# The $F$-pure threshold and squarefree initial ideal
as invariants of linkage

Vaibhav Pandey  Department of Mathematics, Purdue University, 150 N University
St., West Lafayette, IN 47907, USA<EMAIL_ADDRESS>Dedicated to Professor
Bernd Ulrich on the occasion of his seventieth birthday

###### Abstract.

The generic link of an unmixed, radical (even a generic complete intersection)
ideal is radical (in fact, prime). We show that this cannot be strengthened in
general: The squarefreeness of the initial ideal and $F$-purity are not
preserved along generic links. However, for several important cases in liaison
theory, including generic height three Gorenstein ideals and the maximal
minors of a generic matrix, we show that the squarefreeness of the initial
ideal, $F$-purity, and even the $F$-pure threshold are each preserved along
the $n$-th generic link for every $n>0$ by finding a property of such ideals
which propagates along generic links. We use this property to establish the
$F$-regularity of their generic links. Finally, we study the $F$-pure
thresholds of the generic residual intersections of a complete intersection
ideal and answer a related question of Kim–Miller–Niu.

###### Key words and phrases:

generic link, $F$-pure threshold, initial ideal, generic residual
intersection, $F$-regular

###### 2010 Mathematics Subject Classification:

Primary 13C40, 13A35, 14M06; Secondary 14M10, 14M12.

The author was partially supported by the AMS–Simons Travel Grant.

##  1\. Introduction

Given a proper, unmixed ideal $I$ in a polynomial ring $R$ over a field $K$,
its geometric link is an ideal $J$ such that the ideal $I\cap J$ is generated
by a regular sequence, that is, the set theoretic union of the vanishing loci
of $I$ and $J$ is a complete intersection. Clearly, the link of an ideal
depends on the choice of a regular sequence in $I$. When this regular sequence
is chosen in the ‘most general’ manner, that is, as generic combinations of
the generators of the ideal $I$, the link thus obtained, in a polynomial
extension of $R$, is called the (first) generic link $L_{1}(I)$ of $I$.
Evidently, the generic link specializes to any link of $I$ and, for this
reason, is the prototypical link to study. Iterating this process, the $n$-th
generic link of $I$

$L_{n}(I)\colonequals L_{1}(L_{n-1}(I))$

is the generic link of its $(n-1)$-th generic link, and analogously, models
any ideal which is linked to $I$ in $n$ steps.

Since the beginning of the modern study of linkage, or liaison theory, by
Peskine and Szpiro [PS73], a central research theme has been to understand how
the algebraic and geometric properties of an ideal change when one passes to
its link; in particular, one wishes to understand which properties remain
invariant along links. For example, height and the Cohen-Macaulay property are
invariants of linkage, and the canonical module of the link is ‘dual’ to that
of $R/I$ [PS73]. While it is easily seen that the link of an unmixed, radical
ideal may not be radical, by [HU85, Proposition 2.6], the generic link of an
unmixed, radical ideal (even a generically complete intersection ideal) is
indeed radical (in fact, prime). Since one has a conclusion stronger than the
ideal being radical with an assumption which is weaker than the ideal being
radical, it is natural to ask:

######  Question.

Let $I$ be an unmixed ideal in a polynomial ring $R$ over a field and $<$ a
term order in $R$. Let $L_{1}(I)$ denote its (first) generic link in a
polynomial extension $S$ of $R$.

  1. (1)

If the initial ideal $\operatorname{in}_{<}(I)$ of $I$ is squarefree, is the
initial ideal of the generic link $L_{1}(I)$ also squarefree for some term
order in $S$?

  2. (2)

Assume that $K$ has positive characteristic. If $R/I$ is an $F$-pure ring, is
the generic link $S/L_{1}(I)$ also $F$-pure?

Note that if the ideal $I$ has a squarefree initial ideal with respect to some
term order or if it defines an $F$-pure ring $R/I$, then it is radical to
begin with.

We produce an example (Example 5.4) which answers both of these questions in
the negative. In fact, we show that even if the ring $R/I$ is strongly
$F$-regular, its generic link $S/L_{1}(I)$ may not even be $F$-injective. This
strengthens the fact that generic linkage may not preserve rational
singularities [Niu14, Corollary 3.4].

The bulk of this paper, however, deals with understanding under what
hypothesis are the answers to the above questions in the affirmative. A major
roadblock to our enquiry is that the generators of the generic link are known
in very few cases. While in principle a mapping cone construction exists (see
[PS73]) to resolve the link if we know a resolution of the ideal $I$, it is
quite cumbersome to carry out in practice. The mapping cone construction gives
the generators of the generic link of well-behaved ideals whose resolutions
are short, like a perfect ideal of height $2$, a Gorenstein ideal of height
$3$ ([KU92a, §4.10]), and a complete intersection ideal ([HU88, Example 3.4]).

To get around the issue of not knowing the generators of the generic link, for
an unmixed, homogeneous ideal $I$ in a polynomial ring $R$, we find a property
$\mathbf{P}$ of the pair $(R,I)$ (see Definition 3.1) which propagates along
generic links (see Lemma 3.3). Crucially, the property only depends on the
choice of a term order in $R$ and on the regular sequence defining the link,
and not on the generators of the link.

The simple observation that the property $\mathbf{P}$ propagates along generic
links is quite powerful: We show that if the pair $(R,I)$ has the property
$\mathbf{P}$, then the squarefreeness of the initial ideal, $F$-purity, and
the $F$-pure threshold/log canonical threshold (a numerical measure of
singularity in positive characteristic/birational geometry) are each preserved
along the $n$-th generic link of $I$ for every $n>0$ (Theorem 3.5). In fact,
the $F$-pure threshold/log canonical threshold attains its maximum value—the
height of the ideal—along the generic linkage class of the ideal. The property
$\mathbf{P}$ holds in several important cases in liaison theory including
generic height three Gorenstein ideals (Proposition 4.1) and the maximal
minors of a generic matrix (Proposition 5.1). We highlight the key results
which we are able to prove using this point of view:

### (1)

We show that the generic link of a generic height $3$ Gorenstein ideal is
strongly $F$-regular (Theorem 4.3). We remark that the generators of the
generic link of height three Gorenstein ideals are known ([KU92a, §4.10],
[KU92b, Theorem 4.7]); however, due to our approach of viewing the generic
link via the property $\mathbf{P}$, we do not need the generators of the
generic link to establish its $F$-regularity (or its $F$-pure threshold, or
the squarefreeness of its initial ideal). We also show the $n$-th universal
link of a generic height $3$ Gorenstein ideal is $F$-rational (rational
singularities in characteristic $0$) for each $n>0$.

### (2)

In [PT24, Theorem 6.3], the author and Tarasova showed that the generic link
of the maximal minors of a generic matrix is strongly $F$-regular (note that
the generators of the generic link are not known). This result may now be
viewed as an offshoot of the fact that the ideal of maximal minors of a
generic matrix has the property $\mathbf{P}$ (Proposition 5.1). Remarkably,
this perspective allows us to recover the $F$-pure threshold/log canonical
threshold of the maximal minors of a generic matrix calculated in [MSV14,
Theorem 1.2], [Doc13, Theorem 5.6] and that of the generic link of maximal
minors of a generic matrix calculated in [KMN21, Theorem 1] by a simpler
method, as well as to extend these calculations to the $n$-th generic link for
each $n>0$ (see Remark 5.2).

We show that the ideal of non-maximal minors of a generic matrix typically
does not have the property $\mathbf{P}$, and, due to this, each of the three
properties: squarefreeness of the initial ideal, $F$-purity, and the $F$-pure
threshold typically fails to be transferred to its generic link (see Example
5.4).

### (3)

Residual intersections are a vast generalization of linkage, where, under
appropriate technical hypothesis, the regular sequence in consideration is
replaced by an ideal of height $s$, with $s$ atleast the height of $I$. We
show that if $I$ is a complete intersection ideal in a polynomial ring $R$ and
the pair $(R,I)$ has the property $\mathbf{P}$, then the generic $s$-residual
intersections of $I$ also have $\mathbf{P}$ for each $s$ which is atleast as
large as the height of $I$ (Theorem 6.4). In particular, when $s$ is the
height of $I$, this observation helps us to answer [KMN21, Question 4.5(b)] of
Kim–Miller–Niu by finding a class of complete intersection ideals (those
possessing $\mathbf{P}$), for which the $F$-pure threshold is invariant under
generic linkage. We also note that if the complete intersection ideal $I$ does
not have $\mathbf{P}$, then its $F$-pure threshold may not be invariant under
generic linkage (see Remark 6.6).

The paper is organized as follows: In §2, we recall the background of linkage
and positive characteristic techniques needed for the paper. In §3, we discuss
the property $\mathbf{P}$ and its consequences. In §4, we establish the
$F$-regularity of the generic link of generic height $3$ Gorenstein ideals and
discuss some immediate corollaries. In §5, we discuss some subtleties related
to the property $\mathbf{P}$. Finally, in §6, we recall the notion of residual
intersections and study the $F$-pure thresholds of the generic residual
intersections of a complete intersection ideal.

##  2\. Background

### Linkage

While the notion of linkage holds more generally over Cohen–Macaulay rings
(and most of the important results hold true over Gorenstein rings), for the
purpose of this paper, and for simplicity of exposition, we restrict our
attention to the situation where the ambient ring is polynomial (see the
references in this subsection for the general statements).

######  Definition 2.1.

Let $R$ be a polynomial ring, and let $I$ and $J$ be proper $R$-ideals. We say
that $I$ and $J$ are _linked_ (or $R/I$ and $R/J$ are linked) if there exists
an $R$-ideal $\mathfrak{a}$ generated by a regular sequence such that

$J=\mathfrak{a}:I\qquad\text{and}\qquad I=\mathfrak{a}:J,$

and use the notation $I\sim_{\mathfrak{a}}J$. Furthermore we say that the link
is _geometric_ if we have $\operatorname{ht}(I+J)\geq\operatorname{ht}(I)+1$
(where $\operatorname{ht}(-)$ denotes height).

It is clear that the ideal $\mathfrak{a}$ is contained in $I$ and $J$. Note
that the associated primes of $I$ and $J$ have the same height, that is, the
ideals $I$ and $J$ are unmixed. Further, the heights of the ideals $I$, $J$,
and $\mathfrak{a}$ are equal. Moreover, when the link is geometric, it follows
that the ideal $\mathfrak{a}$ is the intersection of $I$ and $J$.

The linked ideal is ‘dual’ to the given ideal in a sense made precise by the
following foundational result of linkage theory:

######  Proposition 2.2.

[PS73] Let $R$ be a polynomial ring, $I$ and $J$ be $R$-ideals, and
$\mathfrak{a}$ is an ideal generated by a regular sequence such that
$I=\mathfrak{a}:J$ and $J=\mathfrak{a}:I$. Suppose that $R/I$ is a
Cohen–Macaulay ring. Then

  1. (1)

$R/J$ is a Cohen–Macaulay ring.

  2. (2)

If $R$ is a local ring, $\omega_{R/I}\cong J/\mathfrak{a}$ and
$\omega_{R/J}\cong I/\mathfrak{a}$, where $\omega_{R/I}$ (respectively
$\omega_{R/J}$) denotes the canonical module of $R/I$ (respectively $R/J$).

  3. (3)

If the ideals $I$ and $J$ are geometrically linked, then
$\operatorname{ht}(I+J)=\operatorname{ht}(I)+1$ and $R/(I+J)$ is a Gorenstein
ring.

######  Definition 2.3.

Let $R$ be a polynomial ring and $I$ an unmixed $R$-ideal of height $g>0$. Let
$\mathbf{f}\colonequals f_{1},\ldots,f_{n}$ be a generating set of $I$. Let
$Y$ be a $g\times n$ matrix of indeterminates, and let $\mathfrak{a}$ be the
ideal generated by the entries of the matrix $Y[f_{1}\dots f_{n}]^{T}$ (where
$[\quad]^{T}$ denotes the transpose of the matrix). The ideal

$L_{1}(I)=L_{1}(\mathbf{f})\colonequals\mathfrak{a}R[Y]:IR[Y]$

is a _generic link_ of $I$.

We point out that, in the above definition, the entries of the matrix
$Y[f_{1}\dots f_{n}]^{T}$ form an $R[Y]$-regular sequence due to [Hoc73]. A
generic link of $I$ is indeed a (geometric) link of $I$:

######  Proposition 2.4.

[PS73] Suppose that $R$ is a polynomial ring and that $I$ is an unmixed ideal
of $R$ of height $g$. Let $\mathfrak{a}$ be an $R$-ideal generated by a length
$g$ regular sequence which is properly contained in the ideal $I$, and let
$J=\mathfrak{a}:I$, then $I\sim_{\mathfrak{a}}J$.

While the definition of generic link depends on a choice of a generating set
of the ideal, it turns out that any two generic links are equivalent:

######  Definition 2.5.

Let $(R,I)$ and $(S,J)$ be pairs where $R$ and $S$ are Noetherian rings and
$I\subseteq R$, $J\subseteq S$ are ideals. We say $(R,I)$ and $(S,J)$ are
_equivalent_ , and write $(R,I)\equiv(S,J)$, if there exist finite sets of
variables, $X$ over $R$ and $Z$ over $S$, and an isomorphism
$\varphi:R[X]\rightarrow S[Z]$ such that $\varphi(IR[X])=JS[Z]$.

######  Lemma 2.6.

[HU85, Proposition 2.4] Let $I$ be an unmixed ideal in a polynomial ring $R$.
If $J_{1}\subseteq R[X_{1}]$ and $J_{2}\subseteq R[X_{2}]$ are generic links
of $I$, then we have $(R[X_{1}],J_{1})\equiv(R[X_{2}],J_{2})$.

Due to the above lemma, we freely use the phrase “the generic link” of an
ideal in this paper.

######  Definition 2.7.

Let $R$ be a polynomial ring and $I$ be an unmixed $R$-ideal of height $g>0$.
Let $\mathbf{f}$ be a generating set of $I$. The $n$-th generic link of $I$ is
the ideal

$L_{n}(I)=L_{1}(L_{n-1}(I))\quad\text{for}\;n>0\quad\text{with}\quad
L_{0}(I)=I\quad\text{and}\quad L_{1}(I)=L_{1}(\mathbf{f}).$

Assume that $R$ is a regular local ring. The first universal link $L^{1}(I)$
of $I$ is obtained similarly as the generic link with respect to invertible
indeterminates and the $n$-th universal link is obtained iteratively:

$L^{n}(I)=L^{1}(L^{n-1}(I))\quad\text{for}\;n>0\quad\text{with}\quad
L^{0}(I)=I\quad\text{and}\quad L^{1}(I)=L^{1}(\mathbf{f}).$

The linkage class of an ideal $I$ in a polynomial ring $R$ is the set of all
$R$-ideals which can be obtained from $I$ by a finite number of links.

The point of defining the $n$-th generic link of $I$ (respectively the $n$-th
universal link of $I$) is that the $n$-th generic link (respectively the
$n$-th universal link) is a deformation (respectively, essentially a
deformation) 111We recall that a ring $A$ is essentially a deformation of
another ring $B$ if it is obtained by a finite sequence of deformations and
localizations at prime ideals beginning from $B$. of any $R$-ideal linked to
$I$ in $n$ steps ([HU87, Proposition 2.14, Theorem 2.17]) and therefore
controls the algebro-geometric properties of any link of $I$ upto deformation.
In fact, under mild assumptions, one can descend from a sequence of universal
links to a sequence of links in the original ring and still preserve most of
the good properties of universal linkage ([HU88, Lemma 2.1]).

### Frobenius splittings and the $F$-pure threshold

For a reduced Noetherian ring $R$ of prime characteristic $p>0$, the
($e$-fold) Frobenius endomorphism on $R$ is the map $F^{e}:R\to R$ with
$F^{e}(r)=r^{p^{e}}$. To avoid any confusion between the the domain and
codomain, we instead use the notation $F^{e}_{*}(R)$ for the codomain and
$F^{e}_{*}(r)$ for its elements. So $F^{e}_{*}(R)$ is the same ring as $R$
with the $R$-module structure obtained from the Frobenius map:

$r.F^{e}_{*}(s)\colonequals F^{e}_{*}(r^{p^{e}}s).$

The ring $R$ is $F$-finite if $F^{e}_{*}(R)$ is a finite $R$-module for some
(equivalently, every) $e>0$. A finitely generated algebra $R$ over a field $K$
is $F$-finite if and only if $F_{*}(K)$ is a finite field extension of $K$—a
fairly mild condition. The ring $R$ is $F$-pure if the Frobenius endomorphism
$F:R\to F_{*}(R)$ with $F(r)=F_{*}(r^{p})$ is pure, that is, for any
$R$-module $M$, the map $F\otimes 1:R\otimes_{R}M\to F_{*}(R)\otimes_{R}M$ is
injective. Note that if $R$ is an $F$-pure ring, the Frobenius map $F:R\to
F^{e}_{*}(R)$ splits as a map of $R$-modules so that the $R$-module $F_{*}(R)$
admits a free $R$-summand. An $F$-finite ring $R$ is strongly $F$-regular if
it has a sufficiently large number of Frobenius splittings, made precise as
follows: For every element $c\in R^{\circ}$, there exists an integer $q=p^{e}$
such that the $R$-linear map $R\longrightarrow F^{e}_{*}(R)$ sending $1$ to
$F^{e}_{*}(c)$ splits as a map of $R$-modules. Clearly, a strongly $F$-regular
ring is $F$-pure.

The rings under consideration in this paper are $\mathbb{N}$-graded; since the
various competing notions of $F$-regularity (like weakly $F$-regular,
$F$-regular, and strongly $F$-regular) coincide in the graded case due to
[LS99, Corollary 4.3], we make no distinction between them throughout the
paper.

The following result will help us in showing that several ideals considered in
this paper define $F$-pure rings.

######  Lemma 2.8.

[PT24, Lemma 3.1, Corollary 3.3] Let $R=K[x_{1},\ldots,x_{n}]$ be a polynomial
ring over the field $K$ of characteristic $p>0$ and let $I$ be an unmixed
homogeneous $R$-ideal. If ${\mathfrak{a}}\subsetneq I$ is an ideal generated
by a regular sequence of length equal to the height of $I$ and
$J={\mathfrak{a}}:I$, then

${\mathfrak{a}}^{[p]}:{\mathfrak{a}}\subseteq(I^{[p]}:I)\cap(J^{[p]}:J).$

In particular, if the ring $S/{\mathfrak{a}}$ is $F$-pure, then so are $S/I$
and $S/J$.

######  Definition 2.9.

Let $R$ be a polynomial ring over a field of characteristic $p>0$ and let
${\mathfrak{m}}$ denote its homogeneous maximal ideal. For a homogeneous
proper ideal $I$ and integer $e>0$, set

$\nu_{I}(p^{e})=\max\\{r\in\mathbb{N}\;|\;I^{r}\nsubseteq{\mathfrak{m}}^{[p^{e}]}\\},$

where ${\mathfrak{m}}^{[p^{e}]}=(a^{p^{e}}\;|\;a\in{\mathfrak{m}})$. If $I$ is
generated by $N$ elements, it is readily seen that

$0\leq\nu_{I}(p^{e})\leq N(p^{e}-1).$

Moreover, if $f\in I^{r}\setminus{\mathfrak{m}}^{[p^{e}]}$, then $f^{p}\in
I^{pr}\setminus{\mathfrak{m}}^{[p^{e+1}]}$. Thus,

$\nu_{I}(p^{e+1})\geq p\nu_{I}(p^{e}).$

It follows that the sequence of real numbers
$\left\\{\nu_{I}(p^{e})/p^{e}\right\\}_{e>0}$ is non-decreasing and bounded
above; its limit is the $F$-pure threshold of $I$, denoted
$\operatorname{fpt}(I)$.

The notion of $F$-pure thresholds is due to Takagi and Watanabe [TW04]. We
point out that the $F$-pure threshold may be defined in a more general setup
(see [MTW05, DSNnBP18, DSNnB18]), however the above definition is adequate for
this paper. The $F$-pure threshold is the positive characteristic analog of
the log canonical threshold: a numerical measure of singularity in birational
geometry. For simplicity of exposition, let $I$ be a homogeneous ideal in a
polynomial ring over the rational numbers. Using “$I$ modulo $p$” to denote
the characteristic $p$ model, one has the inequality

$\operatorname{fpt}(I\,\text{ modulo
}\,p)\leq\operatorname{lct}(I)\quad\text{for all}\;p\gg 0,$

where $\operatorname{lct}(I)$ denotes the log canonical threshold of $I$.
Moreover,

$\lim_{p\to\infty}\operatorname{fpt}(I\,\text{ modulo
}\,p)=\operatorname{lct}(I).$

These facts follow from the work of Hara and Yoshida [HY03]; see [MTW05,
Theorems 3.3, 3.4]. The following result is well-known to experts; we include
a proof for the convenience of the reader.

######  Lemma 2.10.

The $F$-pure threshold of a homogeneous ideal $I$ in a polynomial ring $R$ is
bounded above by its height. Furthermore, if the homogeneous ideal is unmixed
and radical, then

$\operatorname{fpt}(I)=\operatorname{ht}(I)$

implies that the ring $R/I$ is $F$-pure.

###### Proof.

Let ${\mathfrak{p}}$ be a prime ideal in the polynomial ring
$R=K[\underline{x}]$, where the field $K$ has characteristic $p>0$ and
${\mathfrak{m}}$ is the homogeneous maximal ideal of $R$. Then
$R_{{\mathfrak{p}}}$ is a regular local ring of dimension
$\operatorname{ht}({\mathfrak{p}})$. For each $e>0$, the pigeonhole principle
gives

${\mathfrak{p}}_{{\mathfrak{p}}}^{\operatorname{ht}{{\mathfrak{p}}}(p^{e}-1)+1}\subseteq{\mathfrak{p}}_{{\mathfrak{p}}}^{[p^{e}]}.$

Contracting back to $R$, by the flatness of the Frobenius map, we get

${\mathfrak{p}}^{(\operatorname{ht}{p}(p^{e}-1)+1)}\subseteq{\mathfrak{p}}^{[p^{e}]}.$

Therefore

${\mathfrak{p}}^{r}\subseteq{\mathfrak{p}}^{(r)}\subseteq{\mathfrak{p}}^{[p^{e}]}\subseteq{\mathfrak{m}}^{[p^{e}]}\quad\text{for}\quad
r>\operatorname{ht}{{\mathfrak{p}}}(p^{e}-1).$

In particular, let ${\mathfrak{p}}$ be a minimal prime of the ideal $I$ of the
same height. We get

$\operatorname{fpt}(I)\leq\operatorname{fpt}({\mathfrak{p}})\leq\lim_{e\to\infty}\frac{\operatorname{ht}(I)(p^{e}-1)}{p^{e}}=\operatorname{ht}(I).$

The last assertion follows from [Tak04, Theorem 3.11] since the hypothesis

$\operatorname{fpt}(I)=\operatorname{ht}(I)$

implies that the pair $(R,I^{\operatorname{ht}(I)})$ is $F$-pure (see [Tak04,
§3] for the notion of $F$-purity of pairs). ∎

In the next section, we will calculate the $F$-pure thresholds (and log
canonical thresholds) of certain geometric links. We point out out that these
calculations are independent of the choice of a generating set of the ideal
essentially due to lemma 2.6 (alternatively, see [KMN21, Lemma 2.8]).

The next result will be useful in showing that several ideals considered in
this paper have a squarefree initial ideal.

######  Theorem 2.11.

[KV23, Theorem 3.13] Let $R$ be a polynomial ring over a field. Let $I$ be a
radical ideal and $<$ a term order in $R$. Let $g$ be the maximum of the
heights of the associated prime ideals of $I$.

If the initial ideal $\operatorname{in}_{<}(I^{(g)})$ contains a squarefree
monomial, then $\operatorname{in}_{<}(I)$ is a squarefree monomial ideal.

##  3\. The property $\mathbf{P}$

######  Definition 3.1.

Let $R$ be a polynomial ring over a field and $I$ a homogeneous $R$-ideal. Let
$\mathbf{P}$ be the following property of the pair $(R,I)$: For a fixed term
order $<$ in $R$, there exist homogeneous elements
$\underline{\alpha}\colonequals\alpha_{1},\ldots,\alpha_{\operatorname{ht}(I)}$
in the ideal $I$ such that each initial term
$\operatorname{in}_{<}(\alpha_{i})$ is squarefree and each pair of initial
terms $\operatorname{in}_{<}(\alpha_{i}),\operatorname{in}_{<}(\alpha_{j})$ is
mutually coprime for $i\neq j$.

######  Remark 3.2.

Since each pair of initial terms
$\operatorname{in}_{<}(\alpha_{i}),\operatorname{in}_{<}(\alpha_{j})$ is
mutually coprime, the monomials
$\operatorname{in}_{<}(\alpha_{1}),\ldots,\operatorname{in}_{<}(\alpha_{\operatorname{ht}(I)})$
form an $R$-regular sequence. Therefore the polynomials
$\underline{\alpha}=\alpha_{1},\ldots,\alpha_{\operatorname{ht}(I)}$ also form
an $R$-regular sequence (see, for example, [BCRV22, Proposition 1.2.12]).
Since $I$ is homogeneous, as is $(\underline{\alpha})$, we note that
$(\underline{\alpha}):I$ is also a homogeneous $R$-ideal.

An important insight of this paper is that $\mathbf{P}$ propagates along
generic links:

######  Lemma 3.3.

Let $I$ be an unmixed homogeneous ideal in a polynomial ring
$R=K[\underline{x}]$ over the field $K$. Assume that the pair $(R,I)$ has the
property $\mathbf{P}$.

  1. (1)

Let $L_{n}(I)$ denote the $n$-th generic link of $I$ in a polynomial extension
$R[Y]$. For any integer $n\geq 0$, the pair $(R[Y],L_{n}(I))$ has
$\mathbf{P}$.

  2. (2)

If the ideal $I$ is not generated by a regular sequence, then
$(\underline{\alpha}):I$ is a link of $I$ in the polynomial ring $R$. The pair
$(R,(\underline{\alpha}):I)$ has $\mathbf{P}$.

###### Proof.

We first prove item $(1)$. Assume that the ideal $I$ has height $g>0$ and
proceed by induction on $n$. Since the $n$-th generic link of $I$ is the
generic link of the $(n-1)$-th generic link, i.e.,

$\ L_{n}(I)=L_{1}(L_{n-1}(I)),$

it suffices to prove the assertion for $n=1$. Note that the generic link is a
homogeneous ideal (in some polynomial extension of $R$) for each $n>0$.

Let the elements $\alpha_{1},\ldots,\alpha_{g}$ inside the ideal $I$ be as
given by $\mathbf{P}$. Extend it to find generators
$\alpha_{1},\ldots,\alpha_{g},\alpha_{g+1},\ldots,\alpha_{r}$ of $I$ and fix
this generating set. We proceed to find homogeneous elements
$\underline{\beta}\colonequals\beta_{1},\ldots,\beta_{g}$ inside the ideal
$L_{1}(I)$ and a term order $<_{1}$ in the polynomial ring $R[Y]$, where
$Y\colonequals(Y_{i,j})$ is a $g\times r$ matrix of indeterminates such that
the pair $(R[Y],L_{1}(I))$ has $\mathbf{P}$. Recall that

$L_{1}(I)=Y[\alpha_{1}\cdots\alpha_{g}\,\alpha_{g+1}\cdots\alpha_{r}]^{T}R[Y]:(\alpha_{1},\ldots,\alpha_{r})R[Y].$

Define the following variable order in $R[Y]$:

$Y_{1,1}>Y_{2,2}>\cdots>Y_{g,g}>\text{ the remaining }Y_{i,j}>x_{k},$

where the order $<$ on the indeterminates $x_{k}$ is the one given in the
property $\mathbf{P}$ and the order on “the remaining $Y_{i,j}$” is arbitrary.
Consider the lexicographical order $<_{1}$ induced by this variable order in
$R[Y]$.

Let $\beta_{1},\ldots\beta_{g}$ denote the entries of the matrix
$Y[\alpha_{1}\cdots\alpha_{r}]^{T}$. Then,

$\displaystyle\operatorname{in}_{<_{1}}(\beta_{1})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{1,1}\alpha_{1}+Y_{1,2}\alpha_{2}+\ldots+Y_{1,r}\alpha_{r})=\operatorname{in}_{<_{1}}(Y_{1,1}\alpha_{1})=Y_{1,1}\operatorname{in}_{<}(\alpha_{1}),$
$\displaystyle\operatorname{in}_{<_{1}}(\beta_{2})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{2,1}\alpha_{1}+Y_{2,2}\alpha_{2}+\ldots+Y_{2,r}\alpha_{r})=\operatorname{in}_{<_{1}}(Y_{2,2}\alpha_{2})=Y_{2,2}\operatorname{in}_{<}(\alpha_{2}),$
$\displaystyle\vdots$ $\displaystyle\operatorname{in}_{<_{1}}(\beta_{g})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{g,1}\alpha_{1}+Y_{g,2}\alpha_{2}+\ldots+Y_{g,r}\alpha_{r})=\operatorname{in}_{<_{1}}(Y_{g,g}\alpha_{g})=Y_{g,g}\operatorname{in}_{<}(\alpha_{g}).$

Since the pair $(R,I)$ has $\mathbf{P}$, we know that for the elements
$\underline{\alpha}=\alpha_{1},\ldots,\alpha_{g}$, each initial term
$\operatorname{in}_{<}(\alpha_{i})$ is squarefree and each pair of initial
terms $\operatorname{in}_{<}(\alpha_{i}),\operatorname{in}_{<}(\alpha_{j})$ is
mutually coprime for $i\neq j$. It immediately follows from the above display
that the same assertions also hold true for the elements
$\underline{\beta}=\beta_{1},\ldots,\beta_{g}$ of $R[Y]$.

We now prove item $(2)$. Since $I$ is not generated by a regular sequence, the
proper (homogeneous) $R$-ideal $\underline{\alpha}:I$ is a link of the unmixed
ideal $I$ by Proposition 2.4. The pair $(R,(\underline{\alpha}):I)$ has
$\mathbf{P}$ because

$\underline{\alpha}\subseteq(\underline{\alpha}):I.\qed$

######  Remark 3.4.

Note that the above lemma also holds true if the phrase “each initial term
$\operatorname{in}_{<}(\alpha_{i})$ is squarefree” is deleted from the
property $\mathbf{P}$ or if the sequence of elements $\underline{\alpha}$ is
shorter. However, the applications that we have in mind require $\mathbf{P}$
to be as stated.

The next result says that if the pair $(R,I)$ has $\mathbf{P}$, then important
properties of the ideal $I$ like the initial ideal being squarefree,
$F$-purity, and the $F$-pure threshold remain invariant under linkage. In
fact, we get that the $F$-pure threshold (or the log canonical threshold) of
each generic link of $I$, and hence that of $I$ itself, attains its maximal
value. That is, the $F$-pure threshold of the $n$-th generic link is equal to
the height—an invariant of linkage!

######  Theorem 3.5.

Let $R$ be a polynomial ring over a field $K$. Fix an integer $n\geq 0$ and
let $I$ be an unmixed homogeneous $R$-ideal. Assume that the pair $(R,I)$ has
the property $\mathbf{P}$.

  1. (1)

If the field $K$ has positive characteristic, then

$\operatorname{fpt}(L_{n}(I))=\operatorname{ht}(I).$

In particular, the $F$-pure threshold of $I$ is the height of $I$.
Furthermore, the ideal $L_{n}(I)$ defines an $F$-pure ring (in a polynomial
extension of $R$).

  2. (2)

If the field $K$ has characteristic zero, then

$\operatorname{lct}(L_{n}(I))=\operatorname{ht}(I).$

In particular, the log canonical threshold of $I$ is the height of $I$.
Furthermore, the ideal $L_{n}(I)$ defines a log canonical singularity (in a
polynomial extension of $R$)

  3. (3)

The initial ideal of the $n$-th generic link $L_{n}(I)$ is squarefree. In
particular, the initial ideal of $I$ is squarefree.

###### Proof.

Assume throughout that the ideal $I$ has height $g>0$. Recall that since
$L_{n}(I)$ is linked to the ideal $I$, it also has height $g$. Since the pair
$(R,I)$ has the property $\mathbf{P}$, by Lemma 3.3, the pair
$(R[Y],L_{n}(I))$ also has $\mathbf{P}$. Let the elements
$\underline{\alpha}=\alpha_{1},\ldots,\alpha_{g}$ in $L_{n}(I)$ and the term
order $<$ in $R[Y]$ be as stated in $\mathbf{P}$.

Note that item $(2)$ immediately follows from item $(1)$ since $F$-pure rings
in characteristic $p>0$ are log canonical in characteristic zero ([HW02,
Theorem 3.9]) and the log canonical threshold of an ideal is the limit of the
$F$-pure thresholds of its characteristic $p>0$ models. In view of this,
assume that $K$ has characteristic $p>0$. We begin by proving item $(1)$.

Since $\underline{(\alpha)}\subseteq L_{n}(I)$ and, by Lemma 2.10, the
$F$-pure threshold of an ideal is bounded above by its height, we get

$\operatorname{fpt}(\underline{(\alpha)})\leq\operatorname{fpt}(L_{n}(I))\leq
g.$

We claim that $\operatorname{fpt}(\underline{(\alpha)})=g$. This would give us

$g=\operatorname{fpt}(\underline{(\alpha)})\leq\operatorname{fpt}(L_{n}(I))\leq
g,$

so that we have equality throughout. We now show that
$\operatorname{fpt}(\underline{(\alpha)})=g$.

The initial term of

$f\colonequals\alpha_{1}\ldots\alpha_{g}\in\underline{(\alpha)}^{g}$

is squarefree as the pair $(R[Y],L_{n}(I))$ has $\mathbf{P}$. For any $e>0$,
${\mathfrak{m}}^{[p^{e}]}$ is a monomial ideal. So, we have

(1)
$f^{p^{e}-1}=(\alpha_{1}\ldots\alpha_{g})^{p^{e}-1}\in((\underline{\alpha}^{[p^{e}]}):\underline{(\alpha)})\smallsetminus{\mathfrak{m}}^{[p^{e}]}.$

This gives

$(\underline{\alpha})^{g(p^{e}-1)}\nsubseteq{\mathfrak{m}}^{[p^{e}]},\quad\text{while}\quad(\underline{\alpha})^{g(p^{e}-1)+1}\subseteq{\mathfrak{m}}^{[p^{e}]}$

by the pigeonhole principle. We conclude that

$\nu_{({\underline{\alpha}})}(p^{e})=\max\\{r\;|\;(\underline{\alpha})^{r}\nsubseteq{\mathfrak{m}}^{[p^{e}]}\\}=g(p^{e}-1)$

for each $e>0$. Therefore

$\operatorname{fpt}(\underline{\alpha})=g,$

as required; we thus have

$\operatorname{fpt}(L_{n}(I))=\operatorname{ht}(I).$

The $F$-purity of the ring $R[Y]/L_{n}(I)$ follows from Lemma 2.10;
alternatively, it also follows from Equation 1 in view of Lemma 2.8.

We now prove item $(3)$. Since the $n$-th generic link $R[Y]/L_{n}(I)$ is
$F$-pure, we get that $L_{n}(I)$ is a radical ideal. Consider the polynomial

$f\colonequals\alpha_{1}\ldots\alpha_{g}\in L_{n}(I)^{g}.$

Then the initial term

$\operatorname{in}_{<}(f)=\prod_{i=1}^{g}\operatorname{in}_{<}(\alpha_{i})\in\operatorname{in}_{<}((L_{n}(I)^{g})\subseteq\operatorname{in}_{<}((L_{n}(I)^{(g)})$

and is squarefree, as noted above. All the requirements of Theorem 2.11 are
met; we conclude that $L_{n}(I)$ has a squarefree initial ideal. ∎

######  Remark 3.6.

The reader may note that the conclusions of Theorem 3.5 hold for any pair
$(R,I)$ which has $\mathbf{P}$. This will be used in §6, where we show that
the generic residual intersections of a complete intersection ideal inherit
$\mathbf{P}$.

We specialize to a link of $I$ in the ambient polynomial ring for which the
conclusions of Theorem 3.5 hold:

######  Corollary 3.7.

Let $R$ be a polynomial ring over a field $K$ and $I$ an unmixed homogeneous
$R$-ideal which is not generated by a regular sequence. Assume that the pair
$(R,I)$ has the property $\mathbf{P}$. Then the ideal $(\underline{\alpha}):I$
is a link of $I$. Moreover,

  1. (1)

If the field $K$ has positive characteristic, the ideal
$(\underline{\alpha}):I$ defines an $F$-pure ring and

$\operatorname{fpt}((\underline{\alpha}):I)=\operatorname{ht}(I).$

  2. (2)

If the field $K$ has characteristic zero, the ideal $(\underline{\alpha}):I$
defines a log-terminal singularity and

$\operatorname{lct}((\underline{\alpha}):I)=\operatorname{ht}(I).$

  3. (3)

The initial ideal of $(\underline{\alpha}):I$ is squarefree.

###### Proof.

Since $I$ is not generated by a regular sequence, $(\underline{\alpha}):I$ is
a proper $R$-ideal; it is a link of $I$ by Remark 3.2. By the second assertion
of Lemma 3.3, the pair $(R,(\underline{\alpha}):I)$ has $\mathbf{P}$. The
arguments for each of the assertions are precisely the same as those of
Theorem 3.5. ∎

In the next two sections, we list a number of examples, crucial in the study
of linkage, of pairs $(R,I)$ which have the property $\mathbf{P}$. We remind
the reader that each of these examples satisfy the conclusions of Theorem 3.5
and Corollary 3.7.

In [KMN21, Question 4.5 (a)], Kim–Miller–Niu ask if the equality of the log
canonical thresholds of a variety $X$ and that of its generic link implies the
equality of the log canonical threshold of $X$ and those of the higher generic
links of $X$. In view of Theorem 3.5, the examples listed below provide
evidence in favor of an affirmative answer: For each pair $(R,I)$ in the
examples below, the log canonical threshold of the variety and that of its
$n$-th generic link are equal for each $n>0$.

######  Remark 3.8.

In each of the examples discussed in the next two sections, the elements
$\underline{\alpha}$ discussed in the property $\mathbf{P}$ form a part of a
minimal generating set of the ideal of interest. Hence the $n$-th
(homogeneous) generic link that we get in each case is actually the $n$-th
minimal (homogeneous) generic link. Minimal linkage is generally preferable
over arbitrary linkage: The minimal link of an almost complete intersection
ideal defines a Gorenstein ring. For other favorable properties of minimal
links, see [HU87, Remark 2.7].

##  4\. $F$-regularity of the generic link of generic height $3$ Gorenstein
ideals

Let $I$ be a height $3$ ideal in a polynomial ring $R$ such that $R/I$ is a
Gorenstein ring. By the Buchsbaum–Eisenbud structure theorem [BE77], the
generators of $I$ are given by the size $2n$ pfaffians of a
$(2n+1)\times(2n+1)$ alternating matrix. If the entries of this matrix $X$ are
indeterminates, then $I\colonequals\operatorname{Pf}_{2n}(X)$ is said to be a
‘generic’ height $3$ Gorenstein ideal in the ring $R=K[X]$. Note that any
other such ideal can be obtained by appropriately specializing the entries of
$X$.

######  Proposition 4.1.

The pair $(K[X],\operatorname{Pf}_{2n}(X))$ has $\mathbf{P}$.

###### Proof.

Let

$\underline{\alpha}\colonequals[1,2,\ldots,2n],[1,2,\ldots,n,n+2,\ldots,2n+1],[2,3,\ldots,2n+1]$

denote the pfaffians of $X$ with the given rows and columns and fix the
lexicographical order in $K[X]$ induced by the variable order

$x_{1,2n+1}>x_{1,2n}>\cdots>x_{1,2}>x_{2,2n+1}>\cdots>x_{2,3}>x_{3,2n+1}>\cdots>x_{2n,2n+1}$

which moves from left to right along any row of the matrix $X$ and then
vertically downwards. The initial terms

$\displaystyle\operatorname{in}_{<}([1,2,\ldots,2n])$
$\displaystyle=\prod_{i=1}^{n}x_{i,2n+1-i},$
$\displaystyle\operatorname{in}_{<}([1,2,\ldots,n,n+2,\ldots,2n+1])$
$\displaystyle=\prod_{i=1}^{n}x_{i,2n+2-i},$
$\displaystyle\operatorname{in}_{<}([2,3,\ldots,2n+1])$
$\displaystyle=\prod_{i=2}^{n+1}x_{i,2n+3-i}$

are precisely as required for the property $\mathbf{P}$ to hold. Since the
ideal $\operatorname{Pf}_{2n}(X)$ has height $3$, we are done. ∎

######  Corollary 4.2.

Let $I$ be a generic height $3$ Gorenstein ideal in a polynomial ring and
$n\geq 0$. Then,

$\operatorname{fpt}(L_{n}(I))=3.$

In particular, $\operatorname{fpt}(I)=3$. Furthermore, the initial ideal of
$L_{n}(I)$ is squarefree.

###### Proof.

This is immediate from Proposition 4.1 in view of Theorem 3.5. ∎

Proposition 4.1 has a remarkable consequence: The generic link of a generic
height $3$ Gorenstein ideal defines an $F$-regular ring.

######  Theorem 4.3.

Let $X\colonequals(x_{i,j})$ be a $(2n+1)\times(2n+1)$ alternating matrix of
indeterminates, $K$ a field, and $R\colonequals K[X]$. Let
$\operatorname{Pf}_{2n}(X)$ denote the ideal generated by the size $2n$
pfaffians of $X$.

  1. (1)

If $K$ is an $F$-finite field of characteristic $p>0$, the generic link of
$R/\operatorname{Pf}_{2n}(X)$ is $F$-regular.

  2. (2)

If $K$ has characteristic $0$, the generic link of
$R/\operatorname{Pf}_{2n}(X)$ has rational singularities.

###### Proof.

Assertion $(2)$ follows from $(1)$ since $F$-regular rings are $F$-rational
and $F$-rational rings have rational singularities by [Smi97, Theorem 4.3]. We
therefore concentrate on the case where the characteristic of $K$ is positive.

Let $Y\colonequals(Y_{i,j})$ be a generic $3\times(2n+1)$ matrix of
indeterminates and $S\colonequals R[Y]$. The proof will proceed by induction
on $n$. Note that if $n=1$, the ideal
$\operatorname{Pf}_{2n}(X)=\operatorname{Pf}_{2}(X)$ is the homogeneous
maximal ideal of a polynomial ring in $3$ variables. So the $F$-regularity of
the generic link is a special case of [PT24, Theorem 5.1]. For the remainder
of the proof, assume $n\geq 2$. Let $\Delta_{1},\ldots,\Delta_{2n+1}$ denote
the pfaffians of $X$ with

$\Delta_{2n}=[1,3,\ldots,2n+1]\quad\text{and}\quad\Delta_{2n+1}=[2,3,\ldots,2n+1].$

We have,

$J=Y[\Delta_{1},\ldots,\Delta_{2n+1}]^{T}S:(\Delta_{1},\ldots,\Delta_{2n+1})S,$

and $S/J$ is the generic link of $R/\operatorname{Pf}_{2n}(X)$.

We first show that the ring $(S/J)_{x_{1,2}}$ is $F$-regular. To prove the
inductive step $n>1$, we show that the localization $(S/J)_{x_{1,2}}$ is a
faithfully flat extension of the generic link $S^{\prime}/J^{\prime}$ of the
size $2n-2$ pfaffians of a generic alternating $(2n-1)\times(2n-1)$ matrix
$X^{\prime}$, with $S^{\prime}=K[X^{\prime}]$.

Let $\alpha_{i}$ denote the $i$-th entry of the matrix
$Y[\Delta_{1},\ldots,\Delta_{2n+1}]^{T}$ for $1\leq i\leq 2n+1$; set
$\alpha_{i}^{\prime}\colonequals\alpha_{i}/x_{1,2}$ for $i=1,2,3$ and
$\Delta_{j}^{\prime}\colonequals\Delta_{j}/x_{1,2}$ for $j=1,\ldots,2n+1$.
Consider the linear change of coordinates,

$x^{\prime}_{i-2,j-2}\colonequals
x_{i,j}+\frac{x_{1,j}x_{2,i}-x_{1,i}x_{2,j}}{x_{1,2}}\qquad i,j=3,\ldots,2n+1$

and set $X^{\prime}\colonequals(x^{\prime}_{i,j})$, a generic alternating
$(2n-1)\times(2n-1)$ matrix. By [Bar95, Lemma 1.3], there exists an invertible
matrix $E$, with entries in $R_{x_{1,2}}$, such that

$E^{T}XE=\left(\begin{array}[]{@{}c|c@{}}\begin{matrix}0&1\\\
-1&0\end{matrix}&\mbox{\Large 0}\\\ \hline\cr\mbox{\Large
0}&\begin{matrix}0&x^{\prime}_{1,2}&\cdots&x^{\prime}_{1,2n-1}\\\
-x^{\prime}_{1,2}&0&\cdots&x^{\prime}_{2,2n-1}\\\
\vdots&\vdots&\ddots&\vdots\\\
-x^{\prime}_{1,2n-1}&-x^{\prime}_{2,2n-1}&\cdots&0\end{matrix}\end{array}\right).$

A routine, albeit tedious, calculation gives

$\alpha_{i}^{\prime}=Y_{i,1}^{\prime}\Delta_{1}^{\prime}+\ldots+Y_{i,2n-1}^{\prime}\Delta_{2n-1}^{\prime}\qquad
i=1,2,3,$

where $\Delta_{1}^{\prime},\ldots,\Delta_{2n-1}^{\prime}$ are the size $2n-2$
pfaffians of $X^{\prime}$ and the elements $Y^{\prime}_{i,j}$ are
algebraically independent over the field $K$; note that

$\operatorname{Pf}_{2n}(X)=\operatorname{Pf}_{2n}(E^{T}XE)=\operatorname{Pf}_{2n-2}(X^{\prime}).$

Let $Y^{\prime}\colonequals(Y^{\prime}_{i,j})$ be a $3\times(2n-1)$ (generic)
matrix; set $R^{\prime}\colonequals K[X^{\prime}]$, $S^{\prime}\colonequals
R[Y^{\prime}]$, and

$J^{\prime}\colonequals
Y^{\prime}[\Delta_{1}^{\prime},\ldots,\Delta_{2n-1}^{\prime}]S^{\prime}:(\Delta_{1}^{\prime},\ldots,\Delta_{2n-1}^{\prime})S^{\prime}.$

Then we get an isomorphism

$(S/J)_{x_{1,2}}\cong
S^{\prime}/J^{\prime}[x_{1,3},\ldots,x_{1,2n+1},x_{2,3},\ldots,x_{2,2n+1},Y_{i,j}|i=1,2,3;\;j=2n,2n+1],$

as desired (where $\Delta_{i}\mapsto\Delta_{i}^{\prime}$ for
$i=1,\ldots,2n-1$). Therefore, the ring $(S/J)_{x_{1.2}}$ is $F$-regular by
induction.

We now construct an $S/J$-linear splitting of the map

$S/J\to F_{*}(S/J)\qquad\text{where}\qquad 1\mapsto F_{*}(x_{1,2})$

in order to establish the $F$-regularity of the generic link $S/J$. After
reindexing, let

$\Delta_{1}=[1,\ldots,2n],\;\Delta_{2}=[1,\ldots,n,n+2,\ldots,2n+1],\;\text{and}\;\Delta_{3}=[2,\ldots,2n+1]$

denote the pfaffians of $X$ with the given rows and columns and let
$\beta_{1},\beta_{2},\beta_{3}$ denote the first three entries of the matrix
$Y[\Delta_{1},\ldots,\Delta_{2n+1}]^{T}$. By Lemma 3.3 and Proposition 4.1, we
get that the initial term of the polynomial

$f\colonequals\beta_{1}\beta_{2}\beta_{3}$

is squarefree with respect to the term order $<_{1}$ in Lemma 3.3 (constructed
by extending the term order $<$ in Proposition 4.1). Furthermore, we have,

$f^{p-1}\in J^{[p]}:J$

by Lemma 2.8. Note that $x_{1,2}$ does not divide the initial term
$\operatorname{in}_{<_{1}}(f)$. In view of this, consider the polynomial

$g\colonequals\prod_{\begin{subarray}{c}Y_{i,j},x_{k,l}\notin\operatorname{in}_{<_{1}}(f)\\\
(k,l)\neq(1,2)\end{subarray}}Y_{i,j}x_{k,l}f$

and note that $g^{p-1}\in J^{[p]}:J$. The Frobenius trace map

$\operatorname{Tr}(F_{*}(x_{1,2}^{p-2}g^{p-1})):F_{*}(S/J)\to
S/J\quad\text{sends}\quad F_{*}(x_{1,2})\mapsto 1$

to give the required splitting. The $F$-regularity of the generic link $S/J$
now follows from [HH94a, Theorem 5.9(a)]. ∎

######  Remark 4.4.

The generators of the generic link of height $3$ Gorenstein ideals are known:
[KU92a, §4.10] gives the generators of the generic link as nested pfaffians of
varying sizes of an ‘almost alternating’ matrix (see [KU92b, Theorem 4.7] for
an easier proof). We emphasize that due to our approach of viewing the generic
link via the property $\mathbf{P}$, we did _not_ need its generators to
establish the $F$-regularity of the generic link (or its $F$-pure threshold,
or the squarefreeness of its initial ideal).

######  Corollary 4.5.

Let $I$ be a generic height $3$ Gorenstein ideal of $R=K[\underline{x}]$ over
an $F$-finite field $K$ of characteristic $p>0$ and $L^{k}(I)$ the $k$-th
universal link of $I$ in a transcendental extension $S$ of $R$. Then
$S/L^{k}(I)$ is an $F$-rational ring for each $k\geq 0$.

###### Proof.

Due to [HU87, Corollary 2.15], the even universal links $S/L^{2k}(I)$
(respectively the odd universal links $S/L^{2k+1}(I)$) are essentially a
deformation of $R/I$ (respectively that of $S/L^{1}(I)$). That is, the
universal links are obtained by a finite sequence of deformations and
localizations at prime ideals beginning from $R/I$ or $S/L^{1}(I)$, in the
respective cases. Note that generic pfaffian rings are $F$-regular and the
generic links of generic height $3$ Gorenstein ideals are $F$-regular by
Theorem 4.3. Since the universal link is a localization of the corresponding
generic link at a prime ideal, the assertion follows as $F$-regular rings are
$F$-rational and $F$-rationality deforms (and is a local property) due to
[HH94a, Theorem 4.2 (h)]. ∎

If $\underline{\alpha}$ is a regular sequence in a polynomial ring $R$ such
that the pair $(R,\underline{\alpha})$ has $\mathbf{P}$, then, by Theorem 3.5,
any ideal in the linkage class of $(\underline{\alpha})$ has a deformation
(the $n$-th generic link of $(\underline{\alpha})$) which is $F$-pure. By
Proposition 4.1, a class of examples of such licci ideals (i.e., ideals linked
to a complete intersection) are generic height $3$ Gorenstein ideals (see
[Wat73]). Another class of licci ideals is that of generic height $2$ perfect
ideals, subsumed in Proposition 5.1, which we prove next.

##  5\. Minors of a generic matrix: A curious dichotomy

The aim of this section is to discuss some subtleties related to the property
$\mathbf{P}$. We begin by studying the minors of a generic matrix: Let
$X\colonequals(x_{ij})$ be an $m\times n$ matrix of indeterminates for
positive integers $m\leq n$ and $R\colonequals K[X]$. Let $I_{t}(X)$ denote
the $R$-ideal generated by the size $t$-minors of the matrix $X$.

######  Proposition 5.1.

The pair $(K[X]),I_{m}(X))$ has $\mathbf{P}$.

###### Proof.

Let

$\underline{\alpha}\colonequals[1,m],[2,m+1],\ldots,[n-m+1,n]$

denote the size $m$ minors of $X$ with the given adjacent columns and fix the
lexicographical order in $K[X]$ induced by the variable order

$x_{1,1}>x_{1,2}>\dots>x_{1,n}>x_{2,1}>\dots>x_{2,n}>\dots>x_{m,n}.$

For $1\leq i\leq n-m+1$, by determinant expansion along the first row of $X$,
we get

$\operatorname{in}_{<}\big{(}[i,m+i-1]\big{)}=x_{1,i}x_{2,i+1}\dots
x_{m,m+i-1},$

which is the product of the variables in the main diagonal of the minor
$[i,m+i-1]$. Since the determinantal ideal $I_{m}(X)$ has height $n-m+1$, we
are done. ∎

######  Remark 5.2.

In view of Theorem 3.5, Proposition 5.1 recovers the $F$-pure threshold
(respectively log canonical threshold) of determinantal rings of maximal
minors calculated in [MSV14, Theorem 1.2] (respectively [Doc13, Theorem 5.6],
[Joh03]) as well as the log canonical threshold of the generic links of
maximal minors calculated in [KMN21, Theorem 1] by a simpler method and also
extends these calculations to the $n$-th generic link for each $n>0$.

We now illustrate that each of the conclusions of Theorem 3.5 typically fails
for the generic links of the non-maximal minors of a generic matrix. The
following lemma will be useful to this end:

######  Lemma 5.3.

Let $R=K[x_{1},\ldots,x_{n}]$ and $I$ be a homogeneous $R$-ideal of height $g$
for which each generator has degree $d$. Then the $a$-invariant of the (first)
universal link $L^{1}(I)$ is $d(g-1)-n$.

###### Proof.

Let $Y:=(Y_{i,j})$ be a $g\times n$ matrix and
$S:=R_{(x_{1},\ldots,x_{n})}(Y_{i,j})$. Let

$L^{1}(I)\colonequals
Y[f_{1},\ldots,f_{n}]^{T}S:(f_{1},\ldots,f_{n})S=(\underline{\alpha}):IS$

denote the (first) universal link of $I$. Note that, since the ideal $I$ is
generated in the same degree, the factor ring $S/L^{1}(I)$ is the localization
of a graded ring at its homogeneous maximal ideal. Hence its top local
cohomology module

$H^{n-g}_{(x_{1},\ldots,x_{n})}(S/L^{1}(I))$

is a $\mathbb{Z}$-graded $S/L^{1}(I)$-module in a natural manner. By [KU92b,
Lemma 2.3], the graded canonical module of the universal link is

$\omega_{S/L^{1}(I)}\cong
IS/(\underline{\alpha})[-n+\sum_{i=1}^{g}\deg(\alpha_{i})].$

Since the indeterminates $Y_{ij}$ are units in $S$ and so have degree zero, it
follows that the graded canonical module $\omega_{S/L^{1}(I)}$ is generated in
degree

$d-(-n+gd)=n-d(g-1).$

Therefore the $a$-invariant

$a(S/L^{1}(I))=d(g-1)-n.\qed$

######  Example 5.4.

Let $X$ be a $3\times 7$ matrix of indeterminates and $R=K[X]$. The pair
$(R,I_{2}(X))$ clearly does not have the property $\mathbf{P}$: The ideal
$I_{2}(X)$ has height $12$ and clearly does not contain any sequence of
elements $\alpha_{1},\ldots,\alpha_{12}$ as required for $\mathbf{P}$ to hold
since that would require atleast $24$ indeterminates.

By Lemma 5.3, the $a$-invariant of the (first) universal link is

$a([L^{1}(I_{2}(X)))=2(12-1)-21=1>0.$

Since an $F$-injective ring has a non-positive $a$-invariant, it follows that
the universal link $L^{1}(I_{2}(X))$ does not define an $F$-injective ring. As
the universal link is a localization of the generic link at a prime ideal and
since $F$-injectivity is a local property [DM19, Proposition 3.3], it also
follows that the generic link $L_{1}(I_{2}(X))$ is _not_ $F$-injective.
Therefore it is not $F$-pure (though, perhaps interestingly, one may check
that, the $a$-invariant of the generic link is negative). This observation
stands in sharp contrast to the recent result [PT24, Theorem 5.6] which says
that the generic link of the maximal minors of a generic matrix is
$F$-regular.

Since a squarefree initial ideal implies $F$-injectivity (see, for example,
[KV23, Corollary 4.11(2)]), it follows that the generic link $L_{1}(I_{2}(X))$
does _not_ have a squarefree initial ideal. However, the generic determinantal
ideal of minors of any size has a squarefree initial ideal since the minors
form a Gröbner basis with respect to a diagonal term order by [Stu90]. We
conclude that (generic) linkage does _not_ preserve the squarefreeness of the
initial ideal.

In addition, since generic determinantal rings of minors of any size are
$F$-regular ([HH94b, §7]) and hence have rational singularities ([Smi97,
Theorem 4.3]), we recover the fact that (generic) linkage does not preserve
rational singularities (see [Niu14, Corollary 3.4]).

Although the generic link $L_{1}(I_{2}(X))$ is not $F$-pure, its $F$-pure
threshold still makes sense. By [KMN21, Theorem 1] (see also [Doc13, Theorem
5.6]),

$\operatorname{fpt}(L_{1}(I_{2}(X)))=10.5<12=\operatorname{ht}(I_{2}(X)).$

### The natural nullcone of the symplectic group

We illustrate that the choice of the term order appearing in $\mathbf{P}$ may
be quite subtle.

Let $X\colonequals(x_{i,j})$ be a $2t\times n$ matrix of indeterminates for
positive integers $t$ and $n$. Let $\Omega$ denote the size $2t$ standard
symplectic block matrix

$\Omega\colonequals\begin{pmatrix}0&\mathbb{I}\\\ -\mathbb{I}&0\end{pmatrix},$

where $\mathbb{I}$ is the size $t$ identity matrix. Let

$\mathfrak{P}(X)\colonequals(X^{T}\Omega X)$

be the ideal generated by the entries of the alternating matrix $X^{T}\Omega
X$. The ideal $\mathfrak{P}(X)$ is Hilbert’s nullcone ideal for the natural
action of the Symplectic group (see [HJPS23, §6]) given by

$M:X\to MX\quad\text{for}\quad M\in\operatorname{Sp}_{2t}(K).$

The Symplectic nullcone is a Cohen–Macaulay domain with

$\displaystyle\dim K[X]/\mathfrak{P}(X)$ $\displaystyle=$
$\displaystyle\begin{cases}2nt-\displaystyle{\binom{n}{2}}&\text{if }\ n\leq
t+1,\\\ nt+\displaystyle{\binom{t+1}{2}}&\text{if }\ n\geq t\\\ \end{cases}$

by [HJPS23, Theorem 6.8]. We describe the generators of $\mathfrak{P}(X)$. For
ease of notation in the next Proposition, we relabel the entries of the lower
half of the $2t\times n$ matrix $X=(x_{i,j})$ as follows: Let
$w_{i,j}=x_{i+t,n-j+1}$ for $1\leq i\leq t$ and $1\leq j\leq n$. Then

$X=\begin{pmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\ \vdots&\vdots&&\vdots\\\
x_{t,1}&x_{t,2}&\cdots&x_{t,n}\\\ w_{1,n}&w_{1,n-1}&\cdots&w_{1,1}\\\
\vdots&\vdots&&\vdots\\\ w_{t,n}&w_{t,n-1}&\cdots&w_{t,1}\end{pmatrix}.$

The entries of the nullcone ideal $(X^{T}\Omega X)$ are sums of the size two
minors

$d_{i,j}\colonequals\det\begin{pmatrix}x_{1,i}&x_{1,j}\\\
w_{1,n-i+1}&w_{1,n-j+1}\end{pmatrix}+\cdots+\det\begin{pmatrix}x_{t,i}&x_{t,j}\\\
w_{t,n-i+1}&w_{t,n-j+1}\end{pmatrix}$

coming from columns $i$ and $j$ of the matrix $X$.

######  Proposition 5.5.

The pair $(K[X],\mathfrak{P}(X))$ has $\mathbf{P}$.

###### Proof.

Let

$\underline{\alpha}\colonequals\\{d_{i,j}\;|\;1\leq i<j\leq n\;\text{ and
}\;j-i\leq t\\}$

be the subset of the generators of the nullcone ideal $\mathfrak{P}(X)$ with
the given columns. The reader may verify that the above elements (or any such
subset of the generators of $\mathfrak{P}(X))$ do not satisfy $\mathbf{P}$
under the ‘usual’ lex or revlex order on the variables. A more delicate term
order is needed.

To define such a term order $<$ in $K[X]$, we first define an order on the
variables. Sort the entries of the matrix $X$ into blocks
$B_{0},B_{1},\dots,B_{n-1}$ according to the following formula:

$\displaystyle\text{$x_{i,j},w_{i,j}$ are in block $B_{\ell}$ where
}\ell=\begin{cases}2j+i-2&\text{if \; }1\leq j<\dfrac{n-i+1}{2},\\\
2n-2j-i&\text{if \; }\dfrac{n-i+1}{2}\leq j<n-i+1,\\\
0&\text{otherwise.}\end{cases}$

Now, for $a\in B_{\ell}$ and $b\in B_{k}$, set $a<b$ if $\ell<k$. Then, within
each set $B_{\ell}$, fix an arbitrary order among the variables. This gives us
a total variable order in $K[X]$. Our term order $<$ is the reverse
lexicographical order induced by this variable order. The initial terms of the
elements of $\underline{\alpha}$ are as required for the property $\mathbf{P}$
to hold by [PTW23, Lemmas 3.4, 3.5] (see [PTW23, Example 3.3] for an explicit
example). ∎

######  Remark 5.6.

One can show that the generic link of the symplectic nullcone is $F$-regular
in positive prime characteristic. The proof follows along the lines of Theorem
4.3 by making use of [PTW23, Lemma 3.2]. We leave the details to the
interested reader.

##  6\. Generic residual intersections of a complete intersection inherit
$\mathbf{P}$

In this section, we show that the property $\mathbf{P}$ is inherited by the
generic residual intersections of an ideal generated by a regular sequence in
a polynomial ring. This observation is quite surprising in light of the
discussion so far since residual intersections do not preserve height unless
they are links.

###  6.1. Brief recall of residual intersections

The notion of residual intersections essentially goes back to Artin and Nagata
[AN72]. Let $X$ and $Y$ be two irreducible closed subschemes of a Noetherian
scheme $Z$ with $\operatorname{codim}_{Z}X\leq\operatorname{codim}_{Z}Y=s$ and
$Y\nsubseteq X$, then $Y$ is called a residual intersection of $X$ if the
number of equations needed to define $X\cup Y$ as a subscheme of $Z$ is the
smallest possible, namely $s$. However, in order to include the case where $X$
and $Y$ are reducible with $X$ possibly containing some component of $Y$, the
following algebraic definition is more suited:

######  Definition 6.1.

Let $R$ be a polynomial ring and $I$ an $R$-ideal. Given an ideal
$\mathfrak{a}\subsetneq I$ generated by $s$ elements, set $J=\mathfrak{a}:I$.
If $\operatorname{ht}(J)\geq s\geq\operatorname{ht}(I)$, then $J$ is an _$s$
-residual intersection_ of $I$ (or $R/J$ is an $s$-residual intersection of
$R/I$).

If furthermore $I_{{\mathfrak{p}}}={\mathfrak{a}}_{{\mathfrak{p}}}$ for all
${\mathfrak{p}}\in V(I)$ with $\operatorname{ht}{{\mathfrak{p}}}\leq s$, then
$J$ is called a geometric $s$-residual intersection of $I$.

Residual intersections are natural generalizations of linkage: If an ideal $I$
is unmixed of height $g$, then the $g$-residual intersections of $I$ are
precisely the links of $I$ and the geometric $g$-residual intersections are
precisely the geometric links of $I$ by Proposition 2.4. Furthermore, as in
the case of links, we may define the ‘most general’ residual intersections if
the following technical requirement is met:

######  Definition 6.2.

We say that an ideal $I$ in a Noetherian ring satisfies the condition $G_{s}$
if $\mu(I_{{\mathfrak{p}}})\leq\operatorname{ht}({\mathfrak{p}})$ for all
prime ideals ${\mathfrak{p}}$ containing $I$ such that
$\operatorname{ht}({\mathfrak{p}})\leq s-1$ (where $\mu(-)$ denotes the
minimal number of generators). We say that $I$ satisfies $G_{\infty}$ if it
satisfies $G_{s}$ for each $s$. It is clear that this is equivalent to the
condition that $\mu(I_{{\mathfrak{p}}})\leq\operatorname{ht}({\mathfrak{p}})$
for each prime ideal ${\mathfrak{p}}$ containing $I$.

######  Definition 6.3.

Let $R$ be a polynomial ring and $I$ be an ideal of height $g>0$ satisfying
$G_{s+1}$, where $s\geq g$. Choose any generating set
$\underline{f}\colonequals f_{1},\ldots,f_{n}$ of $I$ and let $Y$ be an
$s\times n$ matrix of indeterminates. Let $\mathfrak{a}$ be the ideal
generated by the entries of the matrix $Y[f_{1}\dots f_{n}]^{T}$. Then we set

$\operatorname{RI}(s;I)=\operatorname{RI}(s;\underline{f})\colonequals\mathfrak{a}R[Y]:IR[Y]$

and call this ideal a _generic $s$-residual intersection_ of $I$.

Generic residual intersections are in fact geometric residual intersections
[HU88, Theorem 3.3]. As is the case with generic linkage, the generic residual
intersections are essentially independent of the generating set for the ideal
[HU90, Lemma 2.2]. The generic $s$-residual intersection specializes to any
$s$-residual intersection of the given ideal and tracks the homological
properties of the ideal via deformation.

###  6.2. $F$-pure threshold of the generic residual intersections of a
complete intersection

We layout the setup for this subsection: Let
$\underline{\alpha}\colonequals\alpha_{1},\ldots,\alpha_{g}$ be a homogeneous
regular sequence, with each $\alpha_{i}$ having the same degree, in the
polynomial ring $K[\underline{x}]$ such that the pair
$(K[\underline{x}],\underline{\alpha})$ has $\mathbf{P}$. Fix an integer
$s\geq g$ and let $Y\colonequals(Y_{i,j})$ be a matrix of indeterminates of
size $s\times g$. Let

$\operatorname{RI}(s;\underline{\alpha})\colonequals
Y[\alpha_{1},\cdots,\alpha_{g}]^{T}R[Y]:(\underline{\alpha})R[Y]$

denote the generic $s$-residual intersection of the ideal
$(\underline{\alpha})$; set $M\colonequals
Y[\alpha_{1},\cdots,\alpha_{g}]^{T}$. Since an ideal generated by a regular
sequence satisfies $G_{\infty}$, the residual intersection
$\operatorname{RI}(s;\underline{\alpha})$ exists for each $s\geq g$.

We now show that the generic residual intersections of complete intersections
inherit $\mathbf{P}$. We need the generators of the residual intersection to
do this, which were calculated in [HU88, Example 3.4] (see also [BKM90]).

######  Theorem 6.4.

If the pair $(R,\underline{\alpha})$ has the property $\mathbf{P}$, then
$(R[Y],\operatorname{RI}(s;\underline{\alpha}))$ also has $\mathbf{P}$ for
each $s\geq|\underline{\alpha}|$.

###### Proof.

Since $\underline{\alpha}=\alpha_{1},\ldots,\alpha_{g}$ is a regular sequence,
by the structure theorem for the residual intersections of a complete
intersection ring ([HU88, Example 3.4]), we have

$RI(s,\underline{\alpha})=(\text{entries of the matrix $M$})R[Y]+I_{g}(Y)R[Y]$

is a homogeneous $R[Y]$-ideal of height $s$. We shall construct a term order
$<_{1}$ in $R[Y]$ such that the pair $(R[Y],\underline{\beta})$ satisfies
$\mathbf{P}$ with

$\underline{\beta}\colonequals\\{M_{1,1},\ldots,M_{g-1,1}\\}\bigcup\\{[i,i+1,\ldots,g-i+1]\;|\;1\leq
i\leq s-g+1\\}.$

That is, $\underline{\beta}$ consists of the first $g-1$ entries of the matrix
$M$ and all the $g$-minors of the matrix $Y\colonequals(Y_{i,j})$ with
adjacent rows. Clearly, the set $\underline{\beta}$ has

$(g-1)+(s-g+1)=s=\operatorname{ht}(\operatorname{RI}(s;\underline{\alpha}))$

elements, as required; hereafter, we call these elements
$\beta_{1},\ldots,\beta_{g-1},\beta_{g},\ldots,\beta_{s}$ with the indexing
according to the above display. We now construct the required term order.

Define the following variable order in $R[Y]$:

$\displaystyle Y_{s,g}>Y_{s,g-1}>\cdots>Y_{s,1}>$ $\displaystyle\vdots$
$\displaystyle Y_{g-1,g}>Y_{g-1,g-1}>\cdots>Y_{g-1,1}>$ $\displaystyle
Y_{g-2,g-1}>Y_{g-2,g}>Y_{g-2,g-2}>\cdots>Y_{g-2,1}>$ $\displaystyle
Y_{g-3,g-2}>Y_{g-3,g}>Y_{g-3,g-1}>\cdots>Y_{g-3,1}>$ $\displaystyle\vdots$
$\displaystyle Y_{1,2}>Y_{1,g}>Y_{1,g-1}>\cdots>Y_{1,1}>x_{i.j}.$

That is, we begin the variable order with the last row of the matrix $Y$, move
from right to left, and then vertically up along the rows of $Y$ for (the
bottom) $s-g$ rows, thereafter we follow the same order for the remaining rows
except that the $(i,i+1)$ entry appears first in the ordering of the $i$-th
row. In addition, the ordering on the indeterminates $x_{i,j}$ is the one
given in the property $\mathbf{P}$ for the pair $(R,\underline{\alpha})$. Our
term order is the lexicographical order $<_{1}$ induced by this variable order
in $R[Y]$. Note that the initial terms of the first $g-1$ entries of the
matrix $M$ are

$\displaystyle\operatorname{in}_{<_{1}}(\beta_{1})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{1,1}\alpha_{1}+Y_{1,2}\alpha_{2}+\cdots+Y_{1,g}\alpha_{g})=Y_{1,2}\operatorname{in}_{<}(\alpha_{2}),$
$\displaystyle\operatorname{in}_{<_{1}}(\beta_{2})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{2,1}\alpha_{1}+Y_{2,2}\alpha_{2}+\cdots+Y_{2,g}\alpha_{g})=Y_{2,3}\operatorname{in}_{<}(\alpha_{3}),$
$\displaystyle\vdots$ $\displaystyle\operatorname{in}_{<_{1}}(\beta_{g-1})$
$\displaystyle=\operatorname{in}_{<_{1}}(Y_{g-1,1}\alpha_{1}+Y_{g-1,2}\alpha_{2}+\cdots+Y_{g-1,g}\alpha_{g})=Y_{g-1,g}\operatorname{in}_{<}(\alpha_{g}).$

Further, by determinant expansion along the last row of $Y$, the initial terms
of the remaining $s-g+1$ elements of $\underline{\beta}$

$\operatorname{in}_{<_{1}}(\beta_{i})=\operatorname{in}_{<_{1}}([i,i+1,\cdots,g+i-1])=Y_{i,1}Y_{i+1,2}\ldots
Y_{g+i-1,g},\quad 1\leq i\leq s-g+1$

are just the products of the variables occurring in the main diagonal of the
matrix $[i,i+1,\cdots,g+i-1]$. Since $(R,\underline{\alpha})$ has $\mathbf{P}$
by hypothesis, we get that for the elements
$\underline{\beta}=\beta_{1},\ldots,\beta_{s}$ of $R[Y]$, each initial term is
squarefree and each pair of initial terms is mutually coprime, as required for
$(R[Y],\operatorname{RI}(s;\underline{\alpha}))$ to inherit $\mathbf{P}$. ∎

While the $F$-pure threshold of the generic link of an ideal is atleast as
large as that of the ideal itself by [Niu14, Proposition 3.7], it is clearly
bounded above by the height of the ideal due to Lemma 2.10. On the other hand,
the $F$-pure threshold of the generic residual intersections may behave quite
differently:

######  Corollary 6.5.

If the pair $(R,\underline{\alpha})$ has $\mathbf{P}$, then, for each
$s\geq|\underline{\alpha}|$, we have

$\operatorname{lct}({\operatorname{RI}(s;\underline{\alpha}))}=\operatorname{fpt}({\operatorname{RI}(s;\underline{\alpha}))}=s\geq|\underline{\alpha}|=\operatorname{fpt}{(\underline{\alpha})}=\operatorname{lct}{(\underline{\alpha})}.$

###### Proof.

The assertion immediately follows from Theorem 6.4 in view of Theorem 3.5 and
Remark 3.6. The outer equality is [HY03]. ∎

######  Remark 6.6.

Corollary 6.5 with $s=|\underline{\alpha}|$ answers [KMN21, Question 4.5(b)]
by giving a class of complete intersection ideals—those possessing the
property $\mathbf{P}$—for which the log canonical threshold is preserved under
generic linkage.

We point out that the log canonical threshold, may not be preserved under
generic linkage (and therefore under generic residual intersections) for
complete intersection ideals $I$ in a polynomial ring $R$ if $(R,I)$ does not
have $\mathbf{P}$: Let $R=K[x_{1},x_{2},x_{3}]$ and
$I=(x_{1}^{2}x_{2},x_{3}^{3})$ be a homogeneous $R$-ideal. By [KMN21, Example
2.4(a)] and the linear programming methods in [ST09], we get

$\operatorname{lct}(L_{1}(I))=11/6>5/6=\operatorname{lct}(I).$

We end with a reformulation of [KMN21, Question 4.5(b)]:

######  Question 6.7.

Let $I$ be an unmixed, height $g>0$ ideal in a polynomial ring over a field of
positive characteristic. Let $s>g$ be an integer such that $I$ satisfies
$G_{s+1}$ (so that the generic residual intersection $\operatorname{RI}(s;I)$
exists). When do we have the equality

$\operatorname{fpt}(\operatorname{RI}(s;I))=\operatorname{fpt}(I)?$

## Acknowledgments

I thank Manav Batavia, Linquan Ma, Yevgeniya Tarasova, Bernd Ulrich, and
Matteo Varbaro for several insightful discussions. I am grateful to Sameeksha
Bidhuri for her unwavering support.

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