# Clustering Pseudo Language Family in Multilingual Translation Models with
Fisher Information Matrix

Xinyu Ma Xuebo Liu Min Zhang
Institute of Computing and Intelligence, Harbin Institute of Technology,
Shenzhen, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>Co-first and
Corresponding Author

###### Abstract

In multilingual translation research, the comprehension and utilization of
language families are of paramount importance. Nevertheless, clustering
languages based solely on their ancestral families can yield suboptimal
results due to variations in the datasets employed during the model’s training
phase. To mitigate this challenge, we introduce an innovative method that
leverages the fisher information matrix (FIM) to cluster language families,
anchored on the multilingual translation model’s characteristics. We
hypothesize that language pairs with similar effects on model parameters
exhibit a considerable degree of linguistic congruence and should thus be
grouped cohesively. This concept has led us to define pseudo language
families. We provide an in-depth discussion regarding the inception and
application of these pseudo language families. Empirical evaluations reveal
that employing these pseudo language families enhances performance over
conventional language families in adapting a multilingual translation model to
unfamiliar language pairs. The proposed methodology may also be extended to
scenarios requiring language similarity measurements. The source code and
associated scripts can be accessed at https://github.com/ecoli-
hit/PseudoFamily.

##  1 Introduction

Multilingual neural machine translation (MNMT) aims to construct a single
model to translate multiple languages and has proven its effectiveness
(Aharoni et al., 2019). The application of such models for low-resource
languages has revealed that leveraging supplementary languages during the
fine-tuning phase can yield results that surpass traditional methods (Lakew et
al., 2018; Gu et al., 2018). Nevertheless, the practical challenge lies in the
development of the optimal strategy to identify the most beneficial auxiliary
languages that can bolster the translation of low-resource language pairs.
Current academic investigations in this domain generally fall into two
distinct categories: Firstly, the integration of diverse prior knowledge,
which incorporates the utilization of various forms of existing knowledge and
language family classifications, as evidenced in research such as Bakker et
al. (2009); Chen and Gerdes (2017). Secondly, the exploration and computation
of language similarity center on examining language representations and their
comparative resemblances (Tan et al., 2019; Oncevay et al., 2020; Maurya and
Desarkar, 2022).

Conventional methodologies predominantly rely on supplemental resources or the
alteration of model architecture to categorize language families. However,
those approaches need more adaptability, especially when dealing with recently
developed pre-trained large MNMT models, e.g., m2m100 (Fan et al., 2021),
where the original training data and model architecture are inaccessible. When
it comes to a new scenario this limitation significantly hampers the
effectiveness of existing methodologies for language family clustering.

To address this gap, we introduce a novel approach to pseudo language family
clustering, which inherently depends on the MNMT model itself. Our proposed
methodology hypothesizes that language pairs exhibiting similarities in their
impact on model parameters are likely to possess a high level of congruence
and should, consequently, be grouped together. We employ the FIM to quantify
such similarities between language pairs. We demonstrate the efficacy of our
approach by enhancing low-resource translation with pseudo language family
clustering, yielding superior results compared to traditional language family
clustering.

Our main contributions are as follows:

  * •

We present an innovative methodology for clustering language families uniquely
designed to function without the need for data access or architectural
modifications.

  * •

We unveil three efficient methods anchored in the FIM to comprehensively
compute similarities amongst language pairs.

  * •

Empirical evaluations indicate that our clustered pseudo language families
exhibit superior performance compared to conventional language families.
Furthermore, our approach exhibits versatility, effortlessly adapting to
scenarios that necessitate language clustering.

##  2 Language Clustering with FIM

The accurate representation of language pairs is foundational for computing
their pairwise similarities and ensuring efficacious clustering. As such,
devising profound and robust representations for these pairs becomes
imperative to attain excellence in this arena. Drawing inspiration from
contemporary strides in parameter-efficient fine-tuning where innovative
strategies have emerged to harness model parameters for representation (Ben
Zaken et al., 2022; Gong et al., 2022; Zhang et al., 2023) we propose the
deployment of the FIM. Our approach seeks to capitalize on the task-specific
parameters embedded within the model.

###  2.1 Preliminary

#### Dataset

As an initial step, we curated a common language pool comprising translations
of 17 languages to English (en), sourced from the TED Corpus (Qi et al.,
2018). These language pairs collectively serve as the complete set for
selection and function as the auxiliary language for low-resource languages.
These languages span seven distinct families: Balto-Slavic, Austronesian,
Indo-Iranian, Turkic, Japonic, Koreanic, and Germanic. The languages we
employed are depicted in Figure 1.

#### Model

For empirical validation of our proposed methodologies, we elected to employ
the m2m100_418M model (Fan et al., 2021). It is underpinned by a vast, many-
to-many dataset, encompassing an impressive compilation of 100 languages. The
innovation of this model lies in its unique data mining strategy, it
capitalizes on linguistic similarities, thereby circumventing the need for
exhaustive mining across all conceivable directions. Given the extensive
breadth and innate multilingual capabilities of the model, it offers an apt
platform for our investigative endeavors, furnishing a robust foundation for
the appraisal and affirmation of our methodological propositions. For
consistency, we adhere to the model configuration and training settings
delineated in Section 3.2.

Figure 1: The used language families in our experiments.

#### Fisher Information

The concept of fisher information occupies a pivotal position in information
theory, serving as an instrumental measure in discerning the importance and
prospective value of tuning specific parameters. Essentially, it quantifies
the variability of the first derivative of the likelihood function’s
logarithm. Through gauging the magnitude of this metric, one can deduce the
necessity of fine-tuning particular parameters for subsequent tasks (Xu et
al., 2021).

Formally, the FIM has been defined and employed in prior works (Theis et al.,
2018; Thompson et al., 2019; Zhong et al., 2022) as:

$\begin{split}\rm
F_{\theta}=\mathbb{E}[&(\nabla_{\theta}\log{P(Y|X;\theta)})\\\
&(\nabla_{\theta}\log{P(Y|X;\theta)})^{T}]\end{split}$ (1)

Where $X$ and $Y$ denote the input and output respectively, and $\theta$
denotes the parameters of the model. For the $i$-th parameter in $\theta$ we
empirically estimate its fisher information using the diagonal elements of the
FIM. The formula derives as follows:

$\rm F_{i}=\mathbb{E}{[(\nabla_{i}\log{P(Y|X;\theta)})^{2}]}$ (2)

###  2.2 Calculating Similarity for Language Pairs

While leveraging a diagonal matrix aids in estimating FIM, arriving at precise
probability estimations remains a non-trivial task. In light of this, we
resort to the subsequent equation to approximate FIM:

$\rm F_{i}=\frac{1}{|\rm
D|}\sum_{j}(\nabla_{i}\log{P(Y_{j}|X_{j};\theta)})^{2}$ (3)

Here, $\rm D$ symbolizes the complete dataset, with $|\rm D|$ representing the
data amount, and the entire dataset is divided into $\rm j$ mini-batches.

We input parallel corpora corresponding to language pairs into the model and
estimate the FIM using Equation 3 for just one epoch. We accumulate the FIM
using the formula for each mini-batch during the model training process
without backward propagation. After the completion of one epoch, we calculate
the average FIM obtained from each mini-batch as the final estimation.
Appendix A.1 provides the pseudo-code we use to estimate FIM.

Figure 2: The parameters distribution of FIM for the training languages. We
divide the model into five parts: encoder attention layer ($E_{a}$), encoder
fully connected layer ($E_{f}$), decoder self-attention ($D_{a}$), decoder
cross attention layer ($D_{c}$) and decoder fully connected layer ($D_{f}$).

#### Significance of FFN in FIM Estimation

The model was then divided into five sections, and the distribution of the
highest 40% parameters is depicted in Figure 2. It is pivotal to note that the
feed-forward networks (FFN) contain over 60% of the selected parameters. A
comparison was made with a related study (Meng et al., 2022) regarding the
fine-tuning performance on the MNLI-m
task.111https://paperswithcode.com/task/mnli-m In this study, the researchers
solely fine-tuned a selected type of weight matrix from each transformer
layer, and the best results were achieved when only the FFN was fine-tuned.
From these observations, we conclude that the optimal strategy for this task
involves utilizing the FIM derived from the FFN.

#### Variants for Similarity Calculation

To effectively leverage the results derived from the FIM and judiciously
select auxiliary languages for multilingual training of a specific target
language pair, we devised three distinct strategies for calculating the
similarity or distance between language pairs:

  * 1.

Mean Square Error (MSE): Utilizing MSE offers an intuitive depiction of the
distributional disparities among different language pairs. It is formally
represented as:

$S_{(\rm t,\rm a)}=\frac{\sum(\rm F_{t}-\rm F_{a})^{2}}{|\rm F_{t}|}$ (4)

Here, $\rm t$ denotes the target language direction, and $\rm a$ represents
the auxiliary language pair. Furthermore, $\rm F$ denotes the FIM, $\rm F_{a}$
corresponds to the matrix associated with the auxiliary language, $\rm F_{t}$
corresponds to the target language, and $|\cdot|$ symbolizes the matrix’s
size.

  * 2.

KL Divergence (KL): By leveraging the FIM, we compute the KL divergence from
other languages in relation to the target language, thereby capturing a more
nuanced measure of the distances between language pairs.

$S_{(\rm t,\rm a)}=|\sum\rm F_{a}*\log(\frac{\rm F_{a}}{\rm F_{t}})|$ (5)

The nomenclature used here mirrors that of the MSE, with $|\cdot|$ indicating
absolute value.

  * 3.

Overlap Similarity (Overlap): In further leveraging the FIM, we construct a
fisher information mask. Here, the uppermost K parameters are attributed a
value of 1, whilst the others are designated a value of 0. The similarity
amongst each language pair is ascertained using:

$S_{(\rm t,\rm a)}=\frac{\rm Overlapping(\rm M_{t},\rm M_{a})}{\rm
Activate(\rm M_{t})}$ (6)

Here, $\rm M$ symbolizes the fisher information mask, with $\rm Overlapping$
and $\rm Activate$ respectively quantifying the number of parameters that are
simultaneously activated and those activated in the target direction.

Figure 3: The similarity computed with Overlap method, with darker colors
representing the higher similarity between language pairs.

###  2.3 Clustering Pseudo Language Family

#### Selection of Pseudo Language Families

For the selection of auxiliary languages, we designed a pragmatic algorithm.
Beginning with the sorting of similarities between language pairs, we set an
initial search radius. Within this pre-defined boundary, the language pairs in
closest proximity are integrated into the auxiliary language roster. The
radius is then adjusted, taking cues from the similarity metric of the latest
added language pair. This process is iterated until the list ceases to expand
with new language pairs. As a result, we identify a cluster that we term the
pseudo language family.

This algorithm progressively broadens its selection boundary but at a
decelerating rate, aiming to assimilate languages beneficial for the auxiliary
set while sidestepping those exhibiting pronounced dissimilarities. For a
comprehensive insight into the algorithm, refer to Appendix A.2.

Model | fa | hi | bn | id | ms
---|---|---|---|---|---
KL | id sv de | bg de | sv uk id | bg sv | id bg sv
MSE | tr ko id | fa be tr | hi be fa | bg sv de | id bg ko sk
Overlap | tr ko id | bn fa | hi fa tr | bg ms sv | id bg sv de uk

Table 1: The pseudo language family formed by the similarities computed using
our methods.

#### Analysis

Figure 4: The phylogenetic tree of Slavic languages.

We examined the efficacy of FIM in gauging linguistic similarity by
juxtaposing it with the Slavic language family structure in Figure 4. This
comparison was made using the similarity matrix derived from the Overlap
method, as illustrated in Figure 3. The insights obtained from FIM are
consistent with the findings of phylogenetic tree analysis. For instance,
Serbian (sr) and Croatian (hr), situated on the same terminal branch, are
anticipated to exhibit a higher similarity score.

In contrast, languages such as Ukrainian (uk), which belongs to the East
Slavic linguistic group, are situated remotely on the phylogenetic tree. This
geographical and linguistic distance results in only moderate similarity
scores when compared with non-East Slavic languages in our matrix. These
observations are aligned with the insights presented by Kudugunta et al.
(2019).

##  3 Improving MNMT with Pseudo Family

###  3.1 Searching the Best K for Overlap Method

To ascertain this threshold, we embarked upon a series of experimental
procedures with the range spanning from 30% to 80%. Our meticulous analyses
culminated in the selection of 40% as the default parameter for the Overlap
method, given its superior performance during the evaluation of the overall
performance on the developmental set.

###  3.2 Main Experiments

#### Model and Dataset

We employ the same m2m100_418M model as outlined in (Fan et al., 2021) as the
foundational and utilize the datasets delineated in Section 2.1. For our
experiments, we selected languages from the Indo-Iranian and Austronesian
language families to test our methods, considering all language pairs as
potential auxiliary languages. All the data has been cleaned and tokenized
with fairseq
script222https://github.com/facebookresearch/fairseq/tree/main/examples/m2m_100
and mosesdecoder333https://github.com/moses-smt/mosesdecoder.

#### Baselines

For our evaluation, we considered the following baselines based on the primary
model: 1) Vanilla: Direct translations were conducted using the pre-trained
model for the target language pairs without fine-tuning; 2) FT: The primary
model underwent a fine-tuning phase using bilingual data specific to the
target language pairs; 3) LF: The primary model was fine-tuned by employing
the traditional language family delineated in Figure 1, with temperature
sampling and set temperature at 1.5; 4) LF+FT: Expanding on the LF method, we
integrated an additional fine-tuning phase that used the target language pair
data.

#### Training and Inference

For the training stage, we set the batch size as 4096; for our methods, we up-
sample the training data to be the same to ensure that the proportion of each
language within each mini-batch is equal except Hindi (hi), which is aligned
with LF when using Overlap method. For optimization, we deployed the Adam
optimizer (Kingma and Ba, 2015) with $\beta_{1}=0.9$, $\beta_{2}=0.98$, and
$\epsilon=10e^{-6}$. A learning rate of $lr=3e^{-5}$ was chosen for this
process.

For the inference stage, beam search was employed with a beam size set to 5
and a length penalty 1.0 applied consistently across languages. Evaluation
metrics were based on BLEU scores, calculated using SacreBLEU (Post, 2018).
Our experimental framework was built upon the fairseq toolkit (Ott et al.,
2019).

Model |  fa | hi | bn | id | ms | Avg.
---|---|---|---|---|---|---
Baselines
Vanilla | 25.0 | 19.5 | 10.7 | 28.0 | 28.6 | 22.4
FT | 36.5 | 35.1 | 23.0 | 39.4 | 35.3 | 33.9
LF | 33.9 | 35.6 | 26.3 | 38.4 | 32.5 | 33.3
LF+FT | 36.8 | 35.6 | 26.3 | 39.0 | 32.5 | 34.0
Our Methods
KL | 36.1 | 35.5 | 25.2 | 40.0 | 37.6 | 34.9
MSE | 36.9 | 34.9 | 27.1 | 39.9 | 35.9 | 34.9
Overlap | 36.9 |  35.6 | 27.2 | 39.7 | 35.5 | 35.0

Table 2: The main result of our experiment on baselines and our pseudo family
method with the three different methods to calculate similarities. The
threshold of K in Overlap method is set as 40%.

###  3.3 Result and Analyse

#### Comparison between Language Families

A deeper investigation into language families, as depicted in Table 1, reveals
a critical insight: within the traditional structure of a language family, it
is not guaranteed that all languages will provide superior translation
outcomes, even precipitating a significant compromise in the translation
quality.

Intriguingly, within the pseudo language family, our analysis identified
language pairs that were linguistically divergent from the target language.
For instance, Korean (ko) was selected as an auxiliary language for
translation from Farsi (fa) to  en.  Data similarity metrics show that fa
exhibits a relatively higher similarity with ko than with others, as we have
detailed in Appendix A.3.

This phenomenon highlights that our methodology is proficient in tapping into
dimensions beyond traditional linguistic boundaries, thereby selecting
language pairs or datasets that are more conducive to optimal model training.
We have delved deeper into the comparative analysis substantiating the
superiority of our method in Appendix A.4.

#### Effect of Pseudo Language Family

Our main results, comparing the outcomes between auxiliary languages
identified through our proposed methods and the baselines, are showcased in
Table 2. Our pseudo family strategies delivered notable performance uplifts
across all three methods, clearly outpacing the conventional language family
method.

Averagely, we noted a boost of 1.7 BLEU scores when juxtaposed against the
traditional language family approach, a rise of 1.1 BLEU when compared to pure
fine-tuning, and a 1.0 BLEU elevation against the hybrid approach of fine-
tuning post multilingual training. A notable leap in performance was witnessed
for several low-resourced language pairs, underpinning the efficacy of our
strategy.

Furthermore, we assessed our methodology’s resilience across diverse datasets.
Our proposed method adeptly marries linguistic pertinence with data-related
similarity, facilitating the astute selection of auxiliary languages for
optimized translation outcomes. For comprehensive details, please refer to the
Appendix A.5.

##  4 Conclusion and Future Work

In this study, we introduce a pioneering approach and three methods to
quantify the similarities between language pairs and subsequently propose a
pseudo family, based on this measurement, to enhance the selection of
auxiliary languages for multilingual training in low-resource language
scenarios. Our strategies for assessing language similarity leverage fisher
information to sieve through model parameters, subsequently identifying task-
specific ones and utilizing them to formulate an FIM to gauge the similarity
between language pairs. The experimental outcomes significantly improve BLEU
scores on low-resource languages, affirming our proposed method’s feasibility
and accuracy.

Future works include: 1) expanding the application of the proposed FIM
methodology to other tasks that require the calculation of language similarity
and 2) integrating our findings into curriculum learning frameworks (Liu et
al., 2020; Zhan et al., 2021) and consistency learning frameworks (Li et al.,
2022; Wang et al., 2022; Liu et al., 2023; Li et al., 2023).

## Limitations

In the present study, our methodology and experiments predominantly
concentrated on the analysis of low-resource language pairs. An exploration of
the generalization performance of our approach across a more comprehensive
range of language pairs would be of significant value. Moreover, while our
experiments employed the m2m100_418M model, future investigations could
benefit from incorporating larger m2m100 models. Testing the generalization
capacity of our method with diverse models, such as the NLLB (Costa-jussà et
al., 2022) and SeamlessM4T (Barrault et al., 2023), could yield more valuable
insights.

Furthermore, it is pertinent to note that the FIM retains extensive
information. In forthcoming studies, we intend to implement innovative,
parameter-efficient fine-tuning strategies akin to those propounded by (Gong
et al., 2022). These strategies aim to ameliorate the detection of pertinent
parameters while concurrently managing the total number of parameters with
greater efficacy.

## Acknowledgment

This work was supported in part by the National Natural Science Foundation of
China (Grant No. 62206076), Shenzhen College Stability Support Plan (Grant
Nos. GXWD20220811173340003, GXWD20220817123150002), Shenzhen Science and
Technology Program (Grant No. RCBS20221008093121053). Xuebo Liu was sponsored
by CCF-Tencent Rhino-Bird Open Research Fund. The computational resources of
this research were supported by the Education Center of Experiments and
Innovations at Harbin Institute of Technology, Shenzhen. We would like to
thank the anonymous reviewers and meta-reviewer for their insightful
suggestions.

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##  Appendix A Appendix

###  A.1 Process of Similarity Calculation

Data: $\Theta$ : parameters weight; D: training data; |D|: number of mini-
batches in dataset D; $L(\Theta)$: loss function for parameters $\Theta$;
samplet : mini-batch for t-th step; FIMt: fisher information matrix

1

2initialization: Begin training for epoch one; $\Theta$ is inherent from the
pre-trained model; FIM0 $\leftarrow 0$; $t$ $\leftarrow 0$;

3for  _sample t $\leftarrow$ D_ do

4 $L(\Theta)$ = $step\\_traning(\Theta)$;

5 // training for a mini-batch of dataset D;

6 $g_{t}$= $\frac{\partial{L(\Theta)}}{\partial\theta}$;

7 // Get gradients;

8 FIMt+1 $\leftarrow$ FIMt +$\frac{g_{t}^{2}}{|\rm D|}$;

9 $t\leftarrow t+1$ ;

10 // update FIM;

11 // do not backpropagation;

12

13 end for

return FIMt

Algorithm 1 FIM Estimation

In order to identify pertinent language pairs for multilingual training, we
advocate for the application of the FIM similarity methodology, which has
demonstrated commendable efficacy. Our proposed approach can be delineated
into three phases:

  1. 1.

Construction of Shared Language Pool: A shared language pool, embodying a
plethora of languages across diverse language families, is constructed.

  2. 2.

Derivation of FIM: The fisher information matrix for each language in this
pool is then computed using the estimation strategy detailed in Section 2.2,
and the pseudo-code is shown in Algorithm 1.

  3. 3.

Computation of Language Pair Similarity: When identifying suitable auxiliary
languages for multilingual training with a designated target language pair, we
employ three methods described in Section 2.2 to calculate the similarity
between each language pair.

###  A.2 Algorithm for Language Selection

The algorithmic details for the selection of auxiliary language pairs are as
follows:

  1. 1.

Sort the similarities in descending (ascending for MSE and KL) order to create
a list $L$,

  2. 2.

Initialize $Gap$ as $\lvert L[1]-L[0]\rvert$, add the first language into the
auxiliary list,

  3. 3.

Iterate from $i=2$ to the end of $L$, for each loop we select language as
follow:

     * 1)

If $\lvert L[i-1]-L[i]\rvert<Gap$ add the $i$-th language into the auxiliary
list and update the $Gap=\lvert L[i-1]-L[i]\rvert$

     * 2)

If $\lvert L[i-1]-L[i]\rvert=Gap$ add the $i$-th language into the auxiliary
list and update the $Gap=\frac{Gap}{2}$;

     * 3)

If $\lvert L[i-1]-L[i]\rvert>Gap$, terminate the loop;

  4. 4.

The language pairs in the auxiliary list with the target pair are composed of
the pseudo language family for the target language pair.

Figure 5:  The full similarities computed TED datasets using Overlap method.
We can directly infer the similarity between language pairs by observing the
darkness or lightness of colors. Red boxes cover a language family, yellow
boxes demonstrate the potential subfamily and black boxes contain some
auxiliary languages beyond the language family that can leverage the target
language-pair’s translation.

###  A.3 Data Similarity Analysis of Pseudo Family

From a linguistic vantage point, ko and fa languages demonstrate stark
disparities. They neither share analogous characters nor exhibit proximal
grammatical structures. Despite these disparities, it is pertinent to
highlight the prevalence of word embedding as a method to abstractly represent
characters and sentences, thereby bridging the gap between linguistically
disparate languages. Through word embedding, characters or phrases with
analogous meanings may find themselves in closer proximity within the embedded
space.

Taking this into consideration, our analysis transcends mere linguistic
similarities and delves into the dataset’s similarities as well. We have
employed a sampling methodology to ascertain the data similarity between fa
and both traditional and pseudo language pairs:

  * 1.

Randomly  select 10% data points from each auxiliary language training set,
denoted as $T$.

  * 2.

Randomly  select 10% data points from the training set of fa denoted as S.

  * 3.

For each data point in $T$, compute the cosine distance to all data points in
$S$.

  * 4.

For each data point in $T$, average the cosine distances of its top 5 closest
data points in $S$ as similarity.

  * 5.

Finally, average the calculated similarities for all data points in $T$.

The conclusive outcomes demonstrate that within the dataset, ko and fa share a
similarity of 32.6%, comparable to the likeness between Bengali (bn) at 32.6%
and hi at 33.5%, with the latter duo forming part of the traditional language
family of fa. Significantly, ko boasts a more substantial data volume in
comparison to bn and hi, a factor that could considerably bolster the training
procedure. This observation underscores that our methodology does not
exclusively hinge upon linguistic similarities; instead, it amalgamates
information gleaned from the dataset, a process that is facilitated by the
model’s representational form.

Figure 6: The full similarities computed with OPUS100 and TED datasets. Even
when computing the similarity between language pairs across various datasets,
most similarities remain highest for the same language pair within different
datasets.

###  A.4 Supplementary Analysis of Language Families

Figure 5 elucidates that one can discern subgroups within a vast language
family, thereby alluding to the intricate web of language relationships and
the potential for additional sub-categorizations. Categorizing sub-language
families solely based on linguistic methodology can be a formidable task.

Furthermore, it is apparent that languages chosen due to their intrinsic
mutual similarities may not necessarily be part of the same traditional
language family. This phenomenon echoes our methodology, wherein the model’s
parameters serve as a conduit for representation, thereby facilitating the
amalgamation of multifaceted information spanning languages and data, which
could potentially unveil novel insights that could bolster our comprehension
of pre-trained models.

Moreover, traditional methodologies often employ stringent classifications for
language pairs, which may prove inadequate when navigating languages that
hover near classification boundaries, potentially leading to incongruities.
Additionally, such a rigid approach might fall short when faced with shifts in
the data domain.

In contrast, our approach embraces a more fluid and dynamic selection
algorithm that meticulously curates choices tailored to each target language
pair. Crucially, our method is versatile, accommodating variations across
different models and datasets, thereby ensuring its seamless applicability
across a spectrum of scenarios. This versatility is highlighted by the
observation that numerous languages can seamlessly integrate into multiple
pseudo-language families.

###  A.5 Robustness Analysis Across Different Datasets

To more comprehensively demonstrate the performance of our methodology across
diverse domain-specific datasets, we conducted an in-depth analysis to
scrutinize the variations in similarities generated by our approach across
diverse data combinations. For the purposes of our experimentation and
analysis, we employed the OPUS100 dataset to facilitate our experimental and
research endeavors. Subsequently, we calculated $S_{(\rm t,\rm a)}$ where $\rm
t$ is derived from the TED dataset and $\rm a$ originates from the OPUS100
dataset. Figure 6 documents this full similarity matrix.

Notably, our findings demonstrate that even when languages are derived from
distinct corpora, there remains a pronounced similarity among linguistically
akin ones. This underscores the robustness of our methodology in discerning
intrinsic linguistic affinities.


Integrable (3+1)-dimensional generalization for dispersionless
Davey–Stewartson system Antonio J. Pan-Collantes Departamento de Matemáticas
Universidad de Cádiz - UCA Puerto Real<EMAIL_ADDRESS>
###### Abstract

This paper introduces a (3+1)-dimensional dispersionless integrable system,
utilizing a Lax pair involving contact vector fields, in alignment with
methodologies presented by A. Sergyeyev in 2018. Significantly, it is shown
that the proposed system serves as an integrable (3+1)-dimensional
generalization of the well-studied (2+1)-dimensional dispersionless Davey-
Stewartson system. This way, an interesting new example on integrability in
higher dimensions is presented, with potential applications in modern
mathematical physics. The work lays the foundation for future research into
symmetries, conservation laws, and Hamiltonian structures, offering avenues
for further exploration.

##  1 Introduction

Within the framework of partial differential equations (PDEs), integrable
systems are of great interest in modern mathematics and physics [1, 2, 3, 4].
In particular, taking into account that the vast majority of physical theories
commonly accept that there are three spatial dimensions and one temporal
dimension, the search for integrable systems in (3+1) dimensions becomes
particularly important and challenging, cf. e.g. [5, 3, 6, 7].

The paper [7] gives a systematic approach for constructing (3+1)-dimensional
integrable dispersionless systems using contact Lax pairs. These are
nonisospectral Lax pairs associated with contact vector fields and so, at
least in principle, they are amenable to the inverse scattering transform and
other techniques like the dressing method [4, 8, 9]. Remarkably, the systems
obtained using the method of [7] admit a straightforward reduction to
(2+1)-dimensional systems, which suggests that the said method allows for
finding integrable (3+1)-dimensional generalizations of certain
(2+1)-dimensional systems.

In this paper we implement this strategy for finding a (3+1)-dimensional
integrable generalization for the dispersionless Davey-Stewartson (dDS)
system, which is currently a subject of intense research (see e.g. [10, 11,
12, 13] and references therein), in view of its providing a dispersionless
limit for the the dispersive Davey-Stewartson system. The latter plays a
significant role in mathematical physics, as it represents a (2+1)-dimensional
generalization of the nonlinear Schrödinger equation, and describes many
important wave processes [14, 15, 16, 17].

In what follows, after a summary of relevant facts in Section 2, we will
employ the method from [7] for constructing a new (3+1)-dimensional
dispersionless integrable system (10) in Section 3, and we will show that this
new system is a (3+1)-dimensional generalization of dDS system (see Theorem
1). We note that it could be of interest to study the symmetries, conservation
laws, Hamiltonian operators, and other related structures for (10) using e.g.
the techniques from [18, 19].

##  2 Preliminaries

In the present paper a PDE system in four independent variables will be called
a (3+1)-dimensional dispersionless system (note that dispersionless systems
are also known as hydrodynamic-type systems) [20, 21, 7, 22] if it can be
written in the form

$A_{0}(\mathbf{u})\mathbf{u}_{t}+A_{1}(\mathbf{u})\mathbf{u}_{x}+A_{2}(\mathbf{u})\mathbf{u}_{y}+A_{3}(\mathbf{u})\mathbf{u}_{z}=0,$
(1)

where $x,y,z,t$ are independent variables,
$\mathbf{u}=(u_{1},\ldots,u_{N})^{T}$ is an $N$-dimensional vector of unknown
functions, and $A_{i}(\mathbf{u})$ are $M\times N$ matrices whose entries
depend on the components of $\mathbf{u}$. Both $M$ and $N$ are positive
integers satisfying $M\geq N$. All functions under consideration are tacitly
assumed smooth enough to ensure the validity of all computations. In what
follows a (3+1)-dimensional dispersionless system will be said to be
integrable if it admits a Lax pair.

In the recent paper [7] is presented a systematic approach for constructing
integrable (3+1)-dimensional dispersionless systems, that we briefly review
here for the readers’ convenience.

Consider, following [7], the class of Lax pairs $L\chi=0,M\chi=0$, where

$\displaystyle L$ $\displaystyle=\partial_{y}-X_{f},$ (2) $\displaystyle M$
$\displaystyle=\partial_{t}-X_{g},$

for suitable functions $f=f(p,\mathbf{u})$ and $g=g(p,\mathbf{u})$, and with

$X_{h}=h_{p}\partial_{x}+(ph_{z}-h_{x})\partial_{p}+(h-ph_{p})\partial_{z}.$

Note that $p$ plays the role of the spectral parameter, since
$\mathbf{u}_{p}\equiv 0$ by assumption.

These nonisospectral Lax pairs have a profound connection to contact geometry.
The vector field $X_{h}$ formally looks exactly like a contact vector field
corresponding to a contact Hamiltonian $h$ on a contact 3-manifold with local
coordinates $x,z,p$ and the associated contact one-form $dz+pdx$ (see [23, 24]
and the references therein).

We state without proof the following proposition, which corresponds to
Proposition 1 in [7]:

######  Proposition 1.

A nonisospectral Lax pair of the form (2) is compatible if and only if the
functions $f,g$ satisfy

$f_{t}-g_{y}+\\{f,g\\}=0,$ (3)

with the bracket $\\{,\\}$ defined as

$\\{f,g\\}=f_{p}g_{x}-g_{p}f_{x}-p\left(f_{p}g_{z}-g_{p}f_{z}\right)+fg_{z}-gf_{z}.$

Observe that equation (3) can be rewritten as

$\sum_{i=1}^{N}f_{u_{i}}\left(u_{i}\right)_{t}-g_{u_{i}}\left(u_{i}\right)_{y}+\left(f_{p}g_{u_{i}}-g_{p}f_{u_{i}}\right)\left(u_{i}\right)_{x}+\left(\left(f-pf_{p}\right)g_{u_{i}}-\left(g-pg_{p}\right)f_{u_{i}}\right)\left(u_{i}\right)_{z}=0,$
(4)

and if the dependence of $f$ and $g$ on $p$ is rational, we can bring equation
(4) to a common denominator. Then, vanishing of the expression in question is
equivalent to that of its numerator, and the latter is achieved by equating to
zero the coefficients at all powers of $p$. Consequently, Proposition 1 allows
us to produce integrable systems of the form (1) by making appropriate choices
of functions $f,g$.

Importantly, not every pair of functions $f,g$ is admissible, in the sense
that equation (3) leads to an expression that can be transformed into a system
of Cauchy–Kowalevski type by a suitable change of variables. In this sense, in
[7, 25] it is shown that for any natural numbers $m$ and $n$, the pairs of Lax
functions

$f=p^{n+1}+\sum_{j=0}^{n}v_{j}p^{j},\quad
g=p^{m+1}+\frac{m}{n}v_{n}p^{m}+\sum_{k=0}^{m-1}w_{k}p^{k},$ (5)

and

$f=\sum_{j=1}^{n}\frac{a_{i}}{v_{i}-p},\quad
g=\sum_{k=1}^{m}\frac{b_{k}}{w_{k}-p},$ (6)

are admissible, with $\mathbf{u}$ being in these cases

$\mathbf{u}=(v_{0},\ldots,v_{n},w_{0},\ldots,w_{m-1})^{T}$

and

$\mathbf{u}=(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n},v_{1},\ldots,v_{m},w_{1},\ldots,w_{n})^{T},$

respectively.

Furthermore, in [26] it is presented an example of algebraic, rather than
rational, admissible pair of Lax functions not belonging to any of the classes
above.

######  Remark 1.

As highlighted in [7, 25, 26], under the assumption $\mathbf{u}_{z}=0$, system
(1) reduces to a (2+1)-dimensional dispersionless system. If (1) admits a
contact Lax pair (2) then the system obtained from (1) by putting
$\mathbf{u}_{z}=0$ admits a Lax pair involving Hamiltonian vector fields (as
opposed to contact vector fields) with the Lax operators given by

$L=\partial_{y}-\tilde{X}_{f},\quad M=\partial_{t}-\tilde{X}_{g},$ (7)

where now

$\tilde{X}_{h}=h_{p}\partial_{x}-h_{x}\partial_{p}.$

Detailed discussions on this type of (2+1)-dimensional integrable systems can
be found e.g. in [21, 27].

An important example of a (2+1)-dimensional integrable dispersionless system
is given by the dispersionless Davey-Stewartson system, which is a physically
relevant generalization of the dispersionless nonlinear Schrödinger equation,
cf. e.g. [10]. This system has the form:

$\begin{array}[]{r@{}l}U_{t}+2\left(US_{x}\right)_{x}+2\left(US_{y}\right)_{y}&=0,\\\\[5.69054pt]
S_{t}+S_{x}^{2}+S_{y}^{2}-\delta\phi&=0,\\\\[5.69054pt]
\phi_{xy}-\frac{1}{2}\left(U_{xx}+U_{yy}\right)&=0.\end{array}$ (8)

Observe that we can eliminate $\phi$ from this system, expressing it using the
second equation and substituting the result into the third one, which yields

$\begin{array}[]{r@{}l}U_{t}+2\left(US_{x}\right)_{x}+2\left(US_{y}\right)_{y}&=0,\\\\[5.69054pt]
\displaystyle\frac{1}{\delta}(S_{t}+S_{x}^{2}+S_{y}^{2})_{xy}-\frac{1}{2}\left(U_{xx}+U_{yy}\right)&=0.\end{array}$
(9)

##  3 Integrable (3+1)-dimensional generalization of the (2+1)-dimensional dDS
system

Consider the following system

$\begin{array}[]{rl}u_{t}=&2rv_{x}+vr_{x}+su_{z}-us_{z}+wu_{x}-2vw_{z}+s_{y},\\\\[5.69046pt]
v_{t}=&2qu_{z}-uq_{z}+sv_{z}-2vs_{z}+vw_{x}+wv_{x}+q_{y},\\\\[5.69046pt]
w_{y}=&-2ru_{x}+rv_{z}+2vr_{z}+uw_{z},\\\\[5.69046pt]
r_{y}=&ru_{z}+ur_{z},\\\\[5.69046pt]
q_{x}=&2cvu_{x}+cvv_{z}+\dfrac{q}{v}v_{x},\\\\[5.69046pt]
s_{x}=&\dfrac{q}{v}u_{x}-3cvu_{z}-2cv_{y}+\dfrac{2cuv-2q}{v}v_{z}+2q_{z},\\\
\end{array}$ (10)

where $u,v,w,r,q,s$ are unknown functions of the independent variables
$x,y,z,t$, and $c\in\mathbb{R}$ is a constant.

This system can be transformed into evolutionary form, so that the
Cauchy–Kowalevski theorem could be applied to it (so in particular the system
in question is neither underdetermined nor overdetermined). Indeed, the
coefficients of the variables $u_{z},v_{z},w_{z},q_{z},r_{z},s_{z}$ in the
previous equations constitute a nondegenerate matrix

$\begin{bmatrix}s&0&-2v&0&0&-u\\\ 2q&s&0&-u&0&-2v\\\ 0&r&u&0&2v&0\\\
r&0&0&0&u&0\\\ 0&cv&0&0&0&0\\\ -3cv&\dfrac{2cuv-2q}{v}&0&2&0&0\\\
\end{bmatrix}$

whose determinant is

$cv\left(3cu^{4}v-4qu^{3}-16rv^{3}+4su^{2}v\right)\neq 0,$ (11)

so the system can be put in an evolutionary form with $z$ playing the role of
the evolution parameter.

Our main result establishes that system (10) is integrable and that it is a
(3+1)-dimensional generalization of (9):

######  Theorem 1.

The (3+1)-dimensional dispersionless system (10) admits the Lax pair

$\displaystyle L=\partial_{y}$
$\displaystyle+\dfrac{v}{p^{2}}\partial_{x}-\left(u+\dfrac{2v}{p}\right)\partial_{z}-\left(u_{z}p-u_{x}+v_{z}-\dfrac{v_{x}}{p}\right)\partial_{p},$
(12) $\displaystyle M=\partial_{t}$
$\displaystyle-\left(2rp+w-\dfrac{q}{p^{2}}-\dfrac{2cv^{2}}{p^{3}}\right)\partial_{x}+\left(rp^{2}-s-\dfrac{2q}{p}-\dfrac{3cv^{2}}{p^{2}}\right)\partial_{z}$
$\displaystyle-\left(r_{z}p^{3}+(w_{z}-r_{x})p^{2}+(s_{z}-w_{x})p+q_{z}-s_{x}+\dfrac{2cvv_{z}-q_{x}}{p}-\dfrac{2cvv_{x}}{p^{2}}\right)\partial_{p}.$

Moreover, for $c=-1$ the system (10) is an integrable (3+1)-dimensional
generalization of the (2+1)-dimensional dispersionless Davey-Stewartson system
(9).

######  Remark 2.

Observe that (10) also is a generalization of the (3+1)-dimensionl integrable
system appearing in Example 1 in [7]: if we assume $s=q=0$ and $c=0$ in (10)
we obtain

$\begin{array}[]{rl}u_{t}=&2rv_{x}+vr_{x}+wu_{x}-2vw_{z},\\\
v_{t}=&vw_{x}+wv_{x},\\\ w_{y}=&-2ru_{x}+rv_{z}+2vr_{z}+uw_{z},\\\
r_{y}=&ru_{z}+ur_{z},\\\ \end{array}$ (13)

which coincides with system (38) in [7].

##  4 Proof of Theorem 1

First, observe that the Lax pair (12) is of the form (2), with the functions

$\begin{array}[]{l}f=u+\dfrac{v}{p},\\\
g=rp^{2}+wp+s+\dfrac{q}{p}+c\dfrac{v^{2}}{p^{2}}.\end{array}$ (14)

Now, according to Proposition 1, this Lax pair is compatible if and only if

$f_{t}-g_{y}+\\{f,g\\}=0.$ (15)

Therefore, to show that system (10) implies the compatibility of Lax pair
(12), it is enough to substitute (14) into (15), and check that the resulting
expression is equivalent to (10) upon equating to zero the coefficients at all
powers of $p$, as can be verified by straightforward algebraic manipulations.

To prove the second part of the theorem observe first that, according to
Remark 1, if we assume

$u_{z}=v_{z}=w_{z}=q_{z}=r_{z}=s_{z}=0$ (16)

in system (10) we obtain a (2+1)-dimensional integrable system.

If we further assume, in addition to (16), that $r=1$, system (10) gives rise
to the system

$\begin{array}[]{rl}u_{t}=&wu_{x}+s_{y}+2v_{x},\\\\[5.69046pt]
v_{t}=&vw_{x}+wv_{x}+q_{y},\\\\[5.69046pt] w_{y}=&-2u_{x},\\\\[5.69046pt]
q_{x}=&2cvu_{x}+\dfrac{q}{v}v_{x},\\\\[5.69046pt]
s_{x}=&-2cv_{y}+\dfrac{q}{v}u_{x}.\\\\[5.69046pt] \end{array}$ (17)

Now note that for $c=-1$ system (17) can be reduced, by means of the following
substitutions

$u=S_{y},\quad v=\frac{1}{4}\delta U,\quad w=-2S_{x},\quad
q=-\frac{1}{2}\delta US_{y},\quad s=\frac{1}{2}\delta W-S_{y}^{2},$

where $\delta$ is a constant and $S,U,W$ are functions of $x,y,t$, to the
system

$\displaystyle U_{t}$ $\displaystyle=-2(US_{x})_{x}-2(US_{y})_{y},$ (18a)
$\displaystyle W_{x}$ $\displaystyle=U_{y},$ (18b) $\displaystyle S_{ty}$
$\displaystyle=-2S_{x}S_{xy}+\frac{1}{2}\delta
W_{y}-2S_{y}S_{yy}+\frac{1}{2}\delta U_{x}.$ (18c)

We can rearrange equation (18c) obtaining

$(S_{t})_{y}=-(S_{x}^{2})_{y}-(S_{y}^{2})_{y}+\frac{\delta}{2}(W_{y}+U_{x}),$
(19)

and differentiating (19) with respect to $x$ gives us

$(S_{t})_{xy}=-(S_{x}^{2})_{xy}-(S_{y}^{2})_{xy}+\frac{\delta}{2}(W_{xy}+U_{xx}).$
(20)

Finally, we observe that if $S,U,$ and $W$ satisfy (18) then $S$ and $U$ also
satisfy the system

$\displaystyle U_{t}$ $\displaystyle=-2(US_{x})_{x}-2(US_{y})_{y},$ (21a)
$\displaystyle(S_{t})_{xy}$
$\displaystyle=-(S_{x}^{2})_{xy}-(S_{y}^{2})_{xy}+\frac{\delta}{2}(U_{yy}+U_{xx}),$
(21b)

which is nothing but (9).

So, we see that system (18) is an integrable generalization of (9), system
(17) is an integrable generalization of (18), and finally system (10) is an
integrable (3+1)-dimensional generalization of (17) and hence _a fortiori_ of
(9), and the result follows.

######  Remark 3.

It is worth noting that the pair of functions (14) doesn’t belong to any of
the previously known classes of admissible pairs of rational functions (5) and
(6) yielding (3+1)-dimensional dispersionless integrable systems.

## Acknowledgments

The author thanks the financial support from the University of Cádiz through
the internal funding program ‘Plan Propio de Estímulo y Apoyo a la
Investigación y Transferencia 2022/2023’, and from Junta de Andalucía through
the research group FQM–377.

The author would also like to extend his sincere gratitude to A. Sergyeyev for
stimulating discussions and encouragement.

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# Singularities of pluri-fundamental divisors on Gorenstein Fano varieties of
coindex $4$

Jinhyung Park  Department of Mathematical Sciences, KAIST, 291 Daehak-ro,
Yuseong-gu, Daejeon 34141, Republic of Korea<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension
$n$ with $H$ fundamental divisor. Assume $h^{0}(X,H)\geq n-2$. We prove that a
general element of the linear system $|mH|$ has at worst canonical
singularities for any integer $m\geq 1$. When $X$ has terminal singularities
and $n\geq 5$, we show that a general element of $|mH|$ has at worst terminal
singularities for any integer $m\geq 1$. When $n=4$, we give an example of
Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$
does not have terminal singularities.

###### Key words and phrases:

Fano variety, fundamental divisor, singularity of a pair

J. Park was partially supported by the National Research Foundation (NRF)
funded by the Korea government (MSIT) (NRF-2021R1C1C1005479).

##  1\. Introduction

Throughout the paper, we work over the field $\mathbb{C}$ of complex numbers.
Let $X$ be a Gorenstein Fano variety of dimension $n$ with canonical
singularities. The _index_ of $X$ is

$i_{X}:=\max\\{t\in\mathbb{Z}\mid-K_{X}\sim tH\text{\leavevmode\nobreak\ where
$H$ is an ample Cartier divisor}\\}.$

It is well known that

$1\leq i_{X}\leq n+1.$

The _coindex_ of $X$ is $n+1-i_{X}$. An ample Cartier divisor $H$ on $X$ with
$-K_{X}\sim i_{X}H$ is called the _fundamental divisor_ of $X$. Since
$\operatorname{Pic}(X)$ is torsion-free, $H$ is uniquely determined up to
linear equivalence. It is a natural problem to study singularities of general
members in pluri-fundamental linear systems $|mH|$ for all integers $m\geq 1$.

By Kobayashi–Ochiai, $X$ is a projective space if $i_{X}=n+1$, and $X$ is a
hyperquadric if $i_{X}=n$. A Gorenstein canonical Fano variety $X$ with
$i_{X}=n-1$ is a _del Pezzo variety_ , and del Pezzo varieties were classified
by Fujita [Ft1, Ft3]. If $X$ is a del Pezzo variety, then the base locus
$\operatorname{Bs}|H|$ is empty or consists of a single point neither in
$\operatorname{Sing}X$ nor in $\operatorname{Sing}Y$, where $Y\in|H|$ is a
general member. Thus $Y$ has canonical/terminal singularities if $X$ has
canonical/terminal singularities. A Gorenstein canonical Fano variety $X$ with
$i_{X}=n-2$ is a _Mukai variety_ , and smooth Mukai varieties were classified
by Mukai [Mu] under the assumption that $|H|$ contains a smooth divisor. Mella
[Me] verified this assumption, and moreover, he also proved that if $X$ is a
Gorenstein Mukai variety with canonical/terminal singularities, then a general
member in $|H|$ has canonical/terminal singularities except when $X$ is a
complete intersection in $\mathbb{P}(1,1,1,1,2,3)$ of a quadric defined in the
first four linear variables and a sextic. Finally, note that if $i_{X}\geq
n-2$, then $|mH|$ is base point free for every integer $m\geq 2$ (see [L1,
Remark 4.5]). We can conclude that singularities of general members in $|mH|$
with $m\geq 1$ are well understood when $i_{X}\geq n-2$.

In this paper, we consider the case $i_{X}=n-3$, i.e., $X$ has coindex $4$.
Floris [Fl] proved that a general member of the linear system $|H|$ has
canonical singularities if $X$ is a Gorenstein canonical Fano variety of
coindex $4$ and $h^{0}(X,H)\neq 0$. However, in contrasts to the smaller
coindex cases, there is a smooth Fano fourfold $X$ of coindex $4$ such that
every member in $|H|$ is singular (see [HV, Example 2.12]). Heuberger [H]
proved that if $X$ is a smooth Fano fourfold, then a general member in
$|-K_{X}|$ has only terminal singularities. This is a natural generalization
of a classical result of Shokurov [S] for smooth Fano threefolds. Heuberger’s
theorem together with aforementioned results implies that a general member in
$|H|$ has only terminal singularities if $X$ is smooth.

The main result of this paper is the following.

######  Theorem 1.1.

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension
$n\geq 4$, and $H$ be the fundamental divisor of $X$.

  1. $(1)$

Assume that $h^{0}(X,H)\geq n-2$. Then a general member of the linear system
$|mH|$ has only canonical singularities for every integer $m\geq 1$.

  2. $(2)$

Assume that $X$ has terminal singularities and $h^{0}(X,H)\geq n-2$. Then a
general member of the linear system $|mH|$ has only terminal singularities for
every integer $m\geq 1$ unless $(n,m)=(4,1),(4,2),(4,3)$.

  3. $(3)$

Assume that $X$ is smooth. Then $h^{0}(X,H)\geq n-2$, and a general member of
the linear system $|mH|$ has only terminal singularities for every integer
$m\geq 1$ unless $(n,m)=(4,2)$.

If $X$ is a smooth Fano variety of coindex $4$ and dimension $n$, then Floris
[Fl, Theorem 1.2] and Liu [L1, Theorem 1.2] showed that $h^{0}(X,H)\geq n-2$.
If $X$ is singular, then we do not know whether $H^{0}(X,H)\neq 0$. This
nonvanishing follows from the following:

######  Conjecture 1.2 (Ambro–Kawamata effective nonvanishing conjecture [A],
[K2]).

Let $(X,\Delta)$ be a klt pair, and $D$ be a Cartier divisor on $X$. If $D$ is
nef and $D-(K_{X}+\Delta)$ is nef and big, then $H^{0}(X,D)\neq 0$.

This conjecture has been verified for low dimensional varieties [K2] and Fano
weighted complete intersections [PST]. Especially, [K2, Proposition 4.1 and
Theorem 5.2] say that if $X$ is a Gorenstein Fano fourfold with canonical
singularities, then $h^{0}(X,H)\geq 2$. Although the methods of the present
paper do not yield the results for higher coindex cases directly, we may still
expect that Theorem 1.1 for higher coindex would follow from the effective
nonvanishing conjecture (cf. [HS]).

In Theorem 1.1 $(2)$, when $n=4$, one cannot expect that a general member in
$|H|$ has terminal singularities. We give an example of Gorenstein terminal
Fano fourfold $X$ such that a general member of the linear system $|H|$ does
not have terminal singularities (see Example 2.4 $(1)$). In Theorem 1.1 $(3)$,
we do not know whether there is an example of a smooth Fano fourfold such that
a general element in $|2H|$ does not have terminal singularities. See Remark
3.5 for some partial result.

By [L2, Corollary 3], if $X$ is a Gorenstein Fano variety of coindex $4$ and
dimension $n$ with canonical singularities and $h^{0}(X,H)\geq n-2$, then
$|mH|$ is base point free for any integer $m\geq 4$ (see Remark 2.2). In
particular, if $X$ is a smooth Fano variety of coindex $4$, then a general
member in $|mH|$ is smooth for any integer $m\geq 4$.

One may expect that if a general member in $|H|$ has only mild singularities,
then so does a general member in $|mH|$ for any $m\geq 2$. More generally, we
may ask the following:

######  Question 1.3.

Let $X$ be a smooth projective variety, and $L,M$ be divisors on $X$. Suppose
that general members of $|L|$ and $|M|$ have only canonical/terminal
singularities. Then does a general member of $|L+M|$ have also
canonical/terminal singularities?

The answer is “NO” of course. Some counterexamples are given in Example 3.1.

### Organization

The paper is organized as follows. Section 2 is devoted to proving Theorem 1.1
$(1)$ and $(2)$. We also give some examples of terminal Fano fourfolds in
which a general member of the fundamental linear system does not have terminal
singularities (see Example 2.4). In Section 3, we negatively answer Question
1.3 in Example 3.1, and we prove Theorem 1.1 $(3)$.

### Acknowledgments

The author would like to thank Miles Reid for calling attention to the paper
[CFHR] and In-Kyun Kim and Seung-Jo Jung for useful conversations. The authors
is grateful to the referees for careful reading of the paper and valuable
comments.

##  2\. Pluri-fundamental divisors on singular Fano varieties

In this section, we prove Theorem 1.1 $(1)$ and $(2)$. For the definitions and
basic properties of singularities of pairs, we refer to [KM]. We begin with
fixing some notations. Let $X$ be a Gorenstein Fano variety of coindex $4$ and
dimension $n\geq 4$ with canonical singularities, and $H$ be the fundamental
divisor on $X$. We have $-K_{X}=(n-3)H$. Assume that $h^{0}(X,H)\geq n-2$.
Then $|mH|\neq\emptyset$ for each integer $m\geq 1$. Take a log resolution

$f_{m}\colon X_{m}\longrightarrow X$

of the ideal of the base locus $\operatorname{Bs}|mH|$. We may assume that
$f_{m}$ is obtained by a sequence of blow-ups along smooth centers. We write

$K_{X_{m}}=f_{m}^{*}K_{X}+\sum_{i}a_{m,i}E_{m,i}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
|f_{m}^{*}mH|=|M_{m}|+\sum_{i}r_{m,i}E_{m,i},$

where all $E_{m,i}$ are prime divisors, $|M_{m}|$ is the free part of
$|f_{m}^{*}mH|$, and $\sum_{i}r_{m,i}E_{m,i}$ is the fixed part of
$|f_{m}^{*}mH|$. By [Fl, Theorem 1.1], a general member of $|H|$ is
irreducible. Since $h^{0}(X,H)\geq 2$, it follows that
$\dim\operatorname{Bs}|H|\leq n-2$; thus $\dim\operatorname{Bs}|mH|\leq n-2$
for each integer $m\geq 1$. Hence every $E_{m,i}$ is an $f_{m}$-exceptional
divisor. Since $H$ is a Cartier divisor, all $a_{m,i}$ and $r_{m,i}$ are
nonnegative integers.

######  Lemma 2.1.

$\dim\operatorname{Bs}|mH|\leq 2$ for any integer $m\geq 1$.

###### Proof.

By [Fl, Proposition 4.1], $(X,X_{n-1})$ is a plt($=$purely log terminal) pair,
where $X_{n-1}\in|H|$ is a general member. As $X_{n-1}$ is connected, [KM,
Proposition 5.51] shows that $X_{n-1}$ is irreducible and normal. By [KM,
Theorem 5.50], $X_{n-1}$ has Gorenstein canonical singularities. Note that
$-K_{X_{n-1}}=((n-1)-3)H_{n-1}$, where $H_{n-1}:=H|_{X_{n-1}}$. If $n\geq 5$,
then $X_{n-1}$ is an $(n-1)$-dimensional Gorenstein canonical Fano variety of
index $i_{X_{n-1}}\geq(n-1)-3$ with $h^{0}(X_{n-1},H_{n-1})\geq(n-1)-2$. If
$i_{X_{n-1}}>(n-1)-3$, then $|H_{n-1}|$ is base point free (cf. [L1, Remark
4.5]) so that $(X_{n-1},X_{n-2})$ is a plt pair, where $X_{n-2}\in|H_{n-1}|$
is a general member. If $i_{X_{n-1}}=(n-1)-3$, then by [Fl, Proposition 4.1],
$(X_{n-1},X_{n-2})$ is also a plt pair. Continuing this process, we finally
obtain a Calabi–Yau threefold $X_{3}$ with canonical singularities and
$h^{0}(X_{3},H_{4}|_{X_{3}})\geq 1$. Notice that

$\operatorname{Bs}|H|=\operatorname{Bs}|H_{n-1}|=\cdots=\operatorname{Bs}|H_{4}|=\operatorname{Bs}|H_{4}|_{X_{3}}|.$

This shows $\dim\operatorname{Bs}|H|\leq 2$. Note that
$\operatorname{Bs}|mH|\subseteq\operatorname{Bs}|H|$ for any $m\geq 2$. Then
the lemma follows. ∎

######  Remark 2.2.

$(1)$ If the Ambro–Kawamata effective nonvanishing conjecture is true for
Gorenstein Fano variety of coindex $4$ with canonical singularities, then [Fl,
Proposition 4.1] and the “ladder” argument as in the proof of Lemma 2.1 show
that $h^{0}(X,H)\geq n-2$.
$(2)$ If $X$ is a Gorenstein Fano variety of coindex $4$ with canonical
singularities and $h^{0}(X,H)\geq n-2$, then the “ladder” argument and [L2,
Theorem 2] show that $|mH|$ is base point free for every integer $m\geq 4$.

The following proposition, inspired by [H, Proposition 9], is the key
ingredient of the proof of Theorem 1.1.

######  Proposition 2.3.

For an integer $m\geq 1$, we have

$a_{m,i}\geq\frac{m+n-3}{m}r_{m,i}-1\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ for all $i$}.$

###### Proof.

Suppose that $a_{m,i}-\frac{m+n-3}{m}r_{m,i}<-1$ for some $i$. Let

$c_{0}:=\inf\\{c\mid\text{$a_{m,i}-cr_{m,i}\leq-1$ for some $i$}\\}.$

Then $0<c_{0}<\frac{m+n-3}{m}$. For an integer $k>c_{0}$, choose $k$ general
members $D_{1},\ldots,D_{k}\in|mH|$, and let
$\Delta:=c_{0}\cdot\frac{D_{1}+\cdots+D_{k}}{k}$. Then the pair $(X,\Delta)$
is lc($=$log canonical) but not klt($=$Kawamata log terminal). Let $W$ be a
minimal lc center of the lc pair $(X,\Delta)$. Since
$D_{1},\ldots,D_{k}\in|mH|$ are general, $(X,\Delta)$ is klt outside the base
locus $\operatorname{Bs}|mH|$ (cf. [A, Lemma 5.1]). Thus $W$ is contained in
$\operatorname{Bs}|mH|$, so $\dim W\leq 2$ by Lemma 2.1.

By the generalization of Kawamata’s subadjunction formula [FG, Theorem 1.2],
there exists an effective divisor $\Gamma$ on $W$ such that

$(K_{X}+\Delta)|_{W}\sim_{\mathbb{Q}}K_{W}+\Gamma$

and the pair $(W,\Gamma)$ is klt. Note that

$mH-(K_{X}+\Delta)\sim_{\mathbb{Q}}(m+n-3-c_{0}m)H.$

Since $c_{0}m<m+n-3$, it follows that $mH-(K_{X}+\Delta)$ is ample. Then
$mH|_{W}-(K_{W}+\Gamma)$ is ample. Recall that $\dim W\leq 2$. By [K2, Theorem
3.1], $H^{0}(W,mH|_{W})\neq 0$. Now, since $mH-(K_{X}+\Delta)$ is ample and
$W$ is an lc center of the lc pair $(X,\Delta)$, we can apply [Fj, Theorem
2.2] to see that the restriction map

$H^{0}(X,mH)\longrightarrow H^{0}(W,mH|_{W})$

is surjective. However, $W\subseteq\operatorname{Bs}|mH|$, so this restriction
map is the zero map. We obtain $H^{0}(W,mH|_{W})=0$, which is a contradiction.
Thus the proposition holds. ∎

We are ready to prove Theorem 1.1 $(1)$ and $(2)$.

###### Proof of Theorem 1.1 (1) and (2).

Recall that $X$ is a Gorenstein Fano variety of coindex $4$ and dimension $n$
with canonical singularities. We assume that $h^{0}(X,H)\geq n-2$. Let
$Y_{m}\in|mH|$ be a general element for an integer $m\geq 1$.

$(1)$ We want to prove that $Y_{m}$ has canonical singularities. If
$(X,Y_{m})$ is a plt pair, then [KM, Theorem 5.50 and Proposition 5.51] imply
that $Y_{m}$ has canonical singularities since $Y_{m}$ is Gorenstein. Thus it
is enough to show that the pair $(X,Y_{m})$ is plt. The birational morphism
$f_{m}\colon X_{m}\to X$ is a log resolution of $(X,Y_{m})$. We have

$K_{X_{m}}+f_{m,*}^{-1}Y_{m}=f_{m}^{*}(K_{X}+Y_{m})+\sum_{i}(a_{m,i}-r_{m,i})E_{m,i}.$

If $r_{m,i}=0$, then $a_{m,i}-r_{m,i}\geq 0>-1$. If $r_{m,i}\geq 1$, then
Proposition 2.3 implies that

$a_{m,i}-r_{m,i}\geq\frac{n-3}{m}r_{m,i}-1>-1.$

Thus $(X,Y_{m})$ is a plt pair.

$(2)$ Assume that $X$ has terminal singularities and $n=\dim X\geq 5$ or
$m\geq 4$. We want to show that $Y_{m}$ has terminal singularities. If $m\geq
4$, then [L2, Theorem 2] (see also Remark 2.2 $(2)$) implies that $|mH|$ is
base point free; hence $Y_{m}$ has terminal singularities. From now on, assume
that $n\geq 5$ and $1\leq m\leq 3$. We know that $Y_{m}$ is a normal
projective variety with canonical singularities. Let
$Y_{m}^{\prime}:=f_{m,*}^{-1}Y_{m}$ be the strict transform of $Y_{m}$ under
$f_{m}$. Since $Y_{m}^{\prime}\in|M_{m}|$ is a general element,
$Y_{m}^{\prime}$ is smooth. Then

$f_{m}^{\prime}:=f_{m}|_{Y_{m}^{\prime}}\colon Y_{m}^{\prime}\longrightarrow
Y_{m}$

is a log resolution of $Y_{m}$. We have

$K_{Y_{m}^{\prime}}={f_{m}^{\prime}}^{*}K_{Y_{m}}+\sum_{i}(a_{m,i}-r_{m,i})E_{m,i}|_{Y_{m}^{\prime}}.$

Since $X$ has terminal singularities, we have $a_{m,i}\geq 1$ for all $i$.

Consider the case $m=1$. Note that $\frac{m+n-3}{m}=n-2\geq 3$. If
$r_{1,i}\geq 1$, then Proposition 2.3 implies that $a_{1,i}-r_{1,i}\geq
2r_{1,i}-1>0$. If $r_{1,i}=0$, then $a_{1,i}-r_{1,i}>0$. Thus $Y_{1}$ has
terminal singularities.

Suppose now that $Y_{m}$ does not have terminal singularities for some $2\leq
m\leq 3$. Then there is some $i_{0}$ such that $a_{m,i_{0}}=r_{m,i_{0}}\geq 1$
and $E_{m,i_{0}}|_{Y_{m}^{\prime}}$ is an
$f_{m}|_{Y_{m}^{\prime}}$-exceptional divisor. Since $Y_{m}$ has terminal
singularities outside $\operatorname{Bs}|mH|$, we see that
$f_{m}(E_{m,i_{0}})\subseteq\operatorname{Bs}|mH|$. Since $n\geq 5$ and $2\leq
m\leq 3$, we have $\frac{m+n-3}{m}\geq\frac{5}{3}$. If $r_{m,i}\geq 2$, then
Proposition 2.3 implies that $a_{m,i}-r_{m,i}\geq\frac{2}{3}r_{m,i}-1>0$. Thus
$a_{m,i_{0}}=r_{m,i_{0}}=1$. If
$f_{m}(E_{m,i_{0}})\not\subseteq\operatorname{Sing}X$, then $\dim
f_{m}(E_{m,i_{0}})=n-2$ since $f_{m}$ is a composition of smooth center blow-
ups. This means that $f_{m}(E_{m,i_{0}})$ is a divisor on $Y_{m}$ and
$E_{m,i_{0}}|_{Y_{m}^{\prime}}$ is not an
$f_{m}|_{Y_{m}^{\prime}}$-exceptional divisor. Thus
$f_{m}(E_{m,i_{0}})\subseteq\operatorname{Sing}X$, and $\dim
f_{m}(E_{m,i_{0}})\leq n-3$ because $X$ has terminal singularities. By taking
further blow-ups, we may assume that $f_{1}=f_{m}$ and $X_{1}=X_{m}$. Then
there is an $i_{1}$ such that $E_{1,i_{1}}=E_{m,i_{0}}$. We have
$a_{1,i_{1}}=a_{m,i_{0}}=1$. Now, since
$f_{m}(E_{m,i_{0}})\subseteq\operatorname{Bs}|mH|\subseteq\operatorname{Bs}|H|$,
it follows that $r_{1,i_{1}}\geq 1$. Thus $a_{1,i_{1}}-r_{1,i_{1}}\leq 0$.
Note that $E_{1,i_{1}}|_{Y_{1}^{\prime}}$ is an
$f_{1}|_{Y_{1}^{\prime}}$-exceptional divisor. We get a contradiction to that
$Y_{1}$ has terminal singularities. Hence $Y_{m}$ has terminal singularities
for any $2\leq m\leq 3$. ∎

Finally, we provide some examples of terminal Fano fourfolds in which a
general element in the fundamental linear system does not have terminal
singularities.

######  Example 2.4.

$(1)$ Let $Z:=X_{2,6}$ be a complete intersection in $\mathbb{P}(1,1,1,1,2,3)$
of a general quadric defined in the first four linear variables
$x_{0},\ldots,x_{3}$ and a general sextic (cf. [Me, Theorem 1]). Then $Z$ is a
Gorenstein terminal Fano threefold of index $1$, and
$\operatorname{Sing}Z=\\{p=(0:0:0:0:-1:1)\\}$. A general member of $|H_{Z}|$
is singular at $p$, where $H_{Z}=-K_{Z}$ is the fundamental divisor of $Z$.
Let $X:=Z\times\mathbb{P}^{1}$ so that $X$ is a Gorenstein Fano fourfold of
index $1$ with terminal singularities. Note that
$H=-K_{X}=\pi_{1}^{*}(-K_{Z})+\pi_{2}^{*}(-K_{\mathbb{P}^{1}})$ is the
fundamental divisor of $X$, where $\pi_{1}\colon X\to Z$ and $\pi_{2}\colon
X\to\mathbb{P}^{1}$ are projections. A general element $Y$ in $|H|$ has one
dimensional singular locus $\\{p\\}\times\mathbb{P}^{1}$. Since $\dim Y=3$, it
follows that $Y$ does not have terminal singularities. Here $Y$ is a
Gorenstein Calabi–Yau threefold with canonical singularities.
$(2)$ Let $X:=X_{9}$ be a weighted hypersurface in $\mathbb{P}(1,1,1,1,3,3)$
of degree $9$ (cf. quasismooth Fano 4-fold hypersurfaces ID 8 in [GRDB] based
on [BK]). Then $X$ is a non-Gorenstein $\mathbb{Q}$-Fano fourfold with
terminal singularities such that $-K_{X}$ is a hyperplane with
$(-K_{X})^{4}=1$. Note that $\operatorname{Sing}X$ consists of three terminal
singular points of the type $\frac{1}{3}(1,1,1,1)$. A general element in
$|-K_{X}|$ is a weighted hypersurface $S_{9}$ in $\mathbb{P}(1,1,1,3,3)$ of
degree $9$, and $S_{9}$ is a Gorenstein canonical Calabi–Yau threefold. Note
that $\operatorname{Sing}S_{9}$ consists of three (non-terminal) canonical
singular points of the type $\frac{1}{3}(1,1,1)$.

##  3\. Pluri-fundamental divisors on smooth Fano varieties

In this section, we first answer Question 1.3 by constructing smooth
projective varieties $X$ and divisors $M$ such that general members in $|M|$
are smooth but all members in $|mM|$ are not normal for some $m\geq 2$, and
then prove Theorem 1.1 $(3)$.

######  Example 3.1.

$(1)$ If $E$ is an exceptional divisor on a smooth projective variety, then
$|mE|=\\{mE\\}$ for all $m\geq 1$. Now, $E$ is smooth, but $mE$ is non-reduced
for any $m\geq 2$.
$(2)$ Let $C$ be a smooth projective curve of genus $2$. There are two
distinct points $P,Q$ on $C$ such that $2P\sim 2Q\sim K_{C}$. In particular,
$Q-P\in\operatorname{Pic}^{0}(C)$ is a $2$-torsion. We can also find
$\tau\in\operatorname{Pic}^{0}(C)$ such that
$H^{0}(C,P+\tau)=H^{0}(C,Q-P+\tau)=H^{0}(C,2\tau)=0$. Let
$E:=\mathcal{O}_{C}(P)\oplus\mathcal{O}_{C}(\tau)$, and $S:=\mathbb{P}(E)$
with the natural projection $\pi\colon S\to C$ and the tautological divisor
$H$, i.e., $\mathcal{O}_{S}(H)=\mathcal{O}_{\mathbb{P}(E)}(1)$. Let $A:=H$ and
$B:=H+\pi^{*}(Q-P)$. Then $A,B$ are sections of $\pi$, so they are smooth
irreducible curves isomorphic to $C$. Furthermore, $A,B$ satisfy the
following:

  1. $\bullet$

$A^{2}=B^{2}=A.B=1$,

  2. $\bullet$

$A\not\sim B$ but $2A\sim 2B$,

  3. $\bullet$

$h^{0}(S,A)=h^{0}(S,B)=1$, and

  4. $\bullet$

$h^{0}(S,2A)=h^{0}(S,2B)=2$.

Notice that $A,B$ meet at one point $p$ on $S$ and every member of $|2A|=|2B|$
has multiplicity at least $2$ at $p$. Thus every member in $|2A|=|2B|$ is not
normal.
$(3)$ [PST, Example 5.9] For an integer $m\geq 1$, let

$X=X_{(2m+1)(2m+2)}\subseteq\mathbb{P}(\underbrace{1,\ldots,1}_{1+2m(2m+1)},2m+1,2m+2)$

be a weighted hypersurface of degree $(2m+1)(2m+2)$. Then $X$ is a smooth Fano
variety of index $2$. If $H$ is the fundamental divisor of $X$, then a general
member of $|H|$ is smooth. However, $|-2iH|$ does not contain a smooth member
for any $1\leq i\leq m$. In this case, a general member in $|-2iH|$ has
terminal singularities.

We now turn to the proof of Theorem 1.1 $(3)$.

###### Proof of Theorem 1.1 (3) except the case $(n,m)=(4,3)$.

Let $X$ be a smooth Fano variety of coindex $4$ and dimension $n$ with
fundamental divisor $H$. By Theorem 1.1 $(2)$, we only have to consider the
cases $(n,m)=(4,1),(4,3)$. If $(n,m)=(4,1)$, then $H=-K_{X}$. Now, [H, Theorem
2] says that a general element in $|-K_{X}|$ has terminal singularities. ∎

######  Remark 3.2.

Let $X$ be a smooth Fano variety of coindex $4$, and $H$ be the fundamental
divisor of $X$. By [L2, Theorem 4], $|mH|$ is base point free for any integer
$m\geq 4$; hence a general element $Y_{m}\in|mH|$ is smooth in this case. But
there is a smooth Fano fourfold $X$ of coindex $4$ such that every member in
$|H|$ is singular (see [HV, Example 2.12]).

To finish the proof of Theorem 1.1, it only remains to prove that if $H$ is
the fundamental divisor of a smooth Fano fourfold $X$ of coindex $4$, then a
general element $Y\in|3H|$ has terminal singularities. We know that $Y$ has
canonical singularities. As in Section 2, take a log resolution $f\colon
X_{3}\to X$ of the ideal of the base locus $\operatorname{Bs}|3H|$. We may
assume that $f$ is isomorphic outside $\operatorname{Bs}|3H|$ and it is
obtained by a sequence of blow-ups along smooth centers. We write

$K_{X_{3}}=f^{*}K_{X}+\sum_{i}a_{i}E_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
|f^{*}3H|=|M|+\sum_{i}r_{i}E_{i},$

where all $E_{i}$ are $f$-exceptional prime divisors and $|M|$ is the free
part of $|f^{*}3H|$ and $\sum_{i}r_{i}E_{i}$ is the fixed part of $|f^{*}3H|$.
We may assume that $f(E_{i})\subseteq\operatorname{Bs}|3H|$ for all $i$. All
$a_{i}$ and $r_{i}$ are positive integers.

######  Lemma 3.3.

If a general element $Y$ in $|3H|$ has at worst isolated singularity at $x$
and $\operatorname{mult}_{x}Y\leq 2$, then $Y$ has terminal singularity at
$x$.

###### Proof.

We may assume that $f$ factors through the blow-up of $X$ at $x$ with
exceptional divisor $E_{i_{0}}$. We have $a_{i_{0}}=3$ and $r_{i_{0}}\leq 2$,
so $a_{i_{0}}-r_{i_{0}}>0$. For every $f$-exceptional divisor $E_{i}$ with
$f(E_{i})=\\{x\\}$ but $E_{i}\neq E_{i_{0}}$, we have $a_{i}\geq 4$ since $f$
is a composition of smooth center blow-ups. It is impossible that
$a_{i}=r_{i}\geq 4$ because Proposition 2.3 says that
$a_{i}\geq\frac{4}{3}r_{i}-1>r_{i}$ when $r_{i}\geq 4$. Thus $a_{i}-r_{i}>0$,
and hence, $Y$ has terminal singularity at $x$. ∎

######  Lemma 3.4.

$\dim\operatorname{Bs}|mH|\leq 1$ for any integer $m\geq 2$. In particular,
$\dim\operatorname{Sing}Y\leq 1$.

###### Proof.

Suppose that $\dim\operatorname{Bs}|mH|\geq 2$ for some integer $m\geq 2$. By
Lemma 2.1, we have $\dim\operatorname{Bs}|mH|=2$, so there is an irreducible
surface $S\subseteq\operatorname{Bs}|mH|\subseteq\operatorname{Bs}|H|$. Now,
take two general elements $D_{1},D_{2}\in|H|$. By Proposition 2.3,
$(X,D_{1}+D_{2})$ is an lc pair, and $S$ is an lc center of $(X,D_{1}+D_{2})$.
There is a minimal lc center $C$ of $(X,D_{1}+D_{2})$ contained in $S$. By
[FG, Theorem 1.2], there is an effective divisor $\Gamma$ on $C$ such that

$(K_{X}+D_{1}+D_{2})|_{C}\sim_{\mathbb{Q}}K_{C}+\Gamma$

and $(C,\Gamma)$ is a klt pair. By [K2, Theorem 3.1], $H^{0}(C,mH|_{C})\neq 0$
since $mH-(K_{X}+D_{1}+D_{2})\sim(m-1)H$ is ample. Now, by [Fj, Theorem 2.2],
the restriction map

$H^{0}(X,mH)\longrightarrow H^{0}(S,mH|_{S})$

is surjective. However, $S\subseteq\operatorname{Bs}|mH|$, so this restriction
map is the zero map. We get a contradiction. Therefore,
$\dim\operatorname{Bs}|mH|\leq 1$ for any integer $m\geq 2$. Now, since
$\operatorname{Sing}Y\subseteq\operatorname{Bs}|3H|$, it follows that
$\dim\operatorname{Sing}Y\leq 1$. ∎

###### Proof of Theorem 1.1 (3) for the case $(n,m)=(4,3)$.

We want to prove that a general element $Y\in|3H|$ has terminal singularities.
Note that $H=-K_{X}$ and $\dim\operatorname{Bs}|3H|\leq 1$ by Lemma 3.4. We
know that $h^{0}(X,H)\geq 2$.

First, assume that $H^{4}\geq 2$. Take a general element $Z\in|H|$, which is a
Gorenstein Calabi–Yau threefold with terminal singularities. Suppose that
$\dim\operatorname{Bs}|3H|=1$. Then $Z$ is nonsingular at a general point $x$
in $\operatorname{Bs}|3H|$. By [K1, Theorem 3.1], $|3H|_{Z}|$ is base point
free at $x$. But $x\in\operatorname{Bs}|3H|=\operatorname{Bs}|3H|_{Z}|$, so we
get a contradiction. Thus $\dim\operatorname{Bs}|3H|\leq 0$. Suppose that $Y$
has non-terminal singularity at $x$. By Lemma 3.3,
$\operatorname{mult}_{x}Y\geq 3$. Now, by Proposition 2.3, $(X,Z+Y)$ is an lc
pair. Thus $\operatorname{mult}_{x}Z=1$, so $Z$ is nonsingular at $x$. By [K1,
Theorem 3.1], $3H|_{Z}$ is base point free at $x$, so we get a contradiction
as before. Hence $Y$ has at worst terminal singularities.

Next, assume that $H^{4}=1$. The sectional genus of the polarized pair $(X,H)$
is $g(X,H)=\frac{(K_{X}+3H).H^{3}}{2}+1=2$. By Fujita’s classification [Ft2,
Proposition C], we have $2\leq h^{0}(X,H)\leq 4$, and the following hold:
$\bullet$ $h^{0}(X,H)=4$ $\Leftrightarrow$
$X=X_{10}\subseteq\mathbb{P}(1,1,1,1,2,5)$ is a hypersurface of degree 10.
$\bullet$ $h^{0}(X,H)=3$ $\Leftrightarrow$
$X=X_{6,6}\subseteq\mathbb{P}(1,1,1,2,2,3,3)$ is a complete intersection of
type (6,6).
If $h^{0}(X,H)=4$, then $\operatorname{Bs}|H|=\operatorname{Bs}|3H|=\\{x\\}$
and $|2H|$ is base point free. In this case, $\operatorname{mult}_{x}Y=1$, so
$Y$ is smooth. If $h^{0}(X,H)=3$, then $|3H|$ is base point free so that $Y$
is smooth. We now suppose that $h^{0}(X,H)=2$.111 It is unknown whether there
is a smooth Fano fourfold $X$ with $h^{0}(X,-K_{X})=2$ (cf. [L2, Question 5]).
By Riemann–Roch formula, we have

$h^{0}(X,mH)=\frac{m^{2}(m+1)^{2}}{24}H^{4}+\frac{m(m+1)}{24}H^{2}.c_{2}(X)+1.$

Then $H^{2}.c_{2}(X)=10$, and $h^{0}(X,2H)=5,\leavevmode\nobreak\
h^{0}(X,3H)=12$. Let $Z_{1},Z_{2}\in|H|$ and $W\in|2H|$ be general members.
Then $S:=Z_{1}\cap Z_{2}$ is an irreducible Gorenstein surface with
$K_{S}=H|_{S}$, and $C:=Z_{1}\cap Z_{2}\cap W$ is a Gorenstein curve with
$K_{C}=3H|_{C}$. We have $H^{i}(X,\ell H)=0$ for $1\leq i\leq 3$ and
$\ell\in\mathbb{Z}$, so we get

$\begin{array}[]{l}h^{0}(Z_{1},H|_{Z_{1}})=1,\leavevmode\nobreak\
h^{0}(Z_{1},2H|_{Z_{1}})=3,\leavevmode\nobreak\ h^{0}(Z_{1},3H|_{Z_{1}})=7\\\
h^{0}(S,H|_{S})=0,\leavevmode\nobreak\ h^{0}(S,2H|_{S})=2,\leavevmode\nobreak\
h^{0}(S,3H|_{S})=4\\\ h^{0}(C,H|_{C})=0,\leavevmode\nobreak\
h^{0}(C,2H|_{C})=1,\leavevmode\nobreak\ h^{0}(C,3H|_{C})=4.\end{array}$

Thus $p_{a}(C)=h^{0}(C,3H|_{C})=4$. As $C.H|_{S}=2$, we see that $C$ has at
most two irreducible components. If $C$ is non-reduced, then $C=2H^{\prime}$
for some $H^{\prime}\in|H|_{S}|$. However, since $h^{0}(S,H|_{S})=0$, it
follows that $C$ is reduced.

Suppose that there is an irreducible curve $A$ on $X$ with
$A\subseteq\operatorname{Bs}|2H|\cap\operatorname{Bs}|3H|$. Since
$h^{0}(S,C)=h^{0}(S,2H|_{S})=2$, it follows that $C$ has two irreducible
components. We write $C=A+B$ on $S$. Since the restriction map $H^{0}(X,3H)\to
H^{0}(C,K_{C})$ is surjective, we have $A\subseteq\operatorname{Bs}|K_{C}|$.
Note that

$\deg_{A}(K_{C})=\deg_{B}(K_{C})=3$

since

$\deg_{A}(H|_{C})=A.H=1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ and
}\leavevmode\nobreak\ \leavevmode\nobreak\ \deg_{B}(H|_{C})=B.H=1.$

By [FT, Definition 2.1 and Formula (3)], we have

$4=p_{a}(C)=p_{a}(A)+p_{a}(B)+A\cdot B-1,$

where

$A\cdot B:=\deg_{A}(K_{C})-2p_{a}(A)+2=\deg_{B}(K_{C})-2p_{a}(B)+2.$

If $A\cdot B\geq 2$, then $C$ is numerically 2-connected in the sense of
[CFHR, Definition 3.1]. In this case, by [CFHR, Theorem 3.3], $|K_{C}|$ is
base point free, so we get a contradiction to that
$A\subseteq\operatorname{Bs}|K_{C}|$. Thus $A\cdot B=1$, and then,
$p_{a}(A)=p_{a}(B)=2$. Consider an exact sequence

$0\longrightarrow\omega_{B}\longrightarrow\omega_{C}\longrightarrow\omega_{C}|_{A}\longrightarrow
0,$

which induces the following exact sequence

$0\longrightarrow H^{0}(B,K_{B})\longrightarrow H^{0}(C,K_{C})\longrightarrow
H^{0}(A,K_{C}|_{A})$

Then $h^{0}(A,K_{C}|_{A})\geq 2$, which is a contradiction to that
$A\subseteq\operatorname{Bs}|K_{C}|$. Thus we obtain
$\dim\operatorname{Bs}|2H|\cap\operatorname{Bs}|3H|\leq 0$.

Recall that $Z_{1}$ is a Gorenstein Calabi–Yau threefold with terminal
singularities. Then $\dim\operatorname{Sing}Z_{1}\leq 0$. If $Y$ is singular
along a curve $D$, then $D\subseteq\operatorname{Bs}|3H|$ and
$D\not\subseteq\operatorname{Bs}|2H|$. For a general point $x\in D$, we have
$\operatorname{mult}_{x}|H|=1$ and $\operatorname{mult}_{x}|2H|=0$, so
$\operatorname{mult}_{x}Y=1$ by the upper semicontinuity of the multiplicity.
We get a contradiction because $Y$ is singular at $x$. This means that $Y$
cannot be singular along a curve. By Bertini’s theorem, we see that
$\operatorname{mult}_{x}Y\leq 2$ for all
$x\not\in\operatorname{Bs}|2H|\cup\operatorname{Sing}Z_{1}$. Now, suppose that
$Y$ has an isolated non-terminal singular point $x$. By Lemma 3.3,
$\operatorname{mult}_{x}Y\geq 3$, which implies that
$x\in\operatorname{Bs}|2H|\cup\operatorname{Sing}Z_{1}$. Note that every
general element $Y^{\prime}\in|3H|$ has $\operatorname{mult}_{x}Y^{\prime}\geq
3$. By Proposition 2.3, $(X,Y+Z_{1})$ is an lc pair, so
$\operatorname{mult}_{x}Z_{1}=1$. If $\operatorname{mult}_{x}W=1$, then the
upper semicontinuity of the multiplicity shows that
$\operatorname{mult}_{x}Y^{\prime}\leq 2$, which is a contradiction. Thus
$\operatorname{mult}_{x}W\geq 2$. Now,
$\dim\operatorname{Bs}|2H|\cap\operatorname{Bs}|3H|\leq 0$ implies that $C\cap
Y^{\prime}$ has dimension zero. Notice that $C\cap Y^{\prime}=Z_{1}\cap
Z_{2}\cap W\cap Y^{\prime}$ has length 6, and recall that
$\operatorname{mult}_{x}W\geq 2$ and $\operatorname{mult}_{x}Y^{\prime}\geq
3$. Hence $C\cap Y^{\prime}$ is indeed supported at a single point $x$. Since
$H^{0}(X,3H)\to H^{0}(C,3H|_{C})$ is surjective, every element in $|3H|_{C}|$
has a single support $x$. But this is impossible since $h^{0}(C,3H|_{C})\geq
2$. We can conclude that $Y$ has at worst terminal singularities. ∎

######  Remark 3.5.

Let $X$ be a Fano fourfold of coindex $4$ with fundamental divisor $H=-K_{X}$.
Suppose that $H^{4}\geq 4$, $H^{2}.S\geq 3$ for every irreducible surface $S$,
and $H^{3}.C\geq 2$ for every irreducible curve $C$. Take a general element
$Z\in|H|$. By [K2, Theorem 3.1], $|2H|_{Z}|$ is base point free at every
nonsingular point in $Z$. This implies that $\dim\operatorname{Bs}|2H|\leq 0$.
In this case, we can easily show that a general member in $|2H|$ has terminal
singularities.

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