$ is in $I_{n}(B)$ for all sets
$\Theta\subseteq\\{1,\ldots,t\\}\setminus\\{n\\}$ with $|\Theta|=r-1$. Using
now 5.6 iteratively as described in Example 5.7, it follows that each
$p_{\Psi}$ with $\,\Psi\subseteq\\{A_{1},\ldots,A_{n-1}\\}\,$ is in the ideal
generated by the minors of the form $m_{\Theta}$. Indeed, as in Definition
5.5, $p_{\Psi}=p_{\Theta}^{(1,1)}$ where $\Theta$ is the set of indexes
corresponding to the columns in $\Psi$. Now, apply Eq. 5.3 iteratively,
starting from $p_{\Theta}^{(1,1)}$. The index in first term on the right side
of Eq. 5.3 increases at each iteration, until the term becomes a
$p_{\Theta}^{(s,l_{s}+1)}=m_{\Theta}$. The second term on the right-hand side
of Eq. 5.3 is determined by a set of indexes obtained from one of those
appearing in the previous iteration by replacing an index with a strictly
smaller index. Hence, it eventually coincides with $m_{\Theta^{\star}}$.

To conclude we only have to discuss the case $r\geq n$. Clearly all the
columns $C_{1},\ldots,C_{r}$ in the second part of the matrix are all
equivalent up to permutation of the variables. Hence, similarly as in the
previous case, the result on all the minors involving at least one of the
first $n-1$ columns can be obtained by reducing to the case $r=n-1$. Finally,
we only need to prove the statement for $n\times n$ minors involving only
columns of the form $C_{1},\ldots,C_{r}$. By renaming the variables, it is
sufficient then to consider the matrix

$\footnotesize\begin{bmatrix}u_{11}T_{n+1}&\ldots&u_{n1}T_{2n}\\\
\vdots&\vdots&\ldots\\\ u_{1,n-1}T_{n+1}&\ldots&u_{n,n-1}T_{2n}\\\
(u_{1n}T_{n+1}-T_{n})&\ldots&(u_{nn}T_{2n}-T_{n})\\\ \end{bmatrix}.$

Expanding with respect to the last row, the determinant of this matrix is

$T_{n+1}\cdots
T_{2n}\,U_{\Theta\cup\\{n\\}}+\sum_{i=1}^{n}(-1)^{n+i+1}\left(\dfrac{T_{n}\cdots
T_{2n}}{T_{n+i}}\right)U_{\Theta\cup\\{n+i\\}}=m_{\Theta}$

where $\Theta=\\{1,\ldots,n-1,2n+1,\ldots,t\\}.$ ∎

######  Remark 5.8.

Observe that also the polynomials $m_{\Theta}$ such that $n\in\Theta$ are in
the ideal $I_{n}(B)$. Indeed any of such $m_{\Theta}$ can be expressed as
linear combination with coefficients in $\\{1,-1\\}$ of generators of the form
$m_{\Theta\setminus\\{n\\}\cup\\{k\\}}$, for $k\not\in\Theta$.

######  Remark 5.9.

Similarly as in Definition 5.2, let
$\Theta=\\{i_{1},\ldots,i_{h},j_{1},\ldots,j_{r-1-h}\\}$ be a set of indexes
such that $i_{1},\ldots,i_{h}\leq n$ and $j_{1},\ldots,j_{r-1-h}\geq n+1$.
Call $M$ the submatrix of $U$ of size $h\times(h+1)$ obtained by taking rows
$i_{1},\ldots,i_{h}$ and removing the columns $j_{1},\ldots,j_{r-1-h}$. Then,
the polynomial $m_{\Theta}$ is zero if and only if the rank of $M$ is $<h$.

Indeed, by Definition 5.3, $m_{\Theta}=0$ if and only if for every
$k_{i}\not\in\Theta$, the minor $U_{\Theta\cup\\{k_{i}\\}}=0$. This is
equivalent to say that all the submatrices of $M$ of size $h\times h$ and all
the submatrices of $U$ of size $(h+1)\times(h+1)$ containing $M$ are
simultaneously singular. Hence this means that $\mbox{rank}(M)<h$.

##  6 Linear star configurations of height two

In this section we exploit the results of Section 5 to determine the defining
ideal of the Rees algebra of ideals of linear star configurations of height
two. In particular, in 6.14 we relate the non-linear equations identified in
[13, 3.5 and 4.2] (see 2.6) to the associated primes of the ideal of maximal
minors of the Jacobian dual.

###  6.1 Defining ideal of the Rees algebra

Our first goal is to identify an ideal $\mathcal{P}$, defined in terms of the
polynomials $m_{\Theta}$, as the candidate for the non-linear part of the
defining ideal of the Rees algebra of $I_{2,\mathcal{F}}$. The generators of
this ideal $\mathcal{P}$ are introduced in the following lemma.

######  Lemma 6.1.

Let $\Theta$ and $m_{\Theta}$ be defined as in Definition 5.3. Suppose
$m_{\Theta}\neq 0$. Then $m_{\Theta}=fh_{\Theta}$ where $f$ is either a unit
or a squarefree monomial in the variables $T_{i}$ and $h_{\Theta}$ is an
irreducible nonzero and non-monomial element of $k[T_{1},\ldots,T_{t}]$.

###### Proof.

Let $k_{1},\ldots,k_{n+1}$ be the indexes not belonging to $\Theta$. Clearly
the variable $T_{k_{i}}$ divides $m_{\Theta}$ if and only if
$U_{\Theta\cup\\{k_{i}\\}}=0$. For simplicity rename
$U_{i}:=U_{\Theta\cup\\{k_{i}\\}}$. By relabeling the indexes, we can assume
that there exists $e\geq 2$ such that $U_{i}\neq 0$ for $i\leq e$ and
$U_{i}=0$ for $i>e$. Indeed by assumption $m_{\Theta}\neq 0$ and, by 5.4, it
is in the defining ideal of the Rees algebra of $I_{2,\mathcal{F}}$. Hence
$m_{\Theta}$ cannot be a monomial in the variables $T_{1},\ldots,T_{t}$ and
therefore at least two minors $U_{i}$ are nonzero. Now, if $e=n+1$, then
$h_{\Theta}=m_{\Theta}.$ Otherwise define

$\hypertarget{htheta}{}h_{\Theta}\coloneq\frac{m_{\Theta}}{T_{k_{e+1}}\cdots
T_{k_{n+1}}}.$ (6.1)

We have now that $h_{\Theta}$ can be expressed as $h_{\Theta}=\alpha
T_{k_{1}}+\beta$ where $\alpha=\sum_{i=2}^{e}(T_{k_{2}}\cdots
T_{k_{e}})T_{k_{i}}^{-1}U_{i}$ and $\beta=T_{k_{2}}\cdots T_{k_{e}}U_{1}$.
Since $U_{i}\neq 0$ for every $i\leq e$, we get that $\alpha$ and $\beta$ have
no common factors and $h_{\Theta}$ is irreducible. ∎

######  Definition 6.2.

For every $\Theta$ defined as in Definition 5.3, let $h_{\Theta}$ be defined
as in 6.1. We denote by $\mathcal{P}$ the ideal generated by the $h_{\Theta}$.

Notice that the ideal $\mathcal{P}$ is contained in the non-linear part of the
defining ideal $\mathcal{J}$ of the Rees algebra of $I_{2,\mathcal{F}}$.
Indeed, since $I_{n}(B)\subseteq\mathcal{J}$, by 5.4 and 6.1,
$m_{\Theta}=fh_{\Theta}\in\mathcal{J}$. But $f$ is either a unit or a
squarefree monomial in the variables $T_{1},\ldots,T_{t}$ and cannot be in
$\mathcal{J}$. Since $\mathcal{J}$ is prime, it follows that
$h_{\Theta}\in\mathcal{J}$.

Moreover, observe that if $I_{2,\mathcal{F}}$ satisfies the $G_{n}$ condition,
then $\mathcal{P}=I_{n}(B)$ and this coincides with the non-linear part of
$\mathcal{J}$ by 2.5. In 6.5 we prove that in general
$\mathcal{L}+\mathcal{P}=\mathcal{J}$, however $I_{2,\mathcal{F}}$ may no
longer satisfy $G_{n}$. The following example shows that when the $G_{n}$
condition is not satisfied, one might have a proper containment
$I_{n}(B)\subsetneq\mathcal{P}$.

######  Example 6.3.

Let $R=K[x_{1},x_{2},x_{3},x_{4}]$ and
$\mathcal{F}=\\{x_{1},x_{2},x_{3},x_{4},L_{1},L_{2}\\},$ where
$L_{1}=x_{1}+x_{2}+x_{3}+x_{4}$ and $L_{2}=x_{2}+2x_{3}+3x_{4}$. Then, by 4.6
the ideal $I_{2,\mathcal{F}}$ does not satisfy the $G_{n}$ condition. Notice
that

$U=\begin{bmatrix}1&0\\\ 1&1\\\ 1&2\\\ 1&3\\\
\end{bmatrix}\quad\mathrm{and}\quad B=\begin{bmatrix}T_{1}&0&0&T_{5}&0\\\
-T_{2}&T_{2}&0&T_{5}&T_{6}\\\ 0&-T_{3}&T_{3}&T_{5}&2T_{6}\\\
0&0&-T_{4}&T_{5}-T_{4}&3T_{6}-T_{4}\\\ \end{bmatrix}.$

In this case,

$\mathcal{L}=(x_{1}T_{1}-x_{2}T_{2},x_{2}T_{2}-x_{3}T_{3},x_{3}T_{3}-x_{4}T_{4},x_{4}T_{4}-L_{1}T_{5},x_{4}T_{4}-L_{2}T_{6})$

and $I_{4}(B)$ is generated by:

$\displaystyle m_{6}\\!\\!$ $\displaystyle=$ $\displaystyle
T_{1}T_{2}T_{3}T_{5}-T_{1}T_{2}T_{3}T_{4}+T_{1}T_{2}T_{4}T_{5}+T_{1}T_{4}T_{3}T_{5}+T_{4}T_{2}T_{3}T_{5},$
$\displaystyle m_{5}\\!\\!$ $\displaystyle=$ $\displaystyle
3T_{1}T_{2}T_{3}T_{6}-T_{1}T_{2}T_{3}T_{4}+2T_{1}T_{2}T_{4}T_{6}+T_{1}T_{4}T_{3}T_{6},$
$\displaystyle m_{3}\\!\\!$ $\displaystyle=$ $\displaystyle
T_{1}T_{2}T_{4}T_{5}-2T_{1}T_{2}T_{4}T_{6}-T_{1}T_{2}T_{5}T_{6}+T_{1}T_{4}T_{5}T_{6}+2T_{2}T_{4}T_{5}T_{6},$
$\displaystyle m_{2}\\!\\!$ $\displaystyle=$ $\displaystyle
T_{1}T_{4}T_{3}T_{5}-T_{1}T_{6}T_{3}T_{4}-T_{1}T_{6}T_{4}T_{5}-2T_{1}T_{6}T_{3}T_{5}+T_{4}T_{6}T_{3}T_{5},$
$\displaystyle m_{1}\\!\\!$ $\displaystyle=$ $\displaystyle-
T_{4}T_{2}T_{3}T_{5}+2T_{6}T_{2}T_{4}T_{5}+T_{6}T_{4}T_{3}T_{5}+3T_{6}T_{2}T_{3}T_{5}.$

Also, $\,\displaystyle{h_{1}=\frac{m_{1}}{T_{5}}=\frac{m_{5}}{T_{1}}}\,$ and
$\,\mathcal{P}=(m_{6},m_{3},m_{2},h_{1})\supsetneq I_{4}(B)$. 6.5 will show
that the defining ideal of $\mathcal{R}(I_{2,\mathcal{F}})$ is
$\mathcal{L}+\mathcal{P}$. Each of the polynomials $h_{i}$ has the form
$\partial D_{i}$ for dependency $D_{i}$ among the elements
$x_{1},x_{2},x_{3},x_{4},L_{1},L_{2}$ as in Eq. 2.2 and Eq. 2.3. We have:

$\displaystyle D_{6}\\!\\!\\!$ $\displaystyle\colon$ $\displaystyle
x_{1}+x_{2}+x_{3}+x_{4}-L_{1}=0,$ $\displaystyle D_{5}=D_{1}\\!\\!\\!$
$\displaystyle\colon$ $\displaystyle x_{2}+2x_{3}+3x_{4}-L_{2}=0,$
$\displaystyle D_{3}\\!\\!\\!$ $\displaystyle\colon$ $\displaystyle
2x_{1}+x_{2}-x_{4}-2L_{1}+L_{2}=0,$ $\displaystyle D_{2}\\!\\!\\!$
$\displaystyle\colon$ $\displaystyle x_{1}-x_{3}-2x_{4}-L_{1}+L_{2}=0.$

In the previous example, the non-linear part of the defining ideal of
$\mathcal{R}(I_{2,\mathcal{F}})$ is generated by the polynomials $\partial D$
corresponding to the minimal dependencies $D$ among the elements of
$\mathcal{F}$. This is true in general. Indeed, in 6.5 below, we show that the
defining ideal of $\mathcal{R}(I_{2,\mathcal{F}})$ is
$\mathcal{L}+\mathcal{P}$, where
$\mathcal{L}=(\lambda_{1},\ldots,\lambda_{t-1})$ is the ideal of linear
relations of $I_{2,\mathcal{F}}$. From the presentation matrix of
$I_{2,\mathcal{F}}$ it is clear that
$\displaystyle{\lambda_{i}=F_{i}T_{i}-F_{i+1}T_{i+1}}$ for every
$i=1,\ldots,t-1$.

######  Lemma 6.4.

With the assumptions and notations of 4.9, let $I_{2,\mathcal{F}}$ be a linear
star configuration of height 2. Assume that, up to reordering the variables,
the first row of the matrix $U$ is zero. Set
$\mathcal{G}=\\{x_{2},\ldots,x_{n},L_{1},\ldots,L_{r}\\}.$ Let $B$ and $B^{*}$
be Jacobian dual matrices for $I_{2,\mathcal{F}}$ and $I_{2,\mathcal{G}}$
respectively. Then

$I_{n}(B)=(T_{1})I_{n-1}(B^{*}).$

Moreover, the ideal $\mathcal{P}$ defined in Definition 6.2 for
$I_{2,\mathcal{F}}$ and for $I_{2,\mathcal{G}}$ is the same.

###### Proof.

Expressing $B$ and $B^{*}$ as in Eq. 5.1, it is easy to observe that

$\footnotesize B=\begin{bmatrix}T_{1}&0&\ldots&0\\\ -T_{2}&&&\\\ 0&&B^{*}&\\\
\vdots&&&\\\ 0&&&\end{bmatrix}.$

Both statements now follow from the definitions. ∎

######  Theorem 6.5.

With the assumptions and notations of 4.9, let $I_{2,\mathcal{F}}$ be a the
ideal of a linear star configuration of height two. Then, the ideal
$\mathcal{P}$ is the non-linear part of the defining ideal of the Rees algebra
of $I_{2,\mathcal{F}}$. In particular $\mathcal{J}=\mathcal{L}+\mathcal{P}.$

###### Proof.

By 2.6 it is sufficient to prove that the polynomial $\partial D$ associated
to any dependency $D$ among the elements of $\mathcal{F}$ is in the ideal
$\mathcal{P}$.

First we show that every polynomial $h_{\Theta}$ is of the form $\partial D$
for some dependency $D$. Indeed consider the natural map $\varphi\colon
R[T_{1},\ldots,T_{t}]\to\mathcal{R}(I_{2,\mathcal{F}})$ and let
$G=\prod_{i=1}^{t}F_{i}$. Then, using Eq. 6.1, we have that

$0=\varphi(h_{\Theta})=\sum_{i=1}^{e}(-1)^{\alpha(\Theta,i)}U_{\Theta\cup\\{k_{i}\\}}\varphi\Big{(}\frac{T_{k_{1}}\cdots
T_{k_{e}}}{T_{k_{i}}}\Big{)}=\sum_{i=1}^{e}(-1)^{\alpha(\Theta,i)}U_{\Theta\cup\\{k_{i}\\}}G^{\,e-2}F_{k_{i}}.$

It follows that, after dividing by $G^{e-2}$, the last term is a dependency
$D$ among the elements of $\mathcal{F}$ in the sense of Eq. 2.2. Therefore,
$h_{\Theta}$ is the corresponding polynomial $\partial D$ as defined in Eq.
2.3.

To prove that, for any dependency $D$, $\partial D$ is in $\mathcal{P}$ we
work by induction on $r\geq 1$. If $r=1$, up to multiplying by units, there is
only one dependency $D$ and clearly $\partial D=uh_{\Theta}$ with
$\Theta=\emptyset$ and for some $u\in K$.

Assume now that $r\geq 2$ and notice that any dependency $D$ can be written as

$D\colon a_{1}L_{1}+\ldots+a_{r}L_{r}+b_{1}x_{1}+\ldots+b_{n}x_{n}=0,$

where the coefficients $b_{j}$ are uniquely determined after
$a_{1},\ldots,a_{r}\in K$ are chosen. Using the inductive hypothesis we can
deal with all the dependencies such that at least one of the coefficients
$a_{1},\ldots,a_{r}$ is zero. For simplicity assume that $a_{r}=0$ and
consider the star configuration $I_{2,\mathcal{F^{\prime}}}$ where
$\mathcal{F^{\prime}}:=\mathcal{F}\setminus\\{L_{r}\\}$. Let
$\mathcal{P^{\prime}}$ be the ideal generated by the polynomials $h_{\Theta}$
constructed for $I_{2,\mathcal{F^{\prime}}}$ as in Definition 6.2. We can look
at $\mathcal{P^{\prime}}$ as an ideal of $K[T_{1},\ldots,T_{t}]$. Using 5.1,
5.4 and 6.1, it can be easily checked that
$\mathcal{P^{\prime}}\subseteq\mathcal{P}$. Now, all the dependencies $D$ such
that $a_{r}=0$ are also dependencies among the elements of
$\mathcal{F^{\prime}}$. Hence, by the inductive hypothesis, the corresponding
polynomials $\partial D\in\mathcal{P^{\prime}}\subseteq\mathcal{P}$.

To conclude we can restrict to the case where $a_{1},\ldots,a_{r}\neq 0$, and
since the polynomial $\partial D$ is unique up to multiplying scalars, we can
further assume that $a_{1}=1$. Let now $U$ be the $n\times r$ matrix of the
coefficients $u_{ij}$ as in 4.9. By 6.4 we can always reduce to the case in
which no rows of $U$ are zero. Hence, once fixed such $a_{1},\ldots,a_{r}$,
let $\chi\subseteq\\{1,\ldots,n\\}$ be a (possibly empty) maximal set of
indexes such that $b_{j}=0$ for every $j\in\chi$ and the rows of the matrix
$U$ indexed by the elements of $\chi$ are linearly independent. By definition
$|\chi|\leq n$. We show also that $|\chi|\leq r-1$. Indeed, by way of
contradiction and by relabeling, say that $\chi\supseteq\\{1,\ldots,r\\}$.
Thus $b_{1},\ldots,b_{r}=0$, which implies that the linear forms
$x_{r+1},\ldots,x_{n},L_{1},\ldots,L_{r}$ are not linearly independent. But by
Remark 4.11 the assumption that the first $r$ rows of $U$ are linearly
independent implies that $x_{r+1},\ldots,x_{n},L_{1},\ldots,L_{r}$ form a
regular sequence, a contradiction.

Without loss of generality, say now that $\chi=\\{1,\ldots,h\\}$ with $0\leq
h\leq\mbox{\rm min}\\{r-1,n\\}$. The dependency $D$ becomes $\,D\colon
L_{1}+a_{2}L_{2}+\ldots+a_{r}L_{r}+b_{h+1}x_{h+1}+\ldots+b_{n}x_{n}=0$. If
$h=r-1$, the equations with respect to $x_{1},\ldots,x_{r-1}$ determine a
linear system in $r-1$ equations $\,-u_{1k}=a_{2}u_{2k}+\ldots+a_{r}u_{rk}\,$
for $k=1,\ldots,r-1$ and $r-1$ indeterminates $a_{2},\ldots,a_{r}$. The
assumption that the first $r-1$ rows of $U$ are linearly independent forces
this system to have a unique solution. Hence, up to multiplying units, $D$ is
the only dependency related to such set $\chi$, and necessarily, setting
$\Theta=\chi$, the corresponding polynomial $\partial D$ is $h_{\Theta}$.

Suppose now that $h<r-1$. Since the first $h$ rows of $U$ are linearly
independent, by permuting the columns we may assume that the minor
$W:=U_{\chi\cup\\{n+1,\ldots,n+r-h\\}}$ is nonzero. For $i=1,\ldots,r-h$
define

$\Theta_{i}:=\chi\cup\\{n+1,\ldots,n+r-h\\}\setminus\\{n+i\\}.$

By construction $|\Theta_{i}|=r-1$ and $h_{\Theta_{i}}$ is well-defined. We
claim that

$\partial D=\sum_{i=1}^{r-h}\frac{a_{i}}{W}\left(\frac{T_{n+1}\cdots
T_{n+r-h}}{T_{n+i}}\right)h_{\Theta_{i}}\in\mathcal{P}.$

To do this we need to check that the coefficients of each term coincide. The
quantity on the right-hand side is a sum of terms of the form
$c_{j}(T_{h+1}\cdots T_{n+r})(T_{j}^{-1})$ for $j\geq h+1$. We have to prove
that $c_{j}=b_{j}$ if $j\leq n$ and $c_{j}=a_{j-n}$ if $j\geq n+1$. We
consider different subcases.
Case (i): $n+1\leq j\leq n+r-h$. This term appears only once among the terms
of $h_{\Theta_{j-n}}$. Observe that by Definition 5.3,
$\,\alpha(\Theta_{i},n+i)=-h$. Thus the coefficient of the term we are
considering is $\,(-1)^{h}U_{\Theta_{j-n}\cup\\{j\\}}=(-1)^{h}W$. Hence,
$\,c_{j}=(-1)^{h}a_{j-n}(W^{-1})W=(-1)^{h}a_{j-n}$.
Case (ii): $j>n+r-h$. For $k=1,\ldots,h$, consider the linear system on the
$h$ equations

$a_{r-h+1}u_{r-h+1,k}+\ldots+a_{r}u_{rk}=-(a_{1}u_{1,k}+\ldots+a_{r-h}u_{r-h,k}).$

Set $\sigma_{i,j}\coloneq\alpha(\Theta_{i},n+i)+\alpha(\Theta_{i},j)$. Observe
that by Definition 5.3 $\,\sigma_{i,j}=j-(n+r-h)$. By Cramer’s rule we get

$c_{j}=(-1)^{h}\sum_{i=1}^{r-h}(-1)^{\sigma_{i,j}}\frac{a_{i}}{W}\,U_{\Theta_{i}\cup\\{j\\}}=(-1)^{h}a_{j-n}.$

Case (iii): $h<j\leq n$. Notice that $b_{j}=-(a_{1}u_{1j}+\ldots+a_{r}u_{rj})$
and

$c_{j}=\sum_{i=1}^{r-h}(-1)^{\sigma_{i,j}}\frac{a_{i}}{W}\,U_{\Theta_{i}\cup\\{j\\}},$

where in this case $\sigma_{i,j}=-h+1$. Computing the minor
$U_{\Theta_{i}\cup\\{j\\}}$ with respect to the $j$-th row, we express

$U_{\Theta_{i}\cup\\{j\\}}=u_{ij}W+\sum_{k=r-h+1}^{r}(-1)^{r-h+k}u_{kj}\,U_{\Theta_{i}\cup\\{n+k\\}}.$

Hence, by replacing $U_{\Theta_{i}\cup\\{j\\}}$ in the equation for $c_{j}$
and applying (ii) we get

$\displaystyle c_{j}$ $\displaystyle=$
$\displaystyle(-1)^{1-h}\Big{(}\sum_{i=1}^{r-h}a_{i}u_{ij}+\sum_{i=1}^{r-h}\frac{a_{i}}{W}\sum_{k=r-h+1}^{r}(-1)^{r-h+k}u_{kj}U_{\Theta_{i}\cup\\{n+k\\}}\Big{)}$
$\displaystyle=$
$\displaystyle(-1)^{1-h}\Big{(}\sum_{i=1}^{r-h}a_{i}u_{ij}+\sum_{k=r-h+1}^{r}u_{kj}a_{k}\Big{)}=(-1)^{h}b_{j}.$

This concludes the proof after multiplying all the $c_{j}$ by $(-1)^{h}$. ∎

######  Remark 6.6.

From 6.5 and its proof it follows that the polynomials $h_{\Theta}$ defined in
6.1 are a minimal generating set for the non-linear part $\mathcal{P}$ of the
defining ideal $\mathcal{J}$ of $\mathcal{R}(I_{2,\mathcal{F}})$. Moreover,
Eq. 6.1 provides an explicit formula for each $h_{\Theta}$. In particular, the
degrees of the non-linear equations of the Rees algebra can be explicitly
calculated from the coefficients of the linear forms
$\\{L_{1},\ldots,L_{r}\\}$ and are always between 4 and $n$. In fact, since
any three of $x_{1},\ldots,x_{n},L_{1},\ldots,L_{r}$ are a regular sequence,
there is no dependency of degree at most 3.

######  Corollary 6.7.

Assume $r=1$. Set
$\displaystyle{\mathcal{F}=\\{x_{1},\ldots,x_{n},u_{e+1}x_{e+1}+\ldots+u_{n}x_{n}\\}}$
with $e\geq 0$ and $u_{i}\neq 0$ for all $e+1\leq i\leq n$. Then, the defining
ideal of $\mathcal{R}(I_{2,\mathcal{F}})$ is equal to $\mathcal{L}+(f)$ where

$f=T_{e+1}\cdots T_{n}-\sum_{i=e+1}^{n}\left(\frac{T_{e+1}\cdots
T_{n+1}}{T_{i}}\right)u_{i}.$

###  6.2 Primary decomposition

The aim of this subsection is to interpret the ideal $\mathcal{P}$ in terms of
the primary decomposition of $I_{n}(B)$. We have already observed that when
$I_{2,\mathcal{F}}$ satisfies the $G_{n}$ condition, $I_{n}(B)=\mathcal{P}$ is
the defining ideal of the fiber cone of $I_{2,\mathcal{F}}$, thus a prime
ideal.

When the $G_{n}$ condition is no longer satisfied, $I_{n}(B)$ is not a prime
ideal and we believe that $\mathcal{P}$ is one of its minimal primes. In
particular we state the following conjecture.

######  Conjecture 6.8.

The ideal $\mathcal{P}$ is the only associated prime of $I_{n}(B)$ not
generated by monomials.

We prove that the conjecture is true under an additional assumption on the
matrix of coefficients $U$. We start by some observing some properties of the
height of $I_{n}(B)$.

######  Remark 6.9.

By the Eagon-Northcott Theorem [9, Theorem 1], the ideal $I_{n}(B)$ has height
$\leq r$. This upper bound is met if $I_{2,\mathcal{F}}$ satisfies the $G_{n}$
condition. Indeed, in this case, by 2.5, $I_{n}(B)$ is the defining ideal of
the fiber cone of $I_{2,\mathcal{F}}$. Set
$\mathcal{F}^{\prime}=\mathcal{F}\setminus\\{F_{t}\\}$, and call $B^{\prime}$
the Jacobian dual matrix of $I_{2,\mathcal{F}^{\prime}}$. By 4.6, it is easy
to observe that also $I_{2,\mathcal{F}^{\prime}}$ satisfies condition $G_{n}$
and by 5.4, $I_{n}(B^{\prime})S\subsetneq I_{n}(B)$. The fact that these
ideals are primes and an inductive argument on $r$ imply that $\mbox{\rm
ht}I_{n}(B)=r$.

However, next result shows that if we remove the $G_{n}$ assumption, then the
height of $I_{n}(B)$ can be arbitrarily smaller than $r$. In the case when
$I_{2,\mathcal{F}}$ does not satisfy the $G_{n}$ condition, by 4.10 the matrix
of coefficients $U$ must have some zero minor. Next lemma shows that the
presence of such zero minors corresponds to containments of $I_{n}(B)$ in some
monomial prime ideals of height at most $r$.

######  Lemma 6.10.

Consider a set of indexes $\chi\subseteq\\{1,\ldots,t\\}$ such that
$|\chi|\leq r$. Let $\mathfrak{p}_{\chi}$ be the prime ideal of
$k[T_{1},\ldots,T_{t}]$ generated by the variables $T_{k}$ for $k\in\chi$. The
following are equivalent:

  1. (1)

The ideal $I_{n}(B)\subseteq\mathfrak{p}_{\chi}$.

  2. (2)

Each minor of $U$ of the form $U_{\Omega}$ with $\chi\subseteq\Omega$ is zero.

In particular, in the case $|\chi|=r$ this gives that
$I_{n}(B)\subseteq\mathfrak{p}_{\chi}$ if and only if $U_{\chi}=0$.

###### Proof.

By 5.4, the ideal $I_{n}(B)$ is generated by the polynomials $m_{\Theta}$ for
all the sets of indexes $\Theta$ such that $|\Theta|=r-1,n\not\in\Theta$. If
there exists at least two indexes $k_{i},k_{j}\in\chi\setminus\Theta$, then
each monomial of $m_{\Theta}$ is divisible either by $T_{k_{i}}$ or by
$T_{k_{j}}$ or by both and therefore $m_{\Theta}\in\mathfrak{p}_{\chi}$.

Hence, we need to discuss only the case $\chi\subseteq\Theta\cup\\{k_{i}\\}$
for some $k_{i}\not\in\Theta$. If $k_{i}\in\chi$, all the monomials of
$m_{\Theta}$ except one are divisible by $T_{k_{i}}$ but none of them is
divisible by any other variable $T_{k_{j}}$ generating $\mathfrak{p}_{\chi}$.
The only monomial of $m_{\Theta}$ not divisible by $T_{k_{i}}$ has coefficient
$U_{\Theta\cup\\{k_{i}\\}}$. Hence $m_{\Theta}\in\mathfrak{p}_{\chi}$ if and
only if $U_{\Theta\cup\\{k_{i}\\}}=0$.

If instead we assume $\chi\subseteq\Theta$, we have that none of the monomials
of $m_{\Theta}$ is in $\mathfrak{p}_{\chi}$ and the only possibility to have
$m_{\Theta}\in\mathfrak{p}_{\chi}$ is to have $U_{\Theta\cup\\{j\\}}=0$ for
every $j\not\in\Theta$. The thesis follows since this must hold for every
$\Theta$. ∎

To describe the primary decomposition of $I_{n}(B)$, in order to avoid too
much technicality, we focus on the case when a suitable condition on the
matrix $U$ (but weaker than the $G_{n}$ condition) is satisfied.

######  Remark 6.11.

By Remark 5.9 and 6.10 the following are equivalent:

  1. (1)

$m_{\Theta}\neq 0$ for every $\Theta$.

  2. (2)

For every $h\geq 1$, every submatrix of size $h\times(h+1)$ of $U$ has maximal
rank.

  3. (3)

$I_{n}(B)$ is not contained in any monomial prime ideal of height $<r$.

Notice that if $r=1$, these equivalent conditions are always satisfied, while
if $r=2$ this are satisfied if and only if no row of $U$ is zero. 6.4 shows
that, to study the primary decomposition of $I_{n}(B)$, we can always reduce
to assume that the matrix $U$ has no zero rows. Thus if $r\leq 2$ our proof
covers all the possible cases.

We now need a couple of lemmas to show that, for some distinct sets $\Theta$,
the corresponding $h_{\Theta}$ are associated in the polynomial ring
$K[T_{1},\ldots,T_{t}]$. First we explore further the relation between the
$h_{\Theta}$’s and dependencies pointed out in 6.5.

######  Lemma 6.12.

Let $\Theta$, $m_{\Theta}$, $h_{\Theta}$ be defined as in Definition 5.3 and
6.1. If $h_{\Theta}\neq 0$, then, up to multiplying by a factor in $K$, there
exists a unique dependency
$D:a_{1}L_{1}+\ldots+a_{r}L_{r}+b_{1}x_{1}+\ldots+b_{n}x_{n}=0$ such that
$b_{j}=0$ for $j\in\Theta$, $j\leq n$ and $a_{n-j}=0$ for $j\in\Theta$, $j>n$.
In particular $h_{\Theta}=\partial D$.

###### Proof.

The case $r=1$ is clear. Applying the same inductive argument on $r$ as in the
proof of 6.5, we reduce to prove the statement in the case
$\Theta=\\{j_{1},\ldots,j_{r-1}\\}\subseteq\\{1,\ldots,n\\}$. Since
$h_{\Theta}\neq 0$, also $m_{\Theta}\neq 0$ and by Remark 5.9 this implies
that the rows $j_{1},\ldots,j_{r-1}$ of the matrix $U$ are linearly
independent. Again, as in the proof of 6.5, this condition implies that, up to
multiplying a scalar, there exists a unique dependency $D$ such that
$b_{j_{1}},\ldots,b_{j_{r-1}}=0$. Necessarily $h_{\Theta}=\partial D$. ∎

######  Lemma 6.13.

Let $\Theta$, $m_{\Theta}$, $h_{\Theta}$ be defined as in Definition 5.3 and
6.1. Suppose that $h_{\Theta}\neq m_{\Theta}\neq 0$. Given $j\in\Theta$ and
$k\not\in\Theta$ such that the minor $U_{\Theta\cup\\{k\\}}=0$, define
$\Theta^{\prime}:=\Theta\setminus\\{j\\}\cup\\{k\\}$. Then, if
$m_{\Theta^{\prime}}\neq 0$, there exists a unit $a\in K$ such that
$h_{\Theta}=ah_{\Theta^{\prime}}.$

###### Proof.

Let $\Theta=\\{j_{1},\ldots,j_{r-1}\\}$ and let
$\\{k_{1},\ldots,k_{n+1}\\}=\\{1,\ldots,t\\}\setminus\Theta$. Since $0\neq
m_{\Theta}\neq h_{\Theta}$ by reordering the indexes, there exists $1\leq e<n$
such that $U_{\Theta\cup\\{k_{l}\\}}=0$ for $l\leq e$ and
$U_{\Theta\cup\\{k_{l}\\}}\neq 0$ for $l>e$. Take
$k\in\\{k_{1},\ldots,k_{e}\\}$ and $j\in\\{j_{1},\ldots,j_{r-1}\\}$. Observe
that $U_{\Theta^{\prime}\cup\\{j\\}}=U_{\Theta\cup\\{k\\}}=0$. By definition
of $m_{\Theta}$ we can write

$\frac{m_{\Theta}}{T_{k}}=\sum_{l=1}^{n+1}(-1)^{\alpha(\Theta,l)}\left(\frac{T_{k_{1}}\cdots
T_{k_{n+1}}}{T_{k_{l}}T_{k}}\right)U_{\Theta\cup\\{k_{l}\\}},$
$\frac{m_{\Theta^{\prime}}}{T_{j}}=\sum_{l=1}^{n+1}(-1)^{\alpha(\Theta^{\prime},l)}\left(\frac{T_{k_{1}}\cdots
T_{k_{n+1}}}{T_{k_{l}}T_{k}}\right)U_{\Theta^{\prime}\cup\\{k_{l}\\}}.$

Hence, to conclude we have to show that there exists a unit $a\in K$ such that
for every $l\in\\{k_{1},\ldots,k_{n+1}\\}\setminus\\{k\\}$,

$\hypertarget{signformula}{}U_{\Theta^{\prime}\cup\\{k_{l}\\}}=(-1)^{\alpha(\Theta,k_{l})-\alpha(\Theta^{\prime},k_{l})}aU_{\Theta\cup\\{k_{l}\\}}.$
(6.2)

This equality will also imply that one of these two minors is zero if and only
if so is the other one and the thesis will then follow by 6.1.

Using 6.12, it is sufficient to show that $h_{\Theta}$ and
$h_{\Theta^{\prime}}$ correspond to the same dependency $D$. Let
$D:a_{1}L_{1}+\ldots+a_{r}L_{r}+b_{1}x_{1}+\ldots+b_{n}x_{n}=0$ be the
dependency such that $h_{\Theta}=\partial D$. For simplicity set
$c_{l}\coloneq b_{l}$ for $l\leq n$ and $c_{l}\coloneq a_{n-l}$ for $l>n$.
Hence $c_{j_{l}}=0$ for every $l=1,\ldots,r-1$ and moreover, since
$U_{\Theta\cup\\{k\\}}=0$ then also $c_{k}=0$. Since also
$m_{\Theta^{\prime}}\neq 0$, then $h_{\Theta^{\prime}}\neq 0$ and by 6.12, the
dependency $D^{\prime}$ associated to $h_{\Theta^{\prime}}$ is then equal to
$aD$ for some nonzero $a\in K$. It follows that
$h_{\Theta^{\prime}}=ah_{\Theta}$. ∎

Next theorem shows that, under the assumption that all the $m_{\Theta}$ are
nonzero (which by Remark 6.11 corresponds to a maximal rank condition on
certain submatrices of $U$), the ideal $I_{n}(B)$ can be obtained intersecting
$\mathcal{P}$ with all the monomial primes of height at most $r$ containing
$I_{n}(B)$. Following the notation of 6.10, define $\Lambda$ to be the set of
all the monomial prime ideals of $K[T_{1},\ldots,T_{t}]$ of the form
$\mathfrak{p}_{\chi}$ for $\chi\subseteq\\{1,\ldots,t\\}$, $|\chi|\leq r$ and
such that $I_{n}(B)\subseteq\mathfrak{p}_{\chi}$.

######  Theorem 6.14.

Consider the set $\Lambda$ defined above and call
$Q=\bigcap_{\mathfrak{p}\in\Lambda}\mathfrak{p}.$ Let $\mathcal{P}$ be defined
as in Definition 6.2. Suppose that $m_{\Theta}\neq 0$ for every $\Theta$. Then
$I_{n}(B)$ is radical and its primary decomposition is

$\hypertarget{primary}{}I_{n}(B)=Q\cap\mathcal{P}.$ (6.3)

###### Proof.

By 2.6 and 6.5, the ideal $\mathcal{P}$ is the defining ideal of the fiber
cone of $I_{2,\mathcal{F}}$, hence it is prime of height $r$. The inclusion
$I_{n}(B)\subseteq Q\cap\mathcal{P}$ follows by 6.10 and 6.1. For the other
inclusion consider an element $\alpha=\sum\alpha_{i}h_{\Theta_{i}}\in
Q\cap\mathcal{P}$. Since each $m_{\Theta}\in I_{n}(B)$, we can restrict to
consider only the case in which $h_{\Theta_{i}}\neq m_{\Theta_{i}}$ for every
$i$. By 6.1, this implies that some minor of $U$ of the form
$U_{\Theta_{i}\cup\\{k\\}}=0$. Hence, setting $\chi=\Theta_{i}\cup\\{k\\}$, by
6.10 $\mathfrak{p}_{\chi}\in\Lambda$ and therefore
$Q\subseteq\mathfrak{p}_{\chi}$. But now, none of the monomials of
$h_{\Theta_{i}}$ is in $\mathfrak{p}_{\chi}$. Since $\alpha\in Q$ and
$\mathfrak{p}_{\chi}$ is generated by variables, this clearly implies
$\alpha_{i}\in\mathfrak{p}_{\chi}$. In particular, for every $i$, we get

$\alpha_{i}\in
Q_{\Theta_{i}}:=\bigcap_{\scriptstyle\mathfrak{p}_{\chi}\in\Lambda_{i}}\mathfrak{p}_{\chi},$

where $\Lambda_{i}$ is the set of primes in $\Lambda$ not containing
$h_{\Theta_{i}}.$ Thus we can reduce to fix one set of indexes $\Theta$ such
that $m_{\Theta}\neq h_{\Theta}$, and show that $Q_{\Theta}h_{\Theta}\subseteq
I_{n}(B)$. This would imply that $\alpha\in I_{n}(B)$ and the proof would be
complete.

Call $k_{1},\ldots,k_{n+1}$ the indexes not belonging to $\Theta$ and, by
reordering, consider the positive integer $e$ such that $1\leq e<n$ and
$\,U_{\Theta\cup\\{k_{l}\\}}=0$ if and only if $l\leq e$. Suppose
$\Theta=\\{j_{1},\ldots,j_{r-1}\\}$. Consider the sets
$E=\Theta\cup\\{k_{1},\ldots,k_{e}\\}$ and $\mathcal{E}=\\{T_{i}\mbox{ : }i\in
E\\}$. Let $H:=I_{r,\mathcal{E}}\in K[T_{1},\ldots,T_{t}]$ be the ideal of the
star configuration of height $r$ on the set $\mathcal{E}$.

We first prove that any associated prime $\mathfrak{p}$ of $H$ is in the set
$\Lambda$ and $h_{\Theta}\not\in\mathfrak{p}$. This will imply that
$Q_{\Theta}\subseteq H$ and we later prove that $Hh_{\Theta}\subseteq
I_{n}(B)$.

By 6.10, our first statement is true if we show that for every subset
$\chi\subseteq E$ such that $|\chi|=r$, the minor $U_{\chi}=0$ and
$h_{\Theta}\not\in\mathfrak{p}_{\chi}$. This is true by assumption for the
subsets $\chi$ containing $\Theta$. For the others sets, this can be done
applying 6.13, inductively on the cardinality of
$\chi\cap\\{k_{1},\ldots,k_{e}\\}$. The basis of the induction is the case
when $\Theta\subseteq\chi$. In the case
$|\chi\cap\\{k_{1},\ldots,k_{e}\\}|=2$, we get
$\chi=\Theta^{\prime}\cup\\{k_{l}\\}$ for some
$\Theta^{\prime}=\Theta\setminus\\{j\\}\cup\\{k_{i}\\}$ of the form considered
in 6.13. Since $m_{\Theta^{\prime}}\neq 0$, as a consequence of 6.13, we get
that $h_{\Theta}$ and $h_{\Theta^{\prime}}$ are associated the polynomial ring
$K[T_{1},\ldots,T_{t}]$ and therefore $U_{\chi}=0$, since also
$U_{\Theta\cup\\{k_{l}\\}}=0$. Moreover, by definitions
$h_{\Theta^{\prime}}\not\in\mathfrak{p}_{\chi}$, and thus also
$h_{\Theta}\not\in\mathfrak{p}_{\chi}$. An iterative application of 6.13,
using the fact that all the $m_{\Theta}$’s are nonzero, allows to deal in the
same way with all the other cases.

We now prove that $Hh_{\Theta}\subseteq I_{n}(B)$. The minimal generators of
$H$ have degree $e$ and are of the form $\beta_{\Omega}=\prod_{i\in
E\setminus\Omega}T_{i}$ where $\Omega\subseteq E$ and $|\Omega|=r-1$. For
every of such sets $\Omega$, since $m_{\Omega}\neq 0$, as consequence of 6.1
and of 6.13, we get $m_{\Omega}=\beta_{\Omega}h_{\Omega}$. Moreover,
$h_{\Theta}$ and $h_{\Omega}$ are associated in $K[T_{1},\ldots,T_{t}]$.
Hence, for some unit $u\in K$, we have
$\beta_{\Omega}h_{\Theta}=\beta_{\Omega}uh_{\Omega}=um_{\Omega}\in
I_{n}(B)$.This implies $Hh_{\Theta}\subseteq I_{n}(B)$ and concludes the
proof. ∎

## Acknowledgements

We thank Paolo Mantero for interesting conversations on ideals of star
configurations that partly motivated this project, and Kuei-Nuan Lin for
referring us to the work of Garrousian, Simis and Tohaneanu [13]. We also are
grateful to Alexandra Seceleanu for helpful feedback on a preliminary version
of this preprint. The third author is supported by the NAWA Foundation grant
Powroty "Applications of Lie algebras to Commutative Algebra".

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