# Parton Distribution Functions and Their Impact on Precision of the Current
Theory Calculations ††thanks: Presented at XXX Cracow Epiphany Conference on
Precision Physics at High Energy Colliders - dedicated to the memory of
Staszek Jadach

Maria Ubiali  DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3
0WA, UK

###### Abstract

The unprecedented precision of experimental measurements at the Large Hadron
Collider (LHC) and the increased statistics that will be reached in the High-
Luminosity phase of the LHC (HL-LHC) are pushing the phenomenology community
to a new precision frontier, in which new challenges present themselves and
new questions arise. A key ingredients of theoretical predictions at hadron
colliders are the Parton Distribution Functions (PDFs) of the proton. This
contribution highlights some of the new developments in the determination of
PDFs from a global set of experimental data, from approximate N3LO PDFs and
the inclusion of theory uncertainties in PDF fits, to the realisation of the
non trivial interplay between parton densities at large-$x$ and possible
signals of New Physics in high energy tails of the distributions, which
highlights the synergy between high energy and low energy experimental
programs.

##  1 Introduction

Precision phenomenology at hadron colliders and searches for deviations from
the Standard Model (SM) predictions rely upon a precise and accurate estimate
of the uncertainty in the SM predictions. Most theory predictions for hadron
colliders require Parton Distribution Functions (PDFs) [1, 2, 3, 4, 5, 6, 7,
8, 9], the non-perturbative functions parametrising the subnuclear structure
of the protons in terms of their partonic constituents for high-energy
scattering processes.

PDFs cannot be obtained by perturbative methods, hence they must be fitted
from experimental data. The space of possible parton distributions is an
infinite-dimensional space of functions obeying the DGLAP evolution equations;
as such, given only finite amounts of data, it is an ill-posed problem to
determine the PDFs. Therefore, any PDF fitting attempt must initially restrict
the infinite-dimensional PDF space to a finite-dimensional space instead, by
assuming a functional form for the PDFs at some initial scale $Q_{0}$111This
can be a set of polynomials, that typically involves 30-50 parameters or a
redundant parametrization such as a deep neural network, involving hundreds of
parameters and a regularisation or minimisation stopping criterion, depending
on the methodology., and then DGLAP evolution is used to obtain the PDF at all
scales.

A faithful determination of PDFs and of their uncertainty is a highly non
trivial problem and the precision of experimental data is presenting new
challenges to PDF fitting collaborations. Accurate and precise PDFs are
crucial at the LHC. For example, despite the progress in the precise
determination of PDFs, PDF uncertainty is still one of the largest sources of
theoretical uncertainty affecting the predictions for Higgs boson production
[10, 11, 12]. The experimental PDF uncertainty is not the only component of
the total uncertainty, as there is another component representing the mismatch
in the perturbative order of the PDFs, evaluated at N2LO, and the perturbative
QCD cross sections evaluated at N3LO [13, 14]. The recent publication of
approximate N3LO PDF sets [15, 16, 17], which will be discussed in Sect. 2, is
a crucial step in reducing this source of uncertainty. PDFs do also contribute
to precise extraction of the SM parameters from the LHC data, such as for
example the strong coupling constant [18] and the $W$ boson mass [19], for
which different PDF sets yield results that are significantly different as
compared to the size of statistical and systematic uncertainties.

Finally, PDF uncertainties become sizeable at large-$x$ due to the lack of
precise experimental constraints in that region, and this yields a large
uncertainty in the predictions for the high-energy tails of the measured
distributions, where programs of indirect new physics searches focus, see for
example the discussion in Ref. [20]. A precise and accurate description of the
large-$x$ content of the proton has always been crucial. As an illustration,
the discrepancy that was observed by the CDF collaboration in 1995 between the
SM predictions for jet inclusive cross section at large $p_{T}$ [21] was
mostly due to the lack of an estimate of the uncertainty of the gluon at
large-$x$, and indeed the discrepancy disappeared once the CDF jet
distributions were included by CTEQ in their global analysis of PDFs [22]. The
story taught us the importance of propagating the experimental uncertainty in
the space of PDFs. However, now that uncertainties are much more carefully
estimated, how can we make sure that potential deviations from the SM
predictions in the high energy tails of the distributions are genuinely due to
new physics rather than to our poor understanding of the proton structure at
large $x$? This question will be on of the focus of the discussion in Sect. 2.

##  2 New frontiers and challenges

As data precision increases, so does the challenge of balancing precision and
accuracy. New PDF analyses sometimes lead to significant shifts from previous
sets, potentially larger than nominal uncertainties, yet this doesn’t
undermine accuracy if the source of shifts is identified and the fitting
process is controlled. Growing luminosity in experimental data poses
challenges, especially with increasingly correlated systematics dominating
datasets. Stabilizing covariance matrices and addressing systematic
decorrelation are ongoing tasks crucial for assessing the goodness of fit [23,
24]. Continuous improvements and rigorous tests, ensure robustness,
particularly important for modern PDF sets, see for example discussion on
closure tests in Ref. [25] and future tests in Ref. [26].

In what follows we will select three key aspects: the theory framework
advancements that is behind approximate N3LO PDFs (discussed in Sect. 2.1),
the inclusion of theoretical uncertainty in PDF fits (discussed in Sect. 2.2),
and finally the interplay between PDF and new physics (discussed in Sect.
2.3).

###  2.1 Approximate N3LO PDFs

Calculations of partonic cross sections at N3LO order have been available for
a long time for massless deep-inelastic scattering (DIS), and have become
recently available for an increasingly large set of hadron processes, see Ref.
[27] for an overview. To obtain theoretical predictions for hadronic
observables at the same level of accuracy, the N3LO partonic cross sections
must be convoluted with PDFs determined at the same perturbative order. The
main bottleneck in carrying out this programme is the lack of exact
expressions for the N3LO splitting functions that govern the scale dependence
of the PDFs: for these only partial information is available, see Ref. [17]
for a review. However, a combination of the available partial results is
possible, and an approximate determination of the N3LO splitting functions,
see for example Refs. [28, 29].

Currently, two approximate N3LO PDF sets are available, with the MSHT set [29]
available since a couple of year and the combined QED and N3LO MSHT set [15]
and the NNPDF4.0 set [17] become available only recently. There are several
differences in the approaches, as the MSHTaN3LO set includes all available
theory input at the time of publication, while the NNPDF40aN3LO includes more
theory inputs that were published in between, in particular 6(1) extra
momentum for the $P_{qg}$, $P_{qq}$ ($P_{gq},\,P_{gg}$) splitting functions,
some terms in the large $n_{f}$ limit, and several sub-leading small-$x$ and
large-$x$ terms. A benchmark between the two approaches at the level of
splitting functions is in progress in a Les Houches benchmark exercise.
Another important difference is that in the MSHT approach the incomplete N3LO
terms are added as variation in the prior and estimated by fitting nuisance
parameters to the data, hence the posterior is determined by fitting the data,
while the NNPDF approximation incorporates only theory inputs and their
variations is added via an additional theory covariance matrix associated with
incomplete missing higher orders (IHOU). Moreover in the NNPDF approach the
missing higher order uncertainties (MHOU) associated with the NNLO
contributions are added via a MHOU theory covariance matrix [30, 31].

While the differences in the approach yield slightly discrepant predictions,
overall a good perturbative convergence is observed, with differences
decreasing as the perturbative order increases, and the approximate N3LO
result always compatible with the NNLO within uncertainties. Also, while for
quark PDFs the difference between NNLO and aN3LO results is extremely small,
for the gluon PDF a more significant shift is observed between NNLO and N3LO
(this is much more marked in the MSHT set than in the NNPDF set), thus making
the inclusion of N3LO crucial for precision phenomenology.

###  2.2 Missing higher order uncertainties

As the overall precision in the determination grows, it is paramount to
consistently treat the theoretical uncertainties of the input predictions in
PDF fits. Until very recently PDF uncertainties in NNLO PDF fits only included
experimental component. However such fits are performed by comparing fixed-
order theoretical predictions to experimental data and theoretical predictions
are affected by MHOU, which is usually estimated by means of scale variation.
The inclusion of scale variations in a state-of-the-art NLO PDF fit was
presented for the first time in Refs. [30, 31]. The approach, built upon the
construction of a theory covariance matrix that describes the scale variations
of the processes included in a PDF fit and models their theoretical
correlations, has now been generalised to a NNLO PDF set [32]. The theory
covariance matrix so determined can be added to the experimental covariance
matrix, and the data are fitted using a total covariance matrix, which is the
sum of the two contributions. The inclusion of MHOUs generally improves
perturbative convergence and represents an important step towards achieving
accuracy and improving perturbative convergence. A similar method has been
also been applied to nuclear uncertainties, which are relevant in the
description of data based on nuclear targets [33, 34].

With the availability of PDF sets including MHOUs the total uncertainty of any
theory prediction would combine PDF uncertainty (now including a MHOU
component), with the MHOU uncertainty on the partonic cross section
computations. In order to keep into account the correlation between the scale
variations in the process used for PDF determination, and the scale variation
on the partonic cross section that an external user wishes to compute, one
should compute the cross-correlation of MHO uncertainties between the
predicted process and those used for PDF determination [35, 36]. An
alternative option for the inclusion of MHOUs on predictions that accounts for
MHOUs on PDFs, and their full correlation to MHOUs on the partonic cross
sections, is to include the scale variation in the Monte Carlo sampling, as
proposed in the MCscales approach [37]. Basically the Monte Carlo sampling
method used to propagate the experimental uncertainty of the data into the
PDFs is extended by adding scale fluctuations for each theoretical prediction
on top of the fluctuations on the input data that are used to propagate
experimental uncertainties. The development and the comparison of the various
ways of including MHOUs in the estimate of PDF uncertainties will become
increasingly crucial as the experimental component of PDF uncertainties is set
to decrease thanks to the increased statistics and better understanding of the
experimental systematics in Run III and IV, as well as in the HL-LHC phase.

###  2.3 PDFs and new physics interplay

Recent PDF analyses indicate that the LHC data are increasingly crucial in
pinning down the parton densities, especially for the gluon and sea quarks the
intermediate to large-$x$ regions and its constraining power will become even
more crucial in the HL-LHC phase [38]. As an increasing number of high-energy
data from the LHC are included in PDF fits, the tails of the distributions
that are used in PDF determination are potentially affected by new physics
effects, which are usually parametrised by means of a model independent EFT
expansion [39]. To make sure that new physics is not absorbed or “fitted away”
in the PDFs, one would either have to exclude these data, thus losing
potentially important constraints, or carefully disentangle the SM and new
physics effects.

Several studies have been performed in this direction by both experimental
collaborations and theory groups [40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. In
particular, it has been shown that the determination of SMEFT Wilson
coefficients from a fit of LHC data, like the determination of SM precision
parameters from LHC data [50], might display a non-negligible interplay with
the input set of PDFs used to compute theory predictions. For example, in [41]
it was shown that, while the effect of four-fermion operators on Deep
Inelastic Scattering (DIS) data can be non-negligible, if DIS data were fitted
while taking the effect of such operators into account, the fit quality would
deteriorate proportionally to the energy scale of the data included in the
determination. This makes a fit based on DIS-only data ”safe” from new physics
effects.

On the other hand in the context of high-mass Drell-Yan, especially in the HL-
LHC scenario, neglecting the cross-talk between the large-$x$ PDFs and the
SMEFT effects in the tails could potentially miss new physics manifestations
or misinterpret them [43, 47, 51], as the bounds on SMEFT operators are
significantly broader if the PDFs are fitted by including the effect of the
$\hat{W}$ and $\hat{Y}$ universal operators [52] in the high energy tails of
the data. In Ref. [47] it was shown that the large-$x$ antiquark distributions
can absorb the effect of universal new physics in the tails of the Drell-Yan
distributions by leading to significantly softer antiquark distributions in
the large-$x$ region, far beyond the nominal PDF uncertainties. The study
shows that the only way to avoid this kind of BSM contamination would be to
include more low-energy observables in a PDF fit, thus highlighting the
synergy between low- and high-energy experiments.

Furthermore, the analysis of the top quark sector of Ref. [46] demonstrates
that in that case the bounds of the operators are not broadened by the
interplay with the PDFs, however the correlation between the top sector and
the gluon is manifest in the gluon PDF itself, which becomes softer in the
large-$x$ region if the PDF fit is augmented by the top data and PDFs are
fitted simultaneously along the Wilson coefficients that determine the top
quark sector. This hints to a non-negligible interplay between large-$x$
gluons and hints for new physics signals in the large invariant mass tails of
the $t\bar{t}$ distributions, and possibly in the high-$E_{T}$ jets
distributions.

The recently publicly released tool SIMUnet [48] allows users to assess the
interplay between PDFs and new physics, either by performing simultaneous fits
of SMEFT operators (or operators in any other EFT expansion) alongside the
PDFs of the proton, or by injecting any new physics model in the data and
checking whether a global fit of PDFs can absorb the effects induced by such a
model in the data. SIMUnet, based on the first tagged version of the public
NNPDF code [53], augments it with a number of new features that allow the
exploration of the correlation between a fit of PDFs and BSM degrees of
freedom. The tool allows a global analysis of the SMEFT combining the Higgs,
top, diboson and electroweak sectors, and more data can be added in a rather
straightforward way to combine the sectors presented here alongside Drell-Yan
and flavour observables, for example. Further exploration of the SMEFT and PDF
interplay and of the possible signal of new physics that might be fitted away
by PDFs especially in the top and jets sector would add important pieces of
the mosaic that has just started to be unveiled.

##  3 Conclusions

The success of the ambitious programme of the upcoming Run III at the LHC and
its subsequent High- Luminosity (HL-LHC) upgrade relies not only on achieving
the highest possible accuracy in the experimental measurements and in the
corresponding theoretical predictions, but also on the availability of
statistically robust tools capable of yielding global interpretations of all
subtle deviations from the Standard Model (SM) that the data might indicate.

In this contribution we discussed how the careful inspection and use of novel
theoretical advances such as the recently developed approximate N3LO PDFs, the
inclusion of MHOU and of QED effects are important to achieve the best
accuracy at the LHC. Thanks to the number of newly released data, a powerful
testing ground of the generalisation and extrapolation of the various PDF sets
is available to be explored. Moreover the new light shed on the interplay
between PDFs and new physics can further advance the synergy between new
experiments of the deeply-inelastic scattering (DIS), for example at the
Electron-Ion Collider [54], or the perspective Large Hadron-Electron Collide
[55] or even at the future Forward Physics Facilities planned at CERN [56].
The challenges that we are presented with are certainly a great opportunity to
refine our tools and advance in our understanding of fundamental physics.

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# Gorensteinness for normal tangent cones of elliptic ideals

Tomohiro Okuma  Department of Mathematical Sciences, Yamagata University,
Yamagata, 990-8560, Japan<EMAIL_ADDRESS>,  Kei-ichi Watanabe
Department of Mathematics, College of Humanities and Sciences, Nihon
University, Setagaya-ku, Tokyo, 156-8550, Japan and Organization for the
Strategic Coordination of Research and Intellectual Properties, Meiji
University<EMAIL_ADDRESS>and  Ken-ichi Yoshida  Department of
Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-
ku, Tokyo, 156-8550, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a two-dimensional excellent normal Gorenstein local domain. In this
paper, we characterize elliptic ideals $I\subset A$ for its normal tangent
cone $\overline{G}(I)$ to be Gorenstein. Moreover, we classify all those
ideals in a Gorenstein elliptic singularity in the characteristic zero case.

###### Key words and phrases:

normal tangent cone, Hilbert coefficients, elliptic singularity, normal
reduction number, elliptic ideal, $p_{g}$-ideal

###### 2020 Mathematics Subject Classification:

13G05, 14J17,13H10, 14J27

TO was partially supported by JSPS Grant-in-Aid for Scientific Research (C)
Grant Number 21K03215. KW was partially supported by JSPS Grant-in-Aid for
Scientific Research (C) Grant Number 20K03522 and by GNSAGA INdAM, Italy. KY
was partially supported by JSPS Grant-in-Aid for Scientific Research (C) Grant
Number 19K03430.

## Introduction

Throughout this paper, let $(A,\mathfrak{m})$ be a Noetherian commutative
local ring with the unique maximal ideal $\mathfrak{m}$ and the residue field
$k=A/\mathfrak{m}$. Furthermore, we assume that $A$ is a two-dimensional
excellent normal local domain (which is not regular) containing an
algebraically closed field $k$ unless otherwise specified. Then there exists a
resolution of singularities of $\operatorname{Spec}A$, and any
$\mathfrak{m}$-primary integrally closed ideal of $A$ admits a geometric
representation. That is, for an $\mathfrak{m}$-primary integrally closed ideal
$I\subset A$, there exist a resolution of singularities $f\colon
X\to\operatorname{Spec}A$ and an anti-nef cycle $Z$ on $X$ such that
$I\mathcal{O}_{X}=\mathcal{O}_{X}(-Z)$ and $I=H^{0}(X,\mathcal{O}_{X}(-Z))$.
Then we denote it by $I=I_{Z}$. This enables us to introduce the notion of
‘geometric’ ideals and prove existence theorems of those ideals.

For instance, in [OWY1], the authors introduced the notion of $p_{g}$-ideals.
An integrally closed ideal $I$ is called a $p_{g}$-ideal if
$\dim_{k}H^{1}(X,\mathcal{O}_{X}(-Z))=p_{g}(A)$, where
$p_{g}(A)=\dim_{k}H^{1}(X,\mathcal{O}_{X})$ denotes the geometric genus of
$A$; see Section 2 for more details. Note that any integrally closed ideal is
a $p_{g}$-ideal in a (two-dimensional) rational singularity (see [Li]).
Moreover, they proved that any local ring (resp. Gorenstein) $A$ has a
$p_{g}$-ideal (resp. a good $p_{g}$-ideal) in [OWY1, Theorem 4.1], where an
ideal $I$ is good if and only if $I^{2}=QI$ and $I=Q\colon I$.

In [OWY4], the authors introduced the following two normal reduction numbers.
For any $\mathfrak{m}$-primary integrally closed ideal $I\subset A$ and its
minimal reduction $Q$, we put

$\displaystyle\operatorname{nr}(I)$ $\displaystyle=$
$\displaystyle\min\\{r\in\mathbb{Z}_{\geq
1}\,|\,\overline{I^{r+1}}=Q\overline{I^{r}}\\},$
$\displaystyle\operatorname{\bar{r}}(I)$ $\displaystyle=$
$\displaystyle\min\\{r\in\mathbb{Z}_{\geq
1}\,|\,\overline{I^{n+1}}=Q\overline{I^{n}}\;\text{for all $n\geq r$}\\}.$

Then $I$ is a $p_{g}$-ideal if and only if $\operatorname{\bar{r}}(I)=1$.

The notion of $p_{g}$-ideal is analogous to ideals in rational singularities
in the theory of 2-dimensional normal singularities. Analogous to elliptic
singularities, E. Rossi and the authors introduced the notions of elliptic
ideals and strongly elliptic ideals in [ORWY], in terms of normal reduction
numbers, and discussed characterizations and an existence of those ideals. An
ideal $I$ is an elliptic ideal if and only if $\operatorname{\bar{r}}(I)=2$.
Moreover, $I$ is a strongly elliptic ideal if and only if it is an elliptic
ideal with $\ell_{A}(\overline{I^{2}}/QI)=1$, where $\ell_{A}(M)$ denotes the
length of an $A$-module $M$. This notion characterizes a strongly elliptic
singularity, that is, $p_{g}(A)=1$. Namely, the ring $A$ is a strongly
elliptic singularity if and only if any $\mathfrak{m}$-primary integrally
closed ideal is either a $p_{g}$-ideal or a strongly elliptic ideal.
Similarly, the first-named author showed that any $\mathfrak{m}$-primary
integrally closed ideal $I\subset A$ in an elliptic singularity $A$ is either
a $p_{g}$-ideal or an elliptic ideal ([Ok2]).

For an ideal $I\subset A$, we put

$G(I)=\bigoplus_{n=0}^{\infty}I^{n}/I^{n+1},\qquad\overline{G}(I)=\bigoplus_{n=0}^{\infty}\overline{I^{n}}/\overline{I^{n+1}}.$

Then $G(I)$ (resp. $\overline{G}(I)$) is called the associated graded ring
(resp. the normal tangent cone) of $I$. Many authors have studied the Cohen-
Macaulayness and the Gorensteinness of $G(I)$. But we do not know many
properties of $\overline{G}(I)$ although it seems to be important from
geometric point of view. For instance, if $I$ is a $p_{g}$-ideal, then
$\overline{G}(I)=G(I)$ is Cohen-Macaulay (cf. [OWY2, Theorem 4.1]), and
$\overline{G}(I)$ is Gorenstein if and only if it is good (see Proposition
1.13). Also, if $\overline{G}(I)$ is Cohen-Macaulay, then
$\operatorname{nr}(I)=\operatorname{\bar{r}}(I)$ holds true (see Lemma 1.12).
So it is natural to ask the following question.

Question. Let $I$ be an elliptic ideal.

  1. $(1)$

When is $\overline{G}(I)$ Cohen-Macaulay?

  2. $(2)$

When is $\overline{G}(I)$ Gorenstein?

The first question has a positive answer. Namely, if $I$ is an elliptic ideal,
then $\overline{G}(I)$ is Cohen-Macaulay. One of the main purposes of this
paper is to give an answer to the second question.

Theorem (see Theorem 2.1). Assume that $A$ is Gorenstein and $I=I_{Z}\subset
A$ is an elliptic ideal. For any minimal reduction $Q$ of $I$, the following
conditions are equivalent$:$

  1. $(1)$

$\overline{G}(I)$ is Gorenstein.

  2. $(2)$

$Q\colon I=Q+\overline{I^{2}}$.

  3. $(3)$

$\ell_{A}(\overline{I^{2}}/QI)=\ell_{A}(A/I)$.

  4. $(4)$

$\overline{e}_{2}(I)=\ell_{A}(A/I)$, where $\overline{e}_{2}(I)$ is the second
normal Hilbert coefficient of $I$.

  5. $(5)$

$KZ=-Z^{2}$, that is, $\chi(Z)=0$, where $K$ denotes the canonical divisor on
$X$.

When this is the case, $\ell_{A}(A/I)\leq p_{g}(A)$.

In Section 2, we give a proof of the theorem above and several examples. In
Section 3, we give a complete classification of elliptic ideals in a
Gorenstein elliptic singularities which satisfies the theorem in the
characteristic zero case.

##  1\. Preliminaries

Throughout this section, let $(A,\mathfrak{m})$ be an excellent two-
dimensional normal local domain containing an algebraically closed field
$k=A/\mathfrak{m}$, where $\mathfrak{m}$ denotes the unique maximal ideal of
$A$.

Let $I\subset A$ be an $\mathfrak{m}$-primary ideal. An element $a\in A$ is
said to be integral over $I$ if there exists a monic polynomial

$f(X)=X^{n}+c_{1}X^{n-1}+\cdots+c_{n}\;\;(c_{i}\in I^{i},\,i=1,2,\ldots,n)$

such that $f(a)=0$. The ideal of all elements that are integral over $I$ is
called the integral closure of $I$, denoted by $\overline{I}$. We assume that
$I$ is integrally closed (i.e. $\overline{I}=I$). The ideal $I$ is said to be
normal if $\overline{I^{n}}=I^{n}$ for $n\in\mathbb{Z}_{\geq 1}$. A divisor on
a resolution space whose support is contained in the exceptional set is called
a cycle. Then in our situation, there exist an resolution of singularities
$f\colon X\to\operatorname{Spec}A$ and an anti-nef cycle $Z$ on $X$ such that
$I\mathcal{O}_{X}=\mathcal{O}_{X}(-Z)$ and $I=H^{0}(X,\mathcal{O}_{X}(-Z))$.
Then we say that $I$ is represented by $Z$ on $X$, and denote it by $I=I_{Z}$.

In what follows, let $f\colon X\to\operatorname{Spec}A$ be a resolution of
singularities and $I=I_{Z}$ for some anti-nef cycle $Z$ on $X$. We recall the
following definition of $q(I)$ as follows.

######  Definition 1.1 (cf. [OWY4, Definition 2.1]).

For $I=I_{Z}$ and an integer $k\geq 0$ we define

$q(kI)=\dim_{k}H^{1}(X,\mathcal{O}_{X}(-kZ)).$

Then $q(kI)$ is a non-negative integer for every $k\geq 1$, and there exists a
decreasing sequence

$p_{g}(A)\geq q(I)\geq q(2I)\geq q(3I)\geq\cdots\geq 0,$

where $p_{g}(A)=q(0I)=\dim_{k}H^{1}(X,\mathcal{O}_{X})$ denotes the geometric
genus of $A$. Note that $p_{g}(A)$ is independent of the choice of $X$. We put
$q(\infty I):=q(kI)$ for sufficiently large $k$.

The following formula is called (Kato’s) Riemann-Roch formula. The result was
proved in [Ka] in the complex case, but it holds in any characteristic [WY].

######  Theorem 1.2 (Kato’s Riemann Roch formula).

Assume $I=I_{Z}$. Then we have

$\ell_{A}(A/I)+q(I)=-\dfrac{Z^{2}+KZ}{2}+p_{g}(A),$

where $K=K_{X}$ denotes the canonical divisor on $X$.

Next, we recall the normal Hilbert coefficients and their geometric
representation.

######  Proposition 1.3 (cf. [OWY2, Theorem 3.2]).

For $I=I_{Z}$, let $\overline{P}_{I}(n)$ be the normal Hilbert-polynomial of
$I$$:$

$\overline{P}_{I}(n)=\bar{e}_{0}(I){n+2\choose 2}-\bar{e}_{1}(I){n+1\choose
1}+\bar{e}_{2}(I).$

Then

  1. $(1)$

$\overline{P}_{I}(n)=\ell_{A}(A/\overline{I^{n+1}})$ for all $n\geq
p_{g}(A)-1$.

  2. $(2)$

$\bar{e}_{0}(I)=e_{0}(I)=-Z^{2}$.

  3. $(3)$

$\bar{e}_{1}(I)-\bar{e}_{0}(I)+\ell_{A}(A/I)=p_{g}(A)-q(I)$.

  4. $(4)$

$\bar{e}_{2}(I)=p_{g}(A)-q(nI)=p_{g}(A)-q(\infty I)$ for all $n\geq p_{g}(A)$.

For any ideal $I$, let $Q$ be its minimal reduction. Then we put

$r_{Q}(I)=\min\\{r\in\mathbb{Z}_{\geq 1}\,|\,I^{r+1}=QI^{r}\\}.$

In general, $r_{Q}(I)$ depends on the choice of $Q$. So we define

$r(I)=\min\\{r_{Q}(I)\,|\,\text{$Q$ is a minimal reduction of $I$}\\}.$

Then $r(I)$ is called a reduction number of $I$. We now recall the definition
of two normal reduction numbers.

######  Definition 1.4 (cf. [OWY4, Definition 2.5]).

For $I=I_{Z}$ and its minimal reduction $Q$, we put

$\displaystyle\operatorname{nr}(I)$ $\displaystyle=$
$\displaystyle\min\\{r\in\mathbb{Z}_{\geq
1}\,|\,\overline{I^{r+1}}=Q\overline{I^{r}}\\},$
$\displaystyle\operatorname{\bar{r}}(I)$ $\displaystyle=$
$\displaystyle\min\\{r\in\mathbb{Z}_{\geq
1}\,|\,\overline{I^{n+1}}=Q\overline{I^{n}}\;\text{for all $n\geq r$}\\}.$

A positive integer $\operatorname{nr}(I)$ (resp. $\operatorname{\bar{r}}(I)$)
is called the relative normal reduction number (resp. the normal reduction
number)

######  Remark 1.5 (cf. [Hun, Lemma 4.1]).

Let $Q,Q^{\prime}$ be minimal reductions of $I$. Then
$\overline{I^{n+1}}=Q\overline{I^{n}}$ if and only if
$\overline{I^{n+1}}=Q^{\prime}\overline{I^{n}}$.

Two normal reduction numbers can be calculated in terms of $\\{q(kI)\\}$.

######  Proposition 1.6 (cf. [OWY5, Proposition 2.2]).

For $I=I_{Z}$, the following statements hold.

  1. $(1)$

For any integer $n\geq 1$, we have

$2\cdot
q(nI)+\ell_{A}(\overline{I^{n+1}}/Q\overline{I^{n}})=q((n+1)I)+q((n-1)I).$

  2. $(2)$

We have

$\displaystyle\operatorname{nr}(I)$ $\displaystyle=$
$\displaystyle\min\\{n\in\mathbb{Z}_{\geq
1}\,|\,q((n-1)I)-q(nI)=q(nI)-q((n+1)I)\\},$
$\displaystyle\operatorname{\bar{r}}(I)$ $\displaystyle=$
$\displaystyle\min\\{n\in\mathbb{Z}_{\geq 1}\,|\,q((n-1)I)=q(nI)\\}.$

For instance, if $\operatorname{\bar{r}}(I)=2$, then $q(I)=q(2I)$. Hence we
have

$\ell_{A}(\overline{I^{2}}/QI)=p_{g}(A)-q(I)$

for any minimal reduction $Q$ of $I$.

######  Corollary 1.7.

For $I=I_{Z}$, the following statements hold.

  1. $(1)$

$\operatorname{\bar{r}}(I)\leq p_{g}(A)+1$.

  2. $(2)$

If equality holds in $(1)$, then $\operatorname{nr}(I)=1$.

###### Proof.

(1) Put $g=p_{g}(A)$. Then

$g=p_{g}(A)=q(0I)\geq q(I)\geq q(2I)\geq\cdots\geq q(gI)\geq q((g+1)I)$

is a decreasing sequence of non-negative integers of length $g+2$. Hence there
exists an integer $n$ with $1\leq n\leq g+2$ such that $q((n-1)I)=q(nI)$.
Hence $\operatorname{\bar{r}}(I)\leq n\leq g+1$.

(2) Suppose $\operatorname{\bar{r}}(I)=g+1$. Then

$g=p_{g}(A)=q(0I)>q(I)>q(2I)>\cdots>q(gI)=q((g+1)I)\geq 0$

yields $q(kI)=g-k$ for each $k=1,2,\ldots,g$.

If we substitute $n=1$ in (1) in Proposition 1.6, then we have
$\ell_{A}(\overline{I^{2}}/QI)=0$. Hence $\operatorname{nr}(I)=1$. ∎

Note that there exists an example of $A$ and an $\mathfrak{m}$-primary
integrally closed ideal $I$ which satisfies
$\operatorname{nr}(I)=1<\operatorname{\bar{r}}(I)=p_{g}(A)+1$; see [OWY5,
Example 3.9 and 3.10] and Example 1.11.

Now we recall the definition of normal tangent cones as the main target in
this paper.

######  Definition 1.8.

Let $I=I_{Z}\subset A$ be an $\mathfrak{m}$-primary integrally closed ideal.
Then

  1. $(1)$

$G(I)=\oplus_{n\geq 0}I^{n}/I^{n+1}$ is called the associated graded ring or
the tangent cone of $I$.

  2. $(2)$

$\overline{G}(I)=\oplus_{n\geq 0}\overline{I^{n}}/\overline{I^{n+1}}$ is
called the normal tangent cone of $I$.

In this paper, we want to investigate some ring-theoretic properties of normal
tangent cones of integrally closed ideals. We first consider the case of
$p_{g}$-ideals, which was introduced by the authors in [OWY1].

######  Definition 1.9 (cf. [OWY1]).

Assume that $I=I_{Z}$. Then $I$ is called the $p_{g}$-ideal if one of the
following equivalent conditions is satisfied:

  1. $(1)$

$q(I)=p_{g}(A)$.

  2. $(2)$

$\bar{e}_{1}(I)-\bar{e}_{0}(I)+\ell_{A}(A/I)=0$.

  3. $(3)$

$\bar{e}_{2}(I)=0$.

  4. $(4)$

$\operatorname{\bar{r}}(I)=1$.

  5. $(5)$

$I$ is normal and $I^{2}=QI$ for some minimal reduction $Q$ of $I$.

Then the following results are known.

######  Proposition 1.10 (cf. [ORWY]).

If $I$ is a $p_{g}$-ideal, then $\overline{G}(I)=G(I)$ is Cohen-Macaulay.

Note that $\operatorname{nr}(I)=1$ does not necessarily imply the Cohen-
Macaulayness of $\overline{G}(I)$.

######  Example 1.11 (cf. [OWY5, Example 3.10]).

Let $K$ be a filed of $\operatorname{char}K\neq 2,3$, and let
$A=K[[xy,xz,yz,y^{2},z^{2}]]$ with $x^{2}=y^{6}+z^{6}$. That is,
$A=K[[a,b,c,d,e]]/{\mathfrak{a}}$, where ${\mathfrak{a}}$ is generated by the
following polynomials:

$\begin{array}[]{lll}a^{2}-d^{4}-de^{3},&ab-
cd^{3}-ce^{3},&b^{2}-d^{3}e-e^{4}\\\ ac-bd,&ae-bc,&de-c^{2}\end{array}$

Then $A$ is an excellent normal local domain with
$\dim_{k}\mathfrak{m}/\mathfrak{m}^{2}=e_{0}(\mathfrak{m})+\dim A-1=5$.

If we put $I=(xy,xz,yz,y^{2},z^{4})=(a,b,c,d,e^{2})\supset Q=(c,d-e^{2})$,
then $I=I_{Z}$ and $1=\operatorname{nr}(I)<\operatorname{\bar{r}}(I)=3$. In
particular, $\overline{G}(I)$ is not Cohen-Macaulay by the following lemma.

On the other hand, $G(I)$ is Cohen-Macaulay because $I^{2}=QI$.

######  Lemma 1.12.

Assume that $\overline{G}(I)$ is Cohen-Macaulay. Then

  1. $(1)$

$\operatorname{nr}(I)=\operatorname{\bar{r}}(I)$ holds true.

  2. $(2)$

If $A$ is not a rational singularity, that is, $p_{g}(A)\geq 1$, then
$\operatorname{\bar{r}}(I)\leq p_{g}(A)$.

###### Proof.

(1) Assume that $\overline{G}(I)$ is Cohen-Macaulay and put
$\operatorname{nr}(I)=r\geq 1$. Take a minimal reduction $Q$ of $I$. Then we
have $\overline{I^{n+1}}\subset\overline{I^{r+1}}=Q\overline{I^{r}}\subset Q$
for every integer $n\geq r$. Since $\overline{G}(I)$ is Cohen-Macaulay, we get
$\overline{I^{n+1}}\subset Q\cap\overline{I^{n+1}}=Q\overline{I^{n}}$ for such
an integer $n$ (cf. [VV], [GN]). Therefore $\operatorname{\bar{r}}(I)\leq
r=\operatorname{nr}(I)$, as required.

(2) The assertion follows from (1) and Corollary 1.7. ∎

An $\mathfrak{m}$-primary ideal $I$ is called good if $I^{2}=QI$ and $I=Q:I$
for some (every) minimal reduction $Q$ of $I$.

######  Proposition 1.13.

Assume $A$ is Gorenstein and $I=I_{Z}$ is a $p_{g}$-ideal, that is, $I=I_{Z}$
and $\operatorname{\bar{r}}(I)=1$. Then the following conditions are
equivalent$:$

  1. $(1)$

$\overline{G}(I)(=G(I))$ is Gorenstein.

  2. $(2)$

$I$ is good.

  3. $(3)$

$2\cdot\ell_{A}(A/I)=e_{0}(I)$ $(\text{i.e.}\;\ell_{A}(A/I)=\ell_{A}(I/Q))$.

  4. $(4)$

$KZ=0$, where $K$ is a canonical divisor on $X$.

###### Proof.

$(1)\Longleftrightarrow(2)\Longleftrightarrow(3)$ follows from [GIW, Theorem
6.1]. $(3)\Longleftrightarrow(4)$ follows from Kato’s Riemann-Roch formula
(Theorem 1.2). ∎

Next, we recall the notion of elliptic ideals and strongly elliptic ideals.

######  Definition 1.14 (cf. [ORWY, Theorem 3.2]).

An ideal $I=I_{Z}$ is called the elliptic ideal if one of the following
equivalent conditions is satisfied:

  1. $(1)$

$q(I)=q(\infty I)<p_{g}(A)$.

  2. $(2)$

$\bar{e}_{1}(I)-\bar{e}_{0}(I)+\ell_{A}(A/I)=\bar{e}_{2}(I)>0$.

  3. $(3)$

$\operatorname{\bar{r}}(I)=2$.

######  Definition 1.15 (cf. [ORWY, Theorem 3.9]).

An ideal $I=I_{Z}$ is called the strongly elliptic ideal if one of the
following equivalent conditions is satisfied:

  1. $(1)$

$p_{g}(A)-1=q(I)=q(\infty I)$.

  2. $(2)$

$\bar{e}_{1}(I)-\bar{e}_{0}(I)+\ell_{A}(A/I)=1$ and
$\operatorname{nr}(I)=\operatorname{\bar{r}}(I)$.

  3. $(3)$

$\bar{e}_{2}(I)=1$.

  4. $(4)$

$\operatorname{\bar{r}}(I)=2$ and $\ell_{A}(\overline{I^{2}}/QI)=1$.

######  Proposition 1.16 (cf. [ORWY, Theorem 3.2]).

If $I$ is an elliptic ideal, then $\overline{G}(I)$ is Cohen-Macaulay.

######  Remark 1.17.

If $A$ is an analytically unramified Cohen-Macaulay local domain containing a
field with $\dim A=2$ and $\operatorname{\bar{r}}(I)=2$, then the Cohen-
Macaulayness of $\overline{G}(I)$ also follows from [Hun, Theorems 4.4, 4.5
and Theorem of p.317].

So it is natural to ask the following question.

######  Question 1.18.

Let $I\subset A$ be an elliptic ideal. When is $\overline{G}(I)$ Gorenstein?

In the next section, we give an answer as the main result in this paper.
Before doing that, we recall the following criterion for Gorensteinness of
$G(I)$. Note that if $G(I)$ or $\overline{G}(I)$ is Gorenstein, then so is $A$
(see Proposition 1.6 and 5.17 of [TW]).

######  Theorem 1.19 (cf. [GI, Theorem 1.4]).

Assume that $A$ is a Gorenstein local ring of any dimension. Let $I\subset A$
be an $\mathfrak{m}$-primary ideal with $r(I)=r$, and let $Q$ be a minimal
reduction of $I$. Suppose that $G(I)$ is Cohen-Macaulay. Then the following
conditions are equivalent$:$

  1. $(1)$

$G(I)$ is Gorenstein.

  2. $(2)$

$I^{r}=Q^{r}\colon I^{r}$.

##  2\. Gorensteinness for normal tangent cone of elliptic ideals

In this section, we keep the notation as in the previous section. The main aim
of this section is to prove the following theorem.

######  Theorem 2.1.

Assume that $A$ is Gorenstein and $\operatorname{\bar{r}}(I)=2$. For any
minimal reduction $Q$ of $I$, the following conditions are equivalent$:$

  1. $(1)$

$\overline{G}(I)$ is Gorenstein.

  2. $(2)$

$Q\colon I=Q+\overline{I^{2}}$.

  3. $(3)$

$\ell_{A}(\overline{I^{2}}/QI)=\ell_{A}(A/I)$.

  4. $(4)$

$\overline{e}_{2}(I)=\ell_{A}(A/I)$.

  5. $(5)$

$KZ=-Z^{2}$, that is, $\chi(Z):=-\dfrac{KZ+Z^{2}}{2}=0$

When this is the case, $\ell_{A}(A/I)\leq p_{g}(A)$.

###### Proof.

By Proposition 1.16, $\overline{G}(I)$ is Cohen-Macaulay.

Let $Q=(a,b)$ be a minimal reduction of $I$ and fix it. Then
$a^{*},b^{*}\in\overline{G}(I)$ forms a homogeneous regular sequence of degree
$1$ and we get

$B:=\overline{G}(I)/(a^{*},b^{*})\cong A/I\oplus
I/(Q+\overline{I^{2}})\oplus(Q+\overline{I^{2}})/Q.$

Thus $\overline{G}(I)$ is Gorenstein if and only if $B$ is Gorenstein. $B$ is
an Artinian local ring with the maximal ideal
$\mathfrak{M}_{B}=\mathfrak{m}/I\oplus B_{1}\oplus B_{2}$ and we denote
$\operatorname{Soc}(B)=\operatorname{Ann}_{B}(\mathfrak{M}_{B})$.

$(2)\Longrightarrow(1):$ It suffices to show
$\dim_{k}\operatorname{Soc}(B)=1$. Let $x^{*}$ be a homogeneous element in
$\operatorname{Soc}(B)$. Namely, for $x\in\overline{I^{m}}$, we denote $x^{*}$
the image of $x$ in $B_{m}$.

The case where $x^{*}\in B_{0}$:

Since $x^{*}B_{2}=0$ in $B_{2}$, we have $x\in
Q\colon\overline{I^{2}}=Q\colon(Q+\overline{I^{2}})=Q:(Q:I)=I$. Hence
$x^{*}=0$ in $B_{0}$.

The case where $x^{*}\in B_{1}$:

Since $x^{*}B_{1}=0$ in $B_{2}$, we have $x\in Q:I=Q+\overline{I^{2}}$. Hence
$x^{*}=0$ in $B_{1}$.

The case where $x^{*}\in B_{2}$:

Since $x\in Q:\mathfrak{m}$, we have
$x^{*}\in\operatorname{Soc}((Q:I)/Q)=(Q\colon\mathfrak{m})/Q\cong k$.

Therefore $\dim_{K}\operatorname{Soc}(B)=1$ and thus $B$ is Gorenstein.

$(1)\Longrightarrow(2)\Longleftrightarrow(3):$ Huneke-Itoh’s theorem (cf.
[Hun, It]) yields

$\ell_{A}(B_{2})=\ell_{A}(Q+\overline{I^{2}}/Q)=\ell_{A}(\overline{I^{2}}/Q\cap\overline{I^{2}})=\ell_{A}(\overline{I^{2}}/QI)(\neq
0).$

From Matlis duality, we have

$\ell_{A}(Q:I/Q)=\ell_{A}(\operatorname{Hom}_{A}(A/I,A/Q))=\ell_{A}(K_{A/I})=\ell_{A}(A/I)=\ell_{A}(B_{0})\;(\neq
0).$

As $I\overline{I^{2}}\subset\overline{I^{3}}=Q\overline{I^{2}}\subset Q$, we
have $\overline{I^{2}}\subset Q:I$. In particular, $Q+\overline{I^{2}}\subset
Q:I$. Hence

$\ell_{A}(B_{2})-\ell_{A}(B_{0})=\ell_{A}(\overline{I^{2}}/QI)-\ell_{A}(A/I)=\ell_{A}(Q:I/Q+\overline{I^{2}})\geq
0.$

If $B$ is Gorenstein, then $\ell_{A}(B_{2})-\ell_{A}(B_{0})=0$. Thus the
implication follows from here.

$(3)\Longleftrightarrow(4)\Longleftrightarrow(5):$ Since
$\operatorname{\bar{r}}(I)=2$, we have $q(I)=q(2I)=q(\infty I)$. Then by
Proposition 1.6, we have

$\ell_{A}(\overline{I^{2}}/QI)=p_{g}(A)-q(I).$

Moreover, Proposition 1.3 implies

$\bar{e}_{2}(I)=p_{g}(A)-q(\infty
I)=p_{g}(A)-q(I)=\ell_{A}(\overline{I^{2}}/QI).$

On the other hand, Kato’s Riemann-Roch formula yields

$\ell_{A}(A/I)+q(I)=\chi(Z)+p_{g}(A).$

Hence

$\chi(Z)=\ell_{A}(A/I)-\\{p_{g}(A)-q(I)\\}=\ell_{A}(A/I)-\ell_{A}(\overline{I^{2}}/QI)=\ell_{A}(A/I)-\bar{e}_{2}(I).$

The assertion follows from here. ∎

Using the theorem above, we ask the following question:

######  Question 2.2.

When is $\overline{G}(\mathfrak{m})$ Gorenstein?

If $\operatorname{\bar{r}}(\mathfrak{m})$ is small enough, then
$\overline{G}(\mathfrak{m})$ is Gorenstein. Precisely speaking, if
$\operatorname{\bar{r}}(\mathfrak{m})\leq 2$, then
$\overline{G}(\mathfrak{m})$ is Gorenstein. But there exists an example of $A$
for which $\overline{G}(\mathfrak{m})$ is not Gorenstein (see Example 2.9).

######  Proposition 2.3.

Assume $A$ is Gorenstein which is not regular.

  1. $(1)$

If $\operatorname{\bar{r}}(\mathfrak{m})=1$, then
$\overline{G}(\mathfrak{m})=G(\mathfrak{m})$ is Gorenstein.

  2. $(2)$

If $\operatorname{\bar{r}}(\mathfrak{m})=2$, then $\overline{G}(\mathfrak{m})$
is Gorenstein.

  3. $(3)$

If $p_{g}(A)\leq 2$, then $\overline{G}(\mathfrak{m})$ is Gorenstein.

###### Proof.

(1) Suppose $\operatorname{\bar{r}}(\mathfrak{m})=1$. Since $\mathfrak{m}$ is
a $p_{g}$-ideal, then $\mathfrak{m}^{2}=Q\mathfrak{m}$ and $\mathfrak{m}$ is
normal. As $A$ is Gorenstein, we have $e_{0}(\mathfrak{m})=2$. In particular,
$A$ is a hypersurface. Thus $\overline{G}(\mathfrak{m})=G(\mathfrak{m})$ is
Gorenstein.

(2) Suppose $\operatorname{\bar{r}}(\mathfrak{m})=2$. Since $Q\subsetneq
Q+\overline{\mathfrak{m}^{2}}\subset Q:\mathfrak{m}$ and
$\ell_{A}(Q\colon\mathfrak{m}/Q)=1$, we have
$Q:\mathfrak{m}=Q+\overline{\mathfrak{m}^{2}}$. Then Theorem 2.1 implies that
$\overline{G}(\mathfrak{m})$ is Gorenstein.

(3) If $p_{g}(A)\leq 1$, then $\operatorname{\bar{r}}(\mathfrak{m})\leq
p_{g}(A)+1\leq 2$. If $p_{g}(A)=2$, then $A$ is an elliptic singularity by
[Yau, Theorem B]. Thus [Ok2, Theorem 3.3] implies
$\operatorname{\bar{r}}(\mathfrak{m})\leq 2$. In both cases, the assertion
follows from (1), (2). ∎

A local ring $A$ which satisfies $p_{g}(A)=1$ is called a strongly elliptic
singularity. Then any $\mathfrak{m}$-primary integrally closed ideal $I$ is
either a $p_{g}$-ideal or a strongly elliptic ideal; see [ORWY].

######  Corollary 2.4.

Assume that $A$ is Gorenstein. Let $I$ be a strongly elliptic ideal. Then
$\overline{G}(I)$ is Gorenstein if and only if $I=\mathfrak{m}$.

A Gorenstein local ring $A$ is called a minimally elliptic singularity if it
is a strongly elliptic singularity.

######  Corollary 2.5.

Assume that $A$ is a minimally elliptic singularity. Let $I$ be an integrally
closed $\mathfrak{m}$-primary ideal. Then $\overline{G}(I)$ is Gorenstein if
and only if one of the following cases occurs: $($a$)$ $I=\mathfrak{m}$
$($b$)$ $I$ is normal and good.

In order to give several examples, we recall the following lemma.

######  Lemma 2.6 (cf. [OWY4]).

Let $2\leq a\leq b\leq c$ be integers, and put
$R=\mathbb{C}[x,y,z]_{(x,y,z)}$, and $A=R/(x^{a}+y^{b}+z^{c})$. Then $A$ is an
excellent normal Gorenstein local domain of $\dim A=2$.

  1. $(1)$

$p_{g}(A)=\sharp\\{(i,j,k)\in\mathbb{Z}_{\geq 0}\,|\,0\leq ibc+jac+kab\leq
a(A)\\}$, where $a(A)=abc-(ab+bc+ca)$; this formula follows from Theorem 2.7
below.

  2. $(2)$

$\operatorname{\bar{r}}(\mathfrak{m})=\lfloor\frac{(a-1)b}{a}\rfloor$.

For the convenience of readers, we recall the formula for $p_{g}(S)$ in the
case $S$ is a normal Gorenstein graded ring.

######  Theorem 2.7 (see [Wt2, 2.10], [Wt1, 3.3]).

Let $S=\bigoplus_{i\geq 0}S_{i}$ be a two-dimensional normal graded ring
finitely generated over $S_{0}=k$ and let $S_{+}=\bigoplus_{i\geq 1}S_{i}$.
Then the geometric genus of $(S,S_{+})$ $($or its completion by $S_{+}$$)$ is
given by

$p_{g}(S)=\sum_{i=0}^{a(S)}\dim_{k}S_{i},$

where $a(S)$ is the $a$-invariant of Goto–Watanabe $($[GW]$)$.

######  Example 2.8.

Assume that $A$ is Gorenstein but not regular, and let $g\geq 1$ be an
integer.

  1. $(1)$

The case where $\operatorname{\bar{r}}(\mathfrak{m})=1$.

    1. (a)

If $A$ is a rational singularity, that is, $p_{g}(A)=0$, then
$\operatorname{\bar{r}}(\mathfrak{m})=1$ and
$\bar{G}(\mathfrak{m})=G(\mathfrak{m})$ is a hypersurface.

    2. (b)

If $A=\mathbb{C}[x,y,z]_{(x,y,z)}/(x^{2}+y^{3}+z^{6g+1})$, then $p_{g}(A)=g$,
$\operatorname{\bar{r}}(\mathfrak{m})=1$ and
$\overline{G}(\mathfrak{m})=G(\mathfrak{m})\cong K[X,Y,Z]/(X^{2})$ is
Gorenstein.

  2. $(2)$

The case where $\operatorname{\bar{r}}(\mathfrak{m})=2$.

    1. (a)

If $e_{0}(\mathfrak{m})\geq 3$, then $\mathfrak{m}$ is a strongly elliptic
ideal and $I=\mathfrak{m}$ satisfies the condition of Theorem 2.1(2).

For instance, if $A=\mathbb{C}[x,y,z]_{(x,y,z)}/(x^{3}+y^{3}+z^{3g})$, then
$p_{g}(A)=g$, $\operatorname{\bar{r}}(\mathfrak{m})=2$ and

$\overline{G}(\mathfrak{m})=G(\mathfrak{m})\cong\left\\{\begin{array}[]{ll}\mathbb{C}[X,Y,Z]/(X^{3}+Y^{3}+Z^{3})&\;\text{(if
$g=1$)};\\\\[2.84526pt]
\mathbb{C}[X,Y,Z]/(X^{3}+Y^{3})&\;\text{(otherwise)}.\end{array}\right.$

    2. (b)

If $A=\mathbb{C}[x,y,z]_{(x,y,z)}/(x^{2}+y^{4}+z^{4g})$, then $p_{g}(A)=g$,
$\operatorname{\bar{r}}(\mathfrak{m})=2$ and
$G(\mathfrak{m})\cong\mathbb{C}[X,Y,Z]/(X^{2})$. Moreover, we have

$\overline{G}(\mathfrak{m})\cong\left\\{\begin{array}[]{ll}\mathbb{C}[X,Y,Z]/(X^{2}+Y^{4}+Z^{4})&\;\text{(if
$g=1$)};\\\\[2.84526pt]
\mathbb{C}[X,Y,Z]/(X^{2}+Y^{4})&\;\text{(otherwise)}.\end{array}\right.$

The next example shows that neither $\operatorname{\bar{r}}(\mathfrak{m})=3$
nor $p_{g}(A)=3$ does not imply that $\overline{G}(\mathfrak{m})$ is
Gorenstein.

######  Example 2.9.

Let $A=\mathbb{C}[x,y,z]_{(x,y,z)}/(x^{3}+y^{5}+z^{5})$. Then

  1. $(1)$

$\operatorname{\bar{r}}(\mathfrak{m})=3$.

  2. $(2)$

$p_{g}(A)=3$.

  3. $(3)$

$\overline{G}(\mathfrak{m})$ is not Gorenstein. In fact,
$\overline{G}(\mathfrak{m})/(y^{*},z^{*})\cong\mathbb{C}[X,V]/(X^{2},XV,V^{3})$.

We can probably give a similar criterion as in the case of
$\operatorname{\bar{r}}(I)=2$. Indeed, we have several results for Brieskorn
type hypersurfaces (cf. [OWY4]). Moreover, we have the following example.

######  Example 2.10.

Let $r,n$ be integers with $r\geq 2$ and $n\geq 1$. Let

$A=k[x,y,z]_{(x,y,z)}/(x^{r+1}+y^{r+1}+z^{(r+1)n}).$

Let $I=(x,y,z^{n})$ and $Q=(y,z^{n})$. Then

  1. $(1)$

$\overline{G}(I)=G(I)$ is Gorenstein.

  2. $(2)$

$Q\colon I=Q+I^{r}$.

  3. $(3)$

$\ell_{A}(A/I)=\ell_{A}(I^{r}/QI^{r-1})$.

######  Question 2.11.

Does $\operatorname{\bar{r}}(I)=2$ imply $r(I)\leq 2$?

This is true for strongly elliptic ideals. But we do not know in the case of
general elliptic ideals. Moreover, if this were true, then we can give a
complete answer to the question: when is $G(I)$ Gorenstein for elliptic
ideals?

######  Example 2.12.

Let $A=\mathbb{C}[x,y,z]_{(x,y,z)}/(x^{2}+y^{4}+z^{8})$ and put
$I=(x,y,z^{2})$ and $Q=(y,z^{2})$. Then

  1. $(1)$

$\operatorname{\bar{r}}(I)=2$.

  2. $(2)$

$r(I)=1$ and $Q:I=I$. In particular, $I$ is not normal.

  3. $(3)$

$\overline{G}(I)\cong\mathbb{C}[X,Y,Z,W]/(X^{2}+Y^{4}+W^{4},Z^{2})$ is
Gorenstein.

##  3\. Existence of elliptic ideals for which $\overline{G}(I)$ is Gorenstein

The aim of this section is to give a complete classification of elliptic
ideals $I=I_{Z}$ in a Gorenstein elliptic singularity which satisfy the
conditions of Theorem 2.1. First, we show that such a cycle $Z$ for which
$I=I_{Z},\operatorname{\bar{r}}(I)=2$ and $\bar{G}(I)$ is Gorenstein is
characterized in terms of the “elliptic sequence” on the minimal resolution
$X$ of $\operatorname{Spec}(A)$ and, in particular the possibility of such $Z$
is finite and bounded by $p_{g}(A)$ (Theorem 3.15). Up to this point, our
argument is valid in any characteristics. To guarantee the existence of such
ideals, we need to consider Proposition 3.24. In the case $-Z^{2}=1$, we need
to check the Picard groups of the exceptional curves and have to assume
$\operatorname{char}{k}=0$. We will also give examples of singularities $A,B$
which are hypersurfaces, have the same resolution graph, such that the number
of ideals $I$ for which $\bar{G}(I)$ is Gorenstein is $p_{g}(A)$ for $A$ and
$0$ for $B$ (see Example 3.28).

Since we assume that $\operatorname{\bar{r}}(I)=2$, we have $q(I)<p_{g}(A)$.
We note that $\operatorname{\bar{r}}(I)\leq 2$ for any $\mathfrak{m}$-primary
integrally closed ideal $I$ in an elliptic singularity ([Ok2]).

Let $f\colon X\to\operatorname{Spec}A$ be a resolution with exceptional set
$E$ and let $E=\bigcup E_{i}$ be the decomposition into irreducible
components. A divisor $D$ on $X$ is said to be nef (resp. anti-nef) if
$DE_{i}\geq 0$ (resp. $DE_{i}\leq 0$) for all $E_{i}\subset E$. For a cycle
$D>0$ on $X$, we denote by $\chi(D)$ the Euler characteristic
$\chi(\mathcal{O}_{D})=h^{0}(\mathcal{O}_{D})-h^{1}(\mathcal{O}_{D})$ and by
$Z_{D}$ the fundamental cycle on $\operatorname{Supp}(D)$, i.e., $Z_{D}$ is
the minimal cycle such that
$\operatorname{Supp}(Z_{D})=\operatorname{Supp}(D)$ and $Z_{D}E_{i}\leq 0$ for
all $E_{i}\subset\operatorname{Supp}(D)$. By the Riemann-Roch formula,
$\chi(D)=-(D^{2}+DK_{X})/2$, where $K_{X}$ denotes a canonical divisor on $X$.

######  Definition 3.1 (Wagreich [Wa, p. 428]).

A normal surface singularity $(A,\mathfrak{m})$ is called an elliptic
singularity if one of the following equivalent conditions holds:

  1. $(1)$

$\chi(D)\geq 0$ for all cycles $D>0$ and $\chi(F)=0$ for some cycle $F>0$;

  2. $(2)$

$\chi(Z_{E})=0$.

(For the proof of the implication (2) $\Rightarrow$ (1), see [La, Corollary
4.2], [To, (6.4), (6.5)].)

######  Definition 3.2 (Laufer [La, Definition 3.1 and 3.2]).

If $A$ is an elliptic singularity, then there exists a unique cycle $E_{min}$
such that $\chi(E_{min})=0$ and $\chi(D)>0$ for all cycles $D$ satisfying
$0<D<E_{min}$. The cycle $E_{min}$ is called the minimally elliptic cycle.

From [La, Proposition 3.1, Proposition 3.2, Corollary 4.2], [Wa, p.428], [To,
(6.4), (6.5)], we have the following.

######  Proposition 3.3 (cf. [Ok1, 2.3]).

Assume that $A$ is elliptic and let $D>0$ be an arbitrary cycle on $X$. Then
we have the following.

  1. $(1)$

$\chi(D)\geq 0$. If $\chi(D)=0$, then $D\geq E_{min}$ and $D$ is connected,
i.e., $\operatorname{Supp}(D)$ is connected.

  2. $(2)$

If $D$ is a connected reduced cycle without component of $E_{min}$, then
$E_{min}D\leq 1$ and $\chi(Z_{D})=1$.

######  Lemma 3.4.

Assume that $A$ is an elliptic singularity and let $I\subset A$ be an ideal
represented by a cycle $Z>0$ on $X$. Let $X_{0}\to\operatorname{Spec}A$ be the
minimal resolution and $\phi:X\to X_{0}$ the natural morphism. If $\chi(Z)=0$,
then $I=I_{Z}$ can be represented on $X_{0}$ by $\phi_{*}Z$.

###### Proof.

Let $Z_{0}=\phi_{*}Z$, $F=Z-\phi^{*}Z_{0}$, and
$K_{\phi}=K_{X}-\phi^{*}K_{X_{0}}$. Then $K_{\phi}\equiv K_{X}$ on the
$\phi$-exceptional set, namely, $(K_{\phi}-K_{X})E_{i}=0$ for all
$E_{i}\subset E$ such that $\phi(E_{i})$ is a point. Hence we have

$Z^{2}+ZK_{X}=(\phi^{*}Z_{0}+F)^{2}+(\phi^{*}Z_{0}+F)(\phi^{*}K_{X_{0}}+K_{\phi})=Z_{0}^{2}+Z_{0}K_{X_{0}}+F^{2}+FK_{X}.$

Suppose that $F\neq 0$. Then we have $H^{1}(\mathcal{O}_{F})=0$ and
$\chi(Z)=\chi(Z_{0})+\chi(F)$. Since $A$ is elliptic and $\chi(Z)=0$, we have
$\chi(Z_{0})=\chi(F)=0$; however, it contradicts that
$\chi(F)=h^{0}(\mathcal{O}_{F})>0$. Therefore, $F=0$ and $ZC=0$ for every
$\phi$-exceptional curve $C$. Hence $\mathcal{O}_{X_{0}}(-Z_{0})$ is
generated. ∎

In the rest of this section, we always assume that $A$ is Gorenstein elliptic
singularity. By virtue of Lemma 3.4, for the aim of this section, we may also
assume that $f\colon X\to\operatorname{Spec}A$ is the minimal resolution.

######  Definition 3.5.

We define the elliptic sequence $\\{Z_{0},\dots,Z_{m}\\}$ on $X$ as follows.
Let $B_{0}=E$ and $Z_{0}=Z_{B_{0}}$. If $Z_{0}E_{min}<0$, then the elliptic
sequence is $\\{Z_{0}\\}$. If $Z_{0},\dots,Z_{i}$ are determined and
$Z_{i}E_{min}=0$, then define $B_{i+1}$ to be the maximal reduced connected
cycle containing $\operatorname{Supp}(E_{min})$ such that $Z_{i}B_{i+1}=0$,
and let $Z_{i+1}=Z_{B_{i+1}}$. If we have $Z_{m}E_{min}<0$ for some $m\geq 0$,
then the elliptic sequence is $\\{Z_{0},\dots,Z_{m}\\}$.

For $0\leq t\leq m$, we define cycles $C_{t}$ and $C^{\prime}_{t}$ by

$C_{t}=\sum_{i=0}^{t}Z_{i}\quad\text{and}\quad
C_{t}^{\prime}=\sum_{i=t}^{m}Z_{i}.$

We put $C_{-1}=C^{\prime}_{m+1}=0$.

######  Example 3.6.

Assume that $\operatorname{char}(k)\neq 2$. Let $R=k[x,y,z]_{(x,y,z)}$ and put
$A=R/(x^{2}+y^{4}+z^{4m+4})$ (cf. Example 2.8 (2)(b)). Then $A$ is elliptic
(see [OWY4, 4.4]) and its resolution graph is as in Figure 1, where $E_{m}$ is
an elliptic curve and other components are nonsingular rational curves. Then
$Z_{i}=E_{m}+\sum_{j=i}^{m-1}(E_{i}^{(1)}+E_{i}^{(2)})$.

$\cdots$$\cdots$$-2$$-2$$-2$$-2$$-2$$E_{m}$$E_{m-1}^{(1)}$$E_{m-1}^{(2)}$$E_{0}^{(2)}$$E_{0}^{(1)}$
Figure 1.

######  Proposition 3.7 (cf. [Ne, §2]).

We have the following properties$:$

  1. $(1)$

$B_{0}>B_{1}>\dots>B_{m}$, $Z_{m}=E_{min}$.

  2. $(2)$

$Z_{i}Z_{j}=0$ for $i\not=j$ and $-Z_{0}^{2}\geq\cdots\geq-Z_{m}^{2}$.

  3. $(3)$

$C_{t}$ is anti-nef for $0$ $\leq t\leq m$.

  4. $(4)$

For $0\leq t\leq m$, $C^{\prime}_{t}$ is the canonical cycle on $B_{t}$,
namely, $(K_{X}+C^{\prime}_{t})E_{i}=0$ for every $E_{i}\leq B_{t}$.

  5. $(5)$

$\chi(C_{t})=\chi(C^{\prime}_{t})=\chi(Z_{t})=0$ for $0\leq t\leq m$.

Since $A$ is Gorenstein, the canonical divisor $K_{X}$ is linearly equivalent
to a cycle. Hence we may assume that $-K_{X}=C^{\prime}_{0}=C_{m}$.

The cycles $C_{t}$ are characterized as follows.

######  Proposition 3.8 (Tomari [To, Theorem (6.4)], Némethi [Ne, 2.13]).

Let $D\geq 0$ be a cycle. If $D\leq C_{m}$ and $D$ is anti-nef, then $D=C_{t}$
for some $-1\leq t\leq m$.

######  Lemma 3.9 ([Ok1, 2.13], cf. [Ne, 2.20]).

For an integer $t$ such that $0\leq t\leq m-1$, the following conditions are
equivalent$:$

  1. $(1)$

there exists a function $f\in H^{0}(\mathcal{O}_{X}(-C_{t}))\setminus
H^{0}(\mathcal{O}_{X}(-C_{t+1}))$.

  2. $(2)$

$\mathcal{O}_{C^{\prime}_{t+1}}(-C_{t})\cong\mathcal{O}_{C^{\prime}_{t+1}}$.

  3. $(3)$

$h^{1}(\mathcal{O}_{X}(-C_{t}))=h^{1}(\mathcal{O}_{X}(-C_{t+1}))+1$.

  4. $(4)$

$\dim_{k}H^{0}(\mathcal{O}_{X}(-C_{t}))/H^{0}(\mathcal{O}_{X}(-C_{t+1}))=1$.

If the conditions above are satisfied, then
$h^{1}(\mathcal{O}_{X}(-C_{t}))=h^{1}(\mathcal{O}_{C^{\prime}_{t+1}})$.

The condition (2) of Lemma 3.9 says that
$\mathcal{O}_{C^{\prime}_{t+1}}(-(Z_{0}+\cdots+Z_{t}))\cong\mathcal{O}_{C^{\prime}_{t+1}}$.
In fact, our main result in this section will need the fact that
$\mathcal{O}_{C^{\prime}_{t+1}}(-Z_{i})\cong\mathcal{O}_{C^{\prime}_{t+1}}(-Z_{j})$
for $0\leq i\leq j\leq t$ and these invertible sheaves have a finite order in
the Picard group of $C^{\prime}_{t+1}$ (see Proposition 3.21, 3.23 and 3.24).

######  Definition 3.10.

We define a set $\mathcal{A}_{f}$ by

$\mathcal{A}_{f}={\left\\{i\,\left|\,0\leq i\leq m-1,\;\text{$i$ satisfies the
conditions in Lemma~{}\ref{l:fexists}}\right.\right\\}}\cup\\{m\\}.$

We put $\beta=\min\mathcal{A}_{f}$.

######  Remark 3.11.

We have $\\#\mathcal{A}_{f}=p_{g}(A)$. In fact, if
$\mathcal{A}_{f}=\\{\beta_{1},\dots,\beta_{\ell}\\}$, $\beta_{i}<\beta_{i+1}$,
then

$\displaystyle\ell_{A}\left(A/H^{0}(\mathcal{O}_{X}(-C_{\beta_{i}}))\right)=i,$
$\displaystyle\ell_{A}\left(A/H^{0}(\mathcal{O}_{X}(-C_{\beta_{\ell}}))\right)=\ell_{A}\left(A/H^{0}(\mathcal{O}_{X}(-C_{m}))\right)=p_{g}(A).$

######  Definition 3.12.

For an element $f\in H^{0}(\mathcal{O}_{X}(-E))=\mathfrak{m}$, let $(f)_{E}$
denote the exceptional part of the divisor $\operatorname{div}_{X}(f)$; hence
$\operatorname{div}_{X}(f)-(f)_{E}$ is the proper transform of the divisor
$\operatorname{div}_{\operatorname{Spec}(A)}(f)$ on $X$.

Let $W>0$ be an anti-nef cycle. A cycle $Y>0$ is called the fixed part of
$\mathcal{O}_{X}(-W)$ if $Y$ is the maximal cycle such that
$H^{0}(\mathcal{O}_{X}(-W-Y))=H^{0}(\mathcal{O}_{X}(-W))$; in this case,
$(f)_{E}=W+Y$ for a general element $f\in H^{0}(\mathcal{O}_{X}(-W))$. An
irreducible component of the fixed part $Y$ is called a fixed component.

Note that if $\mathcal{O}_{X}(-W)$ has no fixed component in a cycle $D>0$ and
$WD=0$, then $\mathcal{O}_{D}(-W)\cong\mathcal{O}_{D}$.

######  Remark 3.13.

From the conditions (1) and (2) of Lemma 3.9, for $i\in\mathcal{A}_{f}$,
$\mathcal{O}_{X}(-C_{i})$ has no fixed component on $B_{i+1}$.

######  Lemma 3.14.

Let $D$ be a nef divisor on $X$. Then we have the following.

  1. $(1)$

$\mathcal{O}_{X}(-C_{m})$ has no fixed component in $B_{m}$. If
$Z_{m}^{2}=-1$, $\mathcal{O}_{Z_{m}}(-C_{m})$ is not generated.

  2. $(2)$

If $DZ_{m}=1$, then $\mathcal{O}_{Z_{m}}(D)$ is not generated.

  3. $(3)$

If $\mathcal{O}_{X}(D)$ has no fixed component in $B_{m}$ and is not
generated, then there exists $0\leq t\leq m$ and a nonsingular point $Q$ of
the scheme $C^{\prime}_{t}$ such that $DC^{\prime}_{t}=1$ and $Q$ is a base
point of $\mathcal{O}_{X}(D)$.

In particular, if $\mathcal{O}_{X}(-C_{t})$ with $t\in\mathcal{A}_{f}$ is not
generated, then $Z_{t}^{2}=-1$.

###### Proof.

The claim (1) follows from (1) and (2) of [Ok1, 3.3]. The claim (2) follows
from [Re, 4.23 Lemma]. The assumption of (3) implies that $\mathcal{O}_{X}(D)$
has only finitely many base points in $B_{m}$. Hence (3) follows from [Ok1,
3.1]. ∎

######  Theorem 3.15.

Let $Z>0$ be an anti-nef cycle on $X$. Assume that $\mathcal{O}_{X}(-Z)$ is
generated and $\chi(Z)=0$. Then
$Z\in{\left\\{C_{t}\,\left|\,t\in\mathcal{A}_{f}\right.\right\\}}$.

###### Proof.

Note that $\ell_{A}(A/H^{0}(\mathcal{O}_{X}(-C_{m})))=p_{g}(A)$ and that
$\ell_{A}(A/I_{Z})=p_{g}(A)-q(I_{Z})\leq p_{g}(A)$ (cf. the proof of Theorem
2.1).

First we consider the case that $Z\geq C_{m}$. If $Z=C_{m}$, we have nothing
to say. Hence assume that $Z=C_{m}+Y$ with $Y>0$. Then since
$\ell_{A}(A/I_{Z})\geq\ell_{A}(A/H^{0}(\mathcal{O}_{X}(-C_{m})))=p_{g}(A)$, we
have $\ell_{A}(A/I_{Z})=p_{g}(A)$ and $q(I_{Z})=0$. Moreover, since
$\operatorname{H}^{0}(X,\mathcal{O}_{X}(-C_{m}))=\operatorname{H}^{0}(X,\mathcal{O}_{X}(-Z))$,
$Y$ is the fixed component of $\mathcal{O}_{X}(-C_{m})$. Also, we have

$0=\chi(Z)=\chi(C_{m})+\chi(Y)-YC_{m}.$

Since $\chi(C_{m})=0$, we have $\chi(Y)=YC_{m}\leq 0$. But from Lemma 3.14
(1), we must have $\chi(Y)>0$. This contradiction shows that $Z=C_{m}$.

Next, we will show that $Z<C_{m}$ if $Z\neq C_{m}$.

Assume that $Z\not\leq C_{m}$ and let $G=\gcd(Z,C_{m})$, namely, $G$ is the
maximal cycle with the property that $G\leq Z$ and $G\leq C_{m}$. Since both
$Z,C_{m}$ are anti-nef, so is $G$. Put $Z=G+F$. Then $G=C_{t}$ for some $0\leq
t<m$ by Proposition 3.8, and $\chi(F)>0$ by Proposition 3.3 (1) since
$F\not\geq Z_{m}$. Hence

$\chi(Z)=\chi(F+G)=\chi(F)+\chi(G)-FG=\chi(F)-FG>0.$

However, this contradicts our assumption that $\chi(Z)=0$. Hence we have
$Z\leq C_{m}$ and our conclusion follows from Lemma 3.8. ∎

######  Corollary 3.16.

Assume that $A$ is an elliptic Gorenstein singularity.

  1. $(1)$

If $t\in\mathcal{A}_{f}$ and $-Z_{t}^{2}\geq 2$, then
$\overline{G}(I_{C_{t}})$ is Gorenstein.

  2. $(2)$

Let $\zeta$ be the number of elliptic ideals $I\subset A$ such that
$\overline{G}(I)$ is Gorenstein. Then $\zeta\leq p_{g}(A)$, and the equality
holds if and only if $-Z_{m}^{2}\geq 2$.

###### Proof.

By Theorem 2.1, Lemma 3.4, and Theorem 3.15, any elliptic ideal $I$ such that
$\overline{G}(I)$ is Gorenstein is represented by some $C_{i}$ on the minimal
resolution. For (1), see Lemma 3.14 (3). For (2), note also that
$\\#\mathcal{A}_{f}=p_{g}(A)$ by Remark 3.11 and that $-Z_{t}^{2}\geq 2$ for
all $t\in\mathcal{A}_{f}$ if and only if $-Z_{m}^{2}\geq 2$ by Proposition 3.7
(2). ∎

######  Definition 3.17.

A numerically Gorenstein normal two-dimensional elliptic singularity is called
a maximally elliptic singularity if its geometric genus equals to the length
of the elliptic sequence ($m+1$, in our case). It is known that every
maximally elliptic singularity is Gorenstein ([Yau]).

######  Example 3.18.

Let $A$ be as in Example 3.6. Then $A$ is maximally elliptic and
$\zeta=p_{g}(A)=m+1$.

We have to classify the cycles $C_{t}$ ($t\in\mathcal{A}_{f}$) such that
$\mathcal{O}_{X}(-C_{t})$ is generated. Lemma 3.14 tells us that if
$\mathcal{O}_{X}(-C_{t})$ is not generated, then $Z_{t}^{2}=-1$; however the
converse does not hold in general (cf. Example 3.28). To analyze the
generation of the invertible sheaves $\mathcal{O}_{X}(-C_{t})$, we have to
know at first the combinatorial structure of the cycles $Z_{t}$ with
$Z_{t}^{2}=-1$, and then some results in [Ok1] which depends on the structure
of the singularity.

######  Lemma 3.19 ([Ok1, 3.5]).

Assume that $0\leq j\leq m$ and $Z_{j}^{2}=-1$. For $i\geq j$, let $F_{i}$
denote the unique irreducible component of $Z_{i}$ such that
$Z_{i}^{2}=Z_{i}F_{i}=-1$ (see Proposition 3.7 (2)).

  1. $(1)$

Assume that $j\leq i<m$. Then $Z_{i}-Z_{m}=F_{m-1}+\cdots+F_{i}$ and this is a
chain of $(-2)$-curves (see Figure 2). Therefore, $C_{m}F_{i}=-K_{X}F_{i}=0$.

  2. $(2)$

If $j>0$, there exists an unique irreducible component $\widetilde{F}\leq
E-B_{j}$ such that $\widetilde{F}B_{j}>0$. Then $\widetilde{F}^{2}=-2$ if and
only if $Z_{j-1}^{2}=-1$.

  3. $(3)$

Assume that $Z_{j-1}^{2}=-1$. Let $D>0$ be a cycle such that $DC=0$ for every
irreducible component $C\subset B_{j}$ and that the coefficient of $F_{j-1}$
in $D$ is $n$. Then $\operatorname{Supp}(D-nZ_{j-1})$ has no component of
$B_{j-1}$.

$B_{m}$$-2$$\cdots$$-2$$F_{m-1}$$F_{j}$$E-B_{j}$ Figure 2. The resolution
graph with $Z_{j}^{2}=-1$

######  Lemma 3.20 ([Ok1, 3.6]).

For $t\in\mathcal{A}_{f}\setminus\\{m\\}$ with $Z_{t}^{2}=-1$, the following
conditions are equivalent$:$

  1. $(1)$

$\mathcal{O}_{X}(-C_{t})$ is generated;

  2. $(2)$

$H^{0}(\mathcal{O}_{X}(-C_{t}-Z_{t}))\subsetneqq
H^{0}(\mathcal{O}_{X}(-C_{t+1}))$;

  3. $(3)$

there exists a function $g\in H^{0}(\mathcal{O}_{X}(-C_{t+1}))$ which
satisfies $(g)_{E}=C_{t+1}$ outside $B_{t+2}$, where we put
$B_{m+1}=\emptyset$.

Actually, the proofs of some results in [Ok1] rely on the complex analytic
setting; however, most of them hold in our algebraic situation. For example,
Lemma 3.22 below enable us to prove Proposition 5.12 and Theorem 5.13 of [Ok1]
in the characteristic zero case, and therefore, we obtain the following.

######  Proposition 3.21 (cf. [Ok1, 5.12, 5.13]111The invariant $\alpha$ in
[Ok1] is zero since $A$ is Gorenstein.).

Assume that $\operatorname{char}(k)=0$. Let $\gamma=\beta+1$ (see Definition
3.10). Then we have the following.

  1. $(1)$

For $0\leq t<u\leq m$, the order of $\mathcal{O}_{C^{\prime}_{u}}(Z_{0})$ in
the Picard group $\operatorname{Pic}C^{\prime}_{u}$ is $\gamma$ and
$\mathcal{O}_{C^{\prime}_{u}}(Z_{0})\cong\mathcal{O}_{C^{\prime}_{u}}(Z_{t})$.

  2. $(2)$

$\gamma\mid m$ and $p_{g}(X,o)=m/\gamma+1$.

  3. $(3)$

$\mathcal{A}_{f}={\left\\{\beta+i\gamma\,\left|\,0\leq
i<m/\gamma\right.\right\\}}\cup\\{m\\}$.

######  Lemma 3.22 (cf. [Ok1, 3.7]).

Assume that $\operatorname{char}(k)=0$. Let $D$ be a cycle such that $D\geq
Z_{m}$. We denote by $\operatorname{Pic}^{0}D$ the subgroup of
$\operatorname{Pic}D$ which consists of numerically trivial elements. Let
$\phi:\operatorname{Pic}^{0}D\to\operatorname{Pic}^{0}Z_{m}$ be the
homomorphism obtained by the restriction. Then we have the following.

  1. $(1)$

$\operatorname{Ker}\phi$ is torsion free.

  2. $(2)$

If $\sigma\in\operatorname{Pic}^{0}D$ is a torsion element, then
$\operatorname{ord}(\sigma)=\operatorname{ord}(\phi(\sigma))$.

###### Proof.

(1) By [Ar, Lemma 1.4], the kernel of the restriction map
$\phi^{\prime}:\operatorname{Pic}D\to\operatorname{Pic}D_{red}$ is torsion
free. If $H^{1}(\mathcal{O}_{D_{red}})=0$, then
$\operatorname{Pic}^{0}D_{red}$ is trivial (cf. [Li, 11.1]) and
$\operatorname{Pic}^{0}D$ is torsion free. Assume
$H^{1}(\mathcal{O}_{D_{red}})\neq 0$. Since $\chi(D_{red})=0$ and $Z_{m}$ is
the minimally elliptic cycle, we have $Z_{m}\leq D_{red}$ by Proposition 3.3.
Let $L$ be an invertible sheaf on $D_{red}$ corresponding to an element of
$\operatorname{Pic}^{0}D_{red}$. We show that $L$ is trivial if and only if so
is $L|_{Z_{m}}$. Let $C=D_{red}-Z_{m}$. We may assume $C>0$. Then we have the
exact sequence

$0\to L\otimes\mathcal{O}_{C}(-Z_{m})\to L\to L|_{Z_{m}}\to 0.$

Since $C^{\prime}Z_{m}=1$ for every connected component $C^{\prime}\leq C$ by
Proposition 3.3, we have $H^{i}(L\otimes\mathcal{O}_{C}(-Z_{m}))=0$ for
$i=0,1$ (cf. [Rh, 1.14]), and $H^{0}(L)=H^{0}(L|_{Z_{m}})$. Since an
invertible sheaf on a connected reduced cycle is trivial if it has non-zero
global sections (cf. [Re, 3.11]), $L$ is trivial if and only if so is
$L|_{Z_{m}}$. Hence
$\operatorname{Ker}\phi=\operatorname{Ker}(\operatorname{Pic}^{0}D\to\operatorname{Pic}^{0}D_{red})\subset\operatorname{Ker}\phi^{\prime}$.

(2) If $n=\operatorname{ord}(\phi(\sigma))$, then $n\sigma$ is a torsion
element in $\operatorname{Ker}\phi$. Hence $n\sigma$ is trivial, and
$\operatorname{ord}(\sigma)=n$. ∎

From Remark 3.11 and Proposition 3.21, we have the following.

######  Proposition 3.23.

For a Gorenstein elliptic singularity $A$ with $\operatorname{char}(k)=0$, the
following are equivalent.

  1. $(1)$

$A$ is maximally elliptic.

  2. $(2)$

$\beta=0$.

  3. $(3)$

$\gamma=1$.

  4. $(4)$

$\mathcal{A}_{f}=\\{0,1,\dots,m\\}$.

######  Proposition 3.24.

Assume that $A$ is a Gorenstein elliptic singularity.

  1. $(1)$

Assume that $\operatorname{char}(k)=0$. If there exists
$t\in\mathcal{A}_{f}\setminus\\{m\\}$ such that $\mathcal{O}_{X}(-C_{t})$ is
not generated, then $A$ is maximally elliptic and $Z_{t}^{2}=-1$.

  2. $(2)$

If $A$ is maximally elliptic, then $\mathcal{O}_{X}(-C_{t})$ is not generated
for every $t\in\mathcal{A}_{f}$ with $Z_{t}^{2}=-1$.

###### Proof.

(1) Assume that $t\in\mathcal{A}_{f}\setminus\\{m\\}$ and
$\mathcal{O}_{X}(-C_{t})$ is not generated. We have $Z_{i}^{2}=-1$ for $i\geq
t$ by Proposition 3.7 (2) and Lemma 3.14 (3), and $C_{t}+Z_{t}=C_{t+1}+F_{t}$,
where $F_{t}$ is as in Lemma 3.19. If $\mathcal{O}_{X}(-C_{m})$ has no fixed
component, then there exists $g\in H^{0}(\mathcal{O}_{X}(-C_{m}))$ such that
$(g)_{E}=C_{m}$, and thus the condition (3) of Lemma 3.20 is satisfied;
however, it contradicts the assumption that $\mathcal{O}_{X}(-C_{t})$ is not
generated. Therefore, $\mathcal{O}_{X}(-C_{m})$ has a fixed part $F$ which
contains $F_{t}$ because $H^{0}(\mathcal{O}_{X}(-C_{m}))\subset
H^{0}(\mathcal{O}_{X}(-C_{t+1}))=H^{0}(\mathcal{O}_{X}(-C_{t+1}-F_{t}))$ by
Lemma 3.20, and $F$ has no components of $B_{m}$ by Lemma 3.14 (1). Since
$\mathcal{O}_{X}(-C_{m}-F)$ has no fixed component, $C_{m}+F$ is anti-nef. If
$F$ does not intersect $Z_{m}$, then we can take $F_{i}$ of Lemma 3.19 with
$i<m$ so that $F_{i}\not\leq F$ and $F_{i}$ intersects $F$; however, it
induces a contradiction that $(C_{m}+F)F_{i}=FF_{i}>0$. Hence $F$ intersects
$Z_{m}$ and $(C_{m}+F)Z_{m}=Z_{m}^{2}+FZ_{m}=-1+FZ_{m}=0$, because $FZ_{m}>0$.
Since
$\mathcal{O}_{Z_{m}}(-C_{m}+Z_{m})\cong\omega_{Z_{m}}\cong\mathcal{O}_{Z_{m}}$
and $\mathcal{O}_{Z_{m}}(-Z_{m-1})\cong\mathcal{O}_{Z_{m}}(-Z_{m}-F)$ (cf.
Lemma 3.19), we have

$\mathcal{O}_{Z_{m}}(-Z_{m-1})\cong\mathcal{O}_{Z_{m}}(-Z_{m}-F-C_{m}+Z_{m})\cong\mathcal{O}_{Z_{m}}(-C_{m}-F)\cong\mathcal{O}_{Z_{m}}.$

It follows from Proposition 3.21 (1) that $\gamma=1$. Therefore, $A$ is
maximally elliptic by Proposition 3.23.

(2) Assume that $A$ is maximally elliptic. Since
$\mathcal{A}_{f}=\\{0,1,\dots,m\\}$ by Proposition 3.23,
$\mathcal{O}_{C^{\prime}_{j}}(-Z_{i})\cong\mathcal{O}_{C^{\prime}_{j}}$ for
$0\leq i<j\leq m$ by Lemma 3.9. It follows from Lemma 3.14 (1) that
$\mathcal{O}_{X}(-C_{m})$ is not generated if $Z_{m}^{2}=-1$. Assume that
$Z_{m-1}^{2}=-1$. Then $Z_{m}^{2}=-1$ by Proposition 3.7 (2). Since
$Z_{m}+F_{m-1}=Z_{m-1}$, we have
$\mathcal{O}_{Z_{m}}(-Z_{m}-F_{m-1})\cong\mathcal{O}_{Z_{m}}(-Z_{m-1})\cong\mathcal{O}_{Z_{m}}$;
this implies that the point $B_{m}\cap F_{m-1}$ is the base point of
$\mathcal{O}_{Z_{m}}(-C_{m})$. If there exists $g\in
H^{0}(\mathcal{O}_{X}(-C_{m}))$ such that $(g)_{E}=C_{m}+Y$, where $Y\geq 0$
and $Y$ has no component of $Z_{m-1}$, then the divisor
$H:=\operatorname{div}_{X}(g)-C_{m}-Y$ satisfies $HF_{m-1}=-YF_{m-1}\leq 0$;
it contradicts that the base point of $\mathcal{O}_{Z_{m}}(-C_{m})$ is
$B_{m}\cap F_{m-1}$. Thus we see that $F_{m-1}$ is a fixed component of
$\mathcal{O}_{X}(-C_{m})$. By Lemma 3.20, $\mathcal{O}_{X}(-C_{m-1})$ is not
generated, because (3) of the lemma is not satisfied for $t=m-1$. Since
$\mathcal{O}_{C^{\prime}_{m-1}}(-C_{m-1}-F_{m-2})=\mathcal{O}_{C^{\prime}_{m-1}}(-C_{m-2}-Z_{m-2})\cong\mathcal{O}_{C^{\prime}_{m-1}}$,
the similar argument as above implies that $F_{m-2}$ is a fixed component of
$\mathcal{O}_{X}(-C_{m-1})$. Inductively, we obtain that
$\mathcal{O}_{X}(-C_{t})$ is not generated for $0\leq t<m$ with
$Z_{t}^{2}=-1$. ∎

######  Theorem 3.25.

Assume that $A$ is an elliptic Gorenstein singularity. Let $I$ be an elliptic
ideal, namely, $\operatorname{\bar{r}}(I)=2$.

  1. $(1)$

Assume that $A$ is maximally elliptic. Then $\overline{G}(I)$ is Gorenstein if
and only if $I$ is represented by $C_{t}$ with $t\in\mathcal{A}_{f}$ such that
$-Z_{t}^{2}\geq 2$.

  2. $(2)$

Assume that $\operatorname{char}(k)=0$ and $A$ is not maximally elliptic. Then
$\overline{G}(I)$ is Gorenstein if and only if $I$ is represented by $C_{t}$
with $t\in\mathcal{A}_{f}$ such that $-Z_{t}^{2}\geq 2$ or that $t<m$.

###### Proof.

As mentioned in the proof of Corollary 3.16, any elliptic ideal $I$ such that
$\overline{G}(I)$ is Gorenstein is represented by some $C_{i}$ on the minimal
resolution. Therefore, the classification of such ideals correspond to that of
cycles $C_{i}$ such that $\mathcal{O}_{X}(-C_{i})$ is generated; it is done by
combination of Lemma 3.14 and Proposition 3.24. ∎

######  Corollary 3.26.

Assume that $A$ is a maximally elliptic singularity with $Z_{0}^{2}=-1$. If
$I\subset A$ is an $\mathfrak{m}$-primary integrally closed ideal and
$\overline{G}(I)$ is Gorenstein, then $\operatorname{\bar{r}}(I)=1$, namely,
$I$ is a $p_{g}$-ideal.

###### Proof.

We have $\operatorname{\bar{r}}(I)\leq 2$ by [Ok2, 3.3]. By Proposition 3.7,
$Z_{t}^{2}=-1$ for $0\leq t\leq m$. Therefore, the assertion immediately
follows from Theorem 3.25. ∎

######  Remark 3.27 (Cf. [Ok2, 4.10]).

In general, for a two-dimensional normal Gorenstein singularity
$(A,\mathfrak{m})$ with $p_{g}(A)>0$, $\mathfrak{m}$ is a $p_{g}$-ideal if and
only if $A$ is a maximally elliptic Gorenstein singularity with
$Z_{0}^{2}=-1$.

The following example shows that the elliptic ideals $I\subset A$ such that
$\overline{G}(I)$ is Gorenstein cannot be characterized by their resolution
graph.

######  Example 3.28 (cf. [Ko, 4.7]).

Let $n$ be a positive integer and consider the polynomials

$f=x^{2}+z(z^{4n+2}+y^{4})\quad\text{and}\quad g=x^{2}+y^{3}+z^{6(2n+1)},$

which are weighted homogeneous polynomials with respect to the weights
$(4n+3,2n+1,2)$ and $(3(2n+1),2(2n+1),1)$, respectively. Let
$R=k[x,y,z]_{(x,y,z)}$, $A=R/(f)$ and $B=R/(g)$. These singularities have the
same resolution graph as in Figure 3, where $g(E_{2n})=1$, $g(E_{i})=0$ for
$0\leq i\leq 2n-1$.

$\cdots$$-1$$-2$$-2$$-2$$E_{2n}$$E_{2n-1}$$E_{1}$$E_{0}$ Figure 3.

It is easy to check that these singularities are Gorenstein elliptic
singularities such that $Z_{i}=\sum_{j=i}^{2n}E_{j}$, $Z_{i}^{2}=-1$, and
$C_{i}E_{j}=-\delta_{ij}$, where $\delta_{ij}$ denotes the Kronecker delta,
for $0\leq i\leq 2n$. Using Theorem 2.7, we easily see that $p_{g}(A)=n+1$ and
$p_{g}(B)=2n+1$.

By Corollary 3.26, $B$ is a maximally elliptic singularity, and hence there is
no elliptic ideal $I$ such that $\overline{G}(I)$ is Gorenstein.

Assume that $\operatorname{char}(k)\neq 2$. Let $\cal G$ be the set of
elliptic ideals $I\subset A$ such that $\overline{G}(I)$ is Gorenstein. We
will show that $\cal G={\left\\{(x,y,z^{j})\,\left|\,1\leq j\leq
n\right.\right\\}}$. First, let us identify the divisors of $x,y,z$ on $X$.
For a divisor $D$ on the resolution space $X$, we denote by $c(D)$ the
coefficient of $E_{2n}$ in $D$. For example, $c(C_{i})=i+1$. Let $H\subset X$
be the proper transform of the curve $\operatorname{Spec}A/(x,z)$. Since
$A/(z)=k[x,y]_{(x,y)}/(x^{2})$ and $c(\operatorname{div}_{X}(z))=\deg(z)=2$,
we have $\operatorname{div}_{X}(z)=2Z_{0}+2H$; we also have $HE=HE_{0}=1$.
Recall that $E_{2n}$ is isomorphic to the curve
$V(f)\subset\mathbb{P}(4n+3,2n+1,2)$. The cone over $V(f)$ has an isolated
singularity at the vertex and the localization of its local ring coincides
with $A$. We take points $p_{0},p_{1},p_{2},p_{3}\in V(f)$ such that

$p_{0}=(0,1,0),\ \ p_{1}=(0,(\sqrt{-1})^{n},\sqrt{-1}),\ \
p_{2}=(0,(\sqrt{-1})^{n+1},\sqrt{-1}),\ \ p_{3}=(1,0,-1).$

Then these points are distinct and

$\\{p_{0},p_{1},p_{2}\\}=V(f,x),\quad\\{p_{3}\\}=V(f,y),\quad\\{p_{0}\\}=V(f,z)\subset\mathbb{P}(4n+3,2n+1,2).$

Let $H_{i}\subset X$ be the proper transform of the curve in
$\operatorname{Spec}A$ corresponding to the cone over the point $p_{i}$.
Clearly, $H=H_{0}$, $H_{i}\cap E_{2n}\neq\emptyset$ and $H_{i}\cap
E_{2n-1}=\emptyset$ for $i=1,2,3$. Since $\deg(y)=2n+1$ and
$H_{3}\equiv-(H_{3}E_{2n})C_{2n}$, we have $H_{3}E_{2n}=1$ and
$\operatorname{div}_{X}(y)=C_{2n}+H_{3}$. Similarly, we have
$\operatorname{div}_{X}(x)=2C_{2n}+Z_{0}+H_{0}+H_{1}+H_{2}$. Thus, for $1\leq
j\leq n$, $\gcd((y)_{E},(z^{j})_{E})=\gcd(C_{2n},2jZ_{0})=C_{2j-1}$. Moreover,
$y$ (resp. $z^{j}$) generates $\mathcal{O}_{X}(-C_{2j-1})$ at every point of
$\sum_{i=0}^{2j-1}E_{i}$ (resp. $\operatorname{Supp}(Z_{2j})\setminus
E_{2j-1}$). In particular, $2j-1\in\mathcal{A}_{f}$. Let $I_{j}=I_{C_{2j-1}}$.
Since $p_{g}(A)=n+1$, it follows from Remark 3.11 that
$\mathcal{A}_{f}={\left\\{2j-1\,\left|\,1\leq j\leq
n\right.\right\\}}\cup\\{2n\\}$ and $\ell_{A}\left(A/I_{j}\right)=j$. As noted
in the proof of Theorem 3.25, the ideals in $\cal G$ correspond to the cycles
$C_{i}$ such that $\mathcal{O}_{X}(-C_{i})$ is generated. Hence $\cal
G={\left\\{I_{j}\,\left|\,1\leq j\leq n\right.\right\\}}$. Since
$(x,y,z^{j})\subset I_{j}$ and
$\dim_{k}k[x,y,z]/(f,x,y,z^{j})=j=\ell_{A}\left(A/I_{j}\right)$, we have
$I_{j}=(x,y,z^{j})$.

By Corollary 3.26, Corollary 3.16 and the argument in Example 3.28, we have
the following.

######  Example 3.29.

Let $R=k[x,y,z]_{(x,y,z)}$ and let $m\geq 0$ be an integer. In each of the
following cases, $A$ is a maximally elliptic singularity with $p_{g}(A)=m+1$,
$\mathcal{A}_{f}=\\{0,1,\dots,m\\}$, the minimally elliptic cycle $Z_{m}$ is
an elliptic curve, and $Z_{0}=E$. Let $\cal G$ be the set of elliptic ideals
$I\subset A$ such that $\overline{G}(I)$ is Gorenstein.

  1. $(1)$

If we put $A=R/(x^{2}+y^{3}+z^{6(m+1)})$, then $-Z_{m}^{2}=-1$ and $\cal
G=\emptyset$. The exceptional set $E$ consists of $Z_{m}$ and a chain of
$(-2)$-curves $\sum_{j=0}^{m-1}E_{j}$ such that $Z_{m}E_{m-1}=E_{j}E_{j-1}=1$
($1\leq j\leq m-1$) (cf. Figure 3).

  2. $(2)$

If we put $A=R/(x^{2}+y^{4}+z^{4(m+1)})$, then $-Z_{m}^{2}=-2$ and

$\cal G={\left\\{(x,y,z^{i})\,\left|\,1\leq i\leq m+1\right.\right\\}}.$

The exceptional set $E$ is described in Example 3.6 and
$(x,y,z^{i})=I_{C_{i-1}}$.

  3. $(3)$

If we put $A=R/(x^{3}+y^{3}+z^{3(m+1)})$, then $-Z_{m}^{2}=-3$ and

$\cal G={\left\\{(x,y,z^{i})\,\left|\,1\leq i\leq m+1\right.\right\\}}.$

The exceptional set $E$ consists of an elliptic curve $Z_{m}$ and three chains
of $(-2)$-curves $\sum_{j=0}^{m-1}E_{j}^{(s)}$ ($s=1,2,3$) and
$(x,y,z^{i})=I_{C_{i-1}}$ as above.

######  Question 3.30.

Let $A$ be an excellent normal Gorenstein local domain of $\dim A=2$. Then is
the set of elliptic ideals $I=I_{Z}$ for which $\overline{G}(I)$ is Gorenstein
a finite set?

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