11institutetext: Course Hero, Inc.

# Question-type Identification for Academic Questions in Online Learning
Platform

Azam Rabiee, Alok Goel, Johnson D’Souza, Saurabh Khanwalkar

###### Abstract

Online learning platforms provide learning materials and answers to students’
academic questions by experts, peers, or systems. This paper explores
question-type identification as a step in content understanding for an online
learning platform. The aim of the question-type identifier is to categorize
question types based on their structure and complexity, using the question
text, subject, and structural features. We have defined twelve question-type
classes, including Multiple-Choice Question (MCQ), essay, and others. We have
compiled an internal dataset of students’ questions and used a combination of
weak-supervision techniques and manual annotation. We then trained a BERT-
based ensemble model on this dataset and evaluated this model on a separate
human-labeled test set. Our experiments yielded an F1-score of 0.94 for MCQ
binary classification and promising results for 12-class multilabel
classification. We deployed the model in our online learning platform as a
crucial enabler for content understanding to enhance the student learning
experience.

###### Keywords:

Question-Type Identification, Content Understanding, Learning Platform, BERT,
Education.

##  1 Introduction

The very idea of adaptive and personalized learning comes from the
understanding that students have individual learning needs based on their
curriculum and level of schooling in addition to their aptitudes, learning
styles, and environments, among other significant factors. Machine learning
practitioners can develop a deep understanding of students’ academic questions
and match them with the optimal content that will allow users to make the best
decisions about how to drive their unique learning journey forward. Over time,
these personalized learning engines will learn how students learn best, what
concepts to focus on, and the temporal relevance of content, eventually
becoming a proactive and assistive educational tool for every learner,
regardless of background, school, or personal learning style.

Figure 1: Question-type identification for a personalized learning engine

A content understanding system is the cornerstone of this ubiquitous and
universal learning experience. It enables personalized learning by attempting
to serve every student’s learning needs across subjects and knowledge levels.
The content understanding system comprises an ensemble of machine learning
models that segment student questions and study resources. One such method of
segmentation is a question-type identifier. Figure 1 shows an example question
flow overview in personalized learning engines. Students ask various types of
questions. As illustrated in Figure 1, the questions first go through a
content understanding system, which includes chunking to separate individual
questions. Then, each question is passed to the _Question Type Identification_
system, which includes feature extraction and model inference. The identifier
provides one or more question-type classes. The identified question-types can
be used in various product use-cases, including accessible and adaptable UI,
search and recommendations, answers and explanations, and optimal expert
selection.

Identifying question types and how they serve different user intents can serve
as a personalization lever for helping students understand the underlying
concepts better. In addition, the personalized learning engine can utilize the
question types for improving question discovery and recommendations. For
example, multiple choice questions (MCQs) may seem more actionable for a
student intending to prepare for an exam [1], while short answer questions
facilitate conceptual learning for completing homework assignments and essays.

The use cases of the identifier are not limited to the above-mentioned items.
Having the feature in our personalized learning engine enables us to built an
infrastructure for smart content understanding with a wide range of
functionalities. Understanding the different types of questions allows us to
help students in the following ways:

  1. 1.

Enables students to access content seamlessly across applications by enabling
user interface (UI) changes based on the question format and length.

  2. 2.

Improves students’ ability to practice by understanding their immediate
learning objective and matching content recommendations accordingly.

  3. 3.

Improves concept explanations by linking students with content from domain-
specific knowledge sources.

  4. 4.

Optimally routes students to expert help based on the complexity and concepts
of their questions.

Our ideas for the identifier have been implemented, evaluated, and deployed in
our online learning platform, which contains course-specific study resources
including user-submitted questions and study guides. The identifier was used
to set different tutor pricing (the cost for tutors to answer a question)
based on question type (MCQ and non-MCQ). A/B testing showed that the pricing
change was successful without any statistically significant difference in
answer rate, tutor response time, or students’ helpfulness ratings. In an
internal analysis of document usage by students on our platform, it was
discovered that documents with answered MCQs are four times more valuable than
documents with answered non-MCQs. This shows another emerging usage of the
question identifier to prioritize answering some question-types over others.
This would enable students to get more answers for the question-types that
they care more about. The identifier can also be used to better understand
students’ and tutors’ preferences in terms of question types, so that students
get question recommendations based on question types, and tutors are assigned
questions based on their past answers and ratings for the question types.
Certain question-types are easier to answer than others; newer tutors can be
assigned easier questions, and students can be recommended questions based on
question type difficulty.

This paper will explore how to build such a question-type identifier with the
following contributions:

  1. 1.

There is no related work to our knowledge that uses machine learning to train
a model for question identification that can be integrated into the real-world
application of the online learning system.

  2. 2.

Lack of enough labeled data is one of the challenges of developing the
identifier. Our experiments rely on data augmentation and a weak supervision
technique (Snorkel [2]).

  3. 3.

We introduced an ensemble and multi-modal architecture for this task, which,
to our knowledge, is novel and gives demonstrably better results. For this
purpose, we extracted handcrafted features alongside BERT model [3] outputs
into another classifier (XGBoost [4]).

The rest of the paper is organized in the following manner. Section 2 reviews
related work in order to clearly illustrate our contribution and development
in existing studies. Section 3 demonstrates the question-type identifier
definitions. We explain our dataset in Section 4. The experimental setup,
model, and results of our identifier are discussed in Section 5. Finally,
Section 6 offers some concluding remarks.

##  2 Related Work

Adaptive and personalized learning has emerged as a fundamental learning
paradigm in the educational technology research community [5]. Machine
learning approaches also enable the personalization of content for a wide
diversity of students [6]. For example, Hypocampus [7] is a personalized
learning engine for children and youth relying on the hypothesis that
introducing time intervals between study sessions using a _repetitive rule_
can improve memory and boost recall. It introduced an adaptive repetitive rule
for knowledge retention in the long-term memory by analyzing students’
learning style, study schedule, knowledge, and performance. Khosravi et al. in
[8] studied another aspect of a personalized learning engine for helping
students to find resources based on their interests and knowledge gaps. These
learning resource recommendations, however, are limited to a few specific
subjects [9] and do not take into account question types.

Question-type identification in contemporary research literature focuses on
the following broad application categories:

  1. 1.

Improved Personalization

  2. 2.

Improved Content Understanding

  3. 3.

Optimal Expert Selection

Sections 2.1, 2.2, and 2.3 review studies aligned with the above three
categories. Then, Section 2.4 surveys technical approaches of question-type
identification.

###  2.1 Improved Personalization

Students’ perception of question types has a significant impact on their
approaches to learning. This hypothesis has been well studied in [10], where
the authors concluded that students preferred short answer questions (SAQs)
for achieving long-term knowledge but preferred the objective format of MCQs
for assessing their knowledge. Similarly, measurement of higher cognitive
skills in undergraduate students has shown that MCQs perform better in student
problem-based learning setups compared to Modified Essay Questions (MEQs)
[11]. Question-types have also been instrumental in measuring the objective
performance of student cohorts. High-achieving and low-achieving students
consistently perform well in SAQs, followed by MCQs and MEQs [12].
Additionally, students with specific learning difficulties find it challenging
to put their thoughts succinctly into essays. Question-types, specifically
MCQs, have been shown to play an impactful role in mitigating those challenges
[13].

###  2.2 Improved Content Understanding

Question classification has been extensively studied in question answering
systems (QASs) as there is a significant link between the said question type
and the expected answer type [14]. These QA systems target question
classification tasks in both the open domain [15, 16, 17, 18] and closed
domain question answering [19, 20, 21]. A lot of the focus here is on finding
answers to questions quickly and efficiently by using extractive and
reasoning-based methods that use semantic markers of the question. Li and Roth
[22] have built popular question classification taxonomies and datasets that
relate to the ”aboutness” of questions, where the emphasis is on uncovering
the meaning behind a question. Our novel work presented in this paper
emphasizes not only the semantic nature of the question text but, more
importantly, the structure of the question. We hypothesize better learning
outcomes for students by focusing on question structure-based applications, as
discussed in Section 2.1. Structure-based question classification has been
partly studied in [18] where in addition to grammar and syntactic features of
the question, the authors have also used question and answer structure
metadata like _Confirmation_ (answers are either Yes or No) and _Choice_
(Similar to MCQ) to inform their downstream User Intent classification tasks
for a QA system.

###  2.3 Optimal Expert Selection

Question classification is capable of helping users of QAS get answers to
their questions promptly by routing user questions to systems with a high
probability of providing the correct answer. These systems could be domain-
specific search engines that use information retrieval (IR) approaches to find
the correct answer [23], systems that use structured knowledge bases [24],
using crowdsourced [25] or expert-sourced answers. Historically, these
approaches excluded question structure-related signals when routing questions
to experts. However, by including the question type as a structural feature,
it becomes increasingly possible to find the optimal expert with the right
skill level to answer a given question.

###  2.4 Technical Background

Question-type identification is a text classification task that was previously
performed by applying typical classifiers such as maximum entropy, decision
trees, or support vector machines to grammatical, domain-expert, and
linguistic features [18, 26, 27]. Recent studies utilize recurrent and
convolutional neural networks (RNNs and CNNs) on the question text embeddings
for the question classification [20, 28]. While ELMo has been shown to
outperform GloVe and Word2Vec for text embedding [20], BERT has outperformed
others in various NLP tasks [3]. In this paper, we investigate BERT-based
architectures in an ensemble model that combins both domain expert and text
embedding features for question-type identification.

##  3 Question-type Identification

Question-type identification aims to classify single- and multi-sentence
students’ questions into predefined question-types, using the question text,
subject, and structural features. We have defined twelve question-types,
summarized in Table 1, appropriate for our content understanding use cases.

Table 1: Summary of question-types Q-Type | Description
---|---
MCQ | Multiple Choice Question, Multi-select
TCQ | Two Choice Question, True/False
FIB | Fill In the Blank
SA | Short Answer, Factoid questions
ES | Essay, Subjective and open-ended questions
CALC | Calculation, Mathematical solution or proof
OR | Ordering
GR | Graphing, Drawing, Sketching
MAT | Matching, Classification
LDGR | Ledger
COD | Coding, Programming, Pseudo-code
TBL | Table completion, Form filling

Regardless of the answer, we have defined the question-types according to
their syntactic patterns, structure, and complexity. For example, the
following question is an essay question-type (ES) that is subjective and open-
ended.

> Discuss the benefits and constraints of different network types and
> standards.

Short Answer (SA) is the question-type that is akin to a factoid question and
has a direct and widely accepted fact-based answer, independent of the length
of the answer. In ordering questions (OR), arranging items in a desired order
is asked, such as:

> Arrange the following groups in order of increasing priority.

Graphing questions (GR) need the sketching or drawing of a diagram, circuit,
flowchart, or histogram. Matching (MAT) is another question type that asks for
the classification of items or the matching of two lists. Accounting questions
that require journalizing transactions and making balance sheets are defined
as ledger question-type (LDGR). Code type (COD) covers all programming,
coding, and debugging questions. Finally, table type (TBL) includes questions
that require completing a table or filling out forms.

####  3.0.1 Multilabeling

In our application, a single question can have multiple question-types
associated with it. For example, if an MCQ includes calculation, it is labeled
as both MCQ and CALC. It is because some question-types, such as MCQ, two
choice question (TCQ) and fill in the blank (FIB), are purely structural and
can be combined with each other and other types. Two examples are given in the
following.

Example 1: MCQ and CALC

> We say that a three-digit number is balanced if the middle digit is the
> arithmetic mean of the other two digits. How many balanced numbers are
> divisible by 18?
> (A) 2
> (B) 3
> (C) 6
> (D) 9
> (E) 18

Example 2: MCQ and FIB

> During the proliferative phase of wound healing, ... build new tissue by
> secreting ... to take the shape of the original tissue.
> A. Fibroblasts, collagen
> B. Platelets, collagen
> C. Mast cells, histamine
> D. Neutrophils, keratin

In addition, some questions are multi-part, with different question types in
each part. For example, the question that asks for a calculation in part (a)
and drawing a graph in part (b) is labeled as CALC and GR. With that in mind,
in this paper, we have explored machine learning models for multilabel
classification with one-versus-all classifiers.

####  3.0.2 Handcrafted Features

We have found that some structural patterns and keywords play an important
role in question-type identification. For example, the presence of an itemized
list (like A., B., C., D.) or some MCQ keywords, such as ”Select all that
apply” and ”What is the correct answer,” are reliable semantic features for
the MCQ type. Similarly, ”Discuss strengths and weaknesses of...” is one of
the phrases related to the ES type. In this paper, we refer to the features
extracted according to the structural patterns and type-based keywords as
_handcrafted_ features. For question-type identification, our classification
approach takes into account handcrafted features as well as question text
embedding. We explored the BERT [3] and DistilBERT [29] models to classify
question text embeddings and the XGBoost algorithm [4] for numeric modality.
We discuss our experiments in Section 5.

##  4 Datasets

Table 2: Summary of Datasets Name | Size | Question Types | Source
---|---|---|---
Test set | 1k | All | Internal, manually annotated
Gold set |  $<$ 1k | All | Internal, manually annotated
Answer-type set | 500k | 10 labels | Internal, labeled according to answers’ format
Silver set | 25k | All | Internal, weak-labeled by Snorkel [2]
Augmented set | 15k | All | Internal, weak-labeled by semantic similarity
TREC [22] | 5.5k | SA | public

We utilized both proprietary data and a public dataset. Our internal datasets
contain students’ questions that were either manually annotated or supervised
using a weak supervision technique. The summary of our data is reported in
Table 2.

Our _test_ and _gold_ sets contain student questions submitted through our
online tutoring platform that are manually annotated. The students’ questions
span over a hundred academic subjects, from economics to literature, biology
to history, accounting to psychology, and other college courses. We didn’t use
students’ personal information in the development of the question-type
identifier.

Our _answer-type_ set contains half a million questions for 14 subjects. This
dataset used 10 prelabeled question types that differ from our type
definitions in Table 1. Instead, these questions were defined based on the
answers’ format, which does not align with our use cases. The questions are
single-labeled and imbalanced in terms of both types and subjects: 44% were
CALC, 41% were SA, and the remaining 15% were distributed across additional
classes. Therefore, this dataset is inappropriate due to its limited and
imbalanced subjects. As a result, we used the dataset for initial fine-tuning
of large models before fine tuning with a superior dataset whose distributions
matched the real-world use cases of content understanding.

We curated an internal dataset for questions over 100 academic subjects and
labeled them using Snorkel [2], as a weak supervision technique. We named the
weakly supervised set of 25k students’ questions as the _silver_ set, as
listed in Table 2. Snorkel uses labeling functions that capture domain
knowledge and resources, which can have unknown accuracy and correlations. The
labeling functions’ outputs are denoised using a machine learning approach to
create much larger training sets much more quickly and at a lower cost than
manual labeling. We compiled a multilabel Snorkel generator using _LabelModel_
in the Snorkel Python library on a development set of 500 manually annotated
questions with 40 labeling functions found during the annotation.

The _augmented_ set was created by weak-labeling questions that are similar to
the gold set using the semantic similarity of the questions’ text embeddings.
We picked 15k unlabeled questions that had 80% to 95% similarity score with
samples in the gold set and labeled them with the same question type as
similar samples. We ensured that there is no overlap between the augmented set
and the test set.

We also included the TREC public dataset [22] in our experiments. TREC is a
dataset for classifying types of questions with 6 semantical classes of
Abbreviation, Entity, Description, Human, Location and Numeric. It provides
questions that mostly consist of one sentence. We categorized the samples in
the dataset as SA in our application since they are all factoid questions that
imply short answers. We empirically found it useful because we did not have
enough labeling function in the Snorkel weak-supervision for the SA class.

##  5 Experiments

We explain the experimental setup, scope, and network architecture in Section
5.1. In Section 5.2, we present the first set of experiments for MCQ
identification. Then, Section 5.3 discusses a multilabel classifier in a
combined model for all twelve question types.

###  5.1 Setup

The scope of our experiments is limited to English language questions and
assumes that the identifier acts on a single question. Thus, student requests
containing multiple questions should be broken down into single questions. An
example of the chunking approach is discussed in [30].

####  5.1.1 BERT-based Ensemble Model

As we mentioned earlier, we explore BERT-based ensemble models in this study.
Figure 2 shows an example of a model that cascades a BERT-based architecture
with another classifier. The former classifier provides the class
probabilities according to the embedding of the question text. Later, the
predicted class probabilities and a handcrafted feature vector are fed to the
second classifier for the final decision.

Figure 2: A BERT-based ensemble model for question-type identification

Manual annotation of our gold training set led to the design of the
handcrafted features discussed earlier in Section 3. Through the annotation
process for making the ground-truth labeled dataset, we have created lists of
numeric features related to each question-type. For example, a feature called
_count_answer_option_ reflects the number of MCQ answer option keywords in the
question text. The answer option keyword list contains 18 phrases such as
”None of the above” or ”All of the above.” In total, we made a handcrafted
13-D feature vector for the MCQ classifier and a 175-D feature vector for the
multilabel classification.

We have examined both BERT and DistilBERT in combination with the well-known
XGBoost algorithm. We also compared the results with a Multimodal Transformer
[31], which similarly relies on a joint training procedure on the two
modalities. The multimodal architecture utilizes BERT and an additional neural
network (NN) architecture to support text as well as numerical and categorical
features.

###  5.2 MCQ Identifier

The first experiment focused only on identifying MCQs using BERT-based models.
We consider MCQ and non-MCQ the highest order question-type categories that
have an imbalanced distribution in our online tutoring platform. If a question
has multiple types, and at least one of them is an MCQ, it will be considered
an MCQ. For example, even if an MCQ requires calculations to find the correct
answer option, it is still considered an MCQ despite CALC itself being a
question-type.

####  5.2.1 Model Training

We started both BERT and DistilBERT networks from HuggingFace BERT Sequence
Classification pre-trained models [32]. Then, we fine-tuned the models on a
training set consisting of 400k samples of answer-type data and less than 1k
manually annotated samples. Training the BERT model on an AWS SageMaker GPU
(ml.p3.2xlarge) took around 14 hours with a batch size of 8 per device. Both
models were trained on the first 512 word-piece tokens of the question text,
converted to lowercase, and tokenized with the HuggingFace provided library.

The MCQ probabilities predicted by the BERT and DistilBERT models were used as
an input feature, along with the 13-D handcrafted feature vector, to train an
XGBoost classifier model. The same gold and answer-type sets are used in
training the XGBoost classifier. _Hyperopt_ library for Bayesian optimization
[33] was utilized for hyperparameter tuning with Recall@Precision = 0.95 as
the optimizer loss function. Finally, the classification probability predicted
by the XGBoost model was compared against a threshold to provide MCQ or non-
MCQ classification.

####  5.2.2 Results

Table 3 reports the results in terms of accuracy (ACC), F0.5 (to weight
precision higher than recall), weighted F1 (WF1, to consider each class’s
support), and recalls at three different precisions (R@95, R@90, and R@85).
F0.5 is calculated only for the MCQ class. For ACC, F0.5, and WF1, the
classification threshold is 0.5; so if the prediction probability is higher
than 0.5, the sample is considered MCQ. For R@N, we select the best threshold
according to the PR curves. Figure 3 compares PR curves for BERT, DistilBERT,
MultiModal, and XGB_DistilBERT in detail.

Table 3: Results of MCQ identifier on the test set Models | Acc | F0.5 | WF1 | R@95 | R@90 | R@85
---|---|---|---|---|---|---
BERT | 93% | 0.81 | 0.92 | 0.43 | 0.63 | 0.65
DistilBERT | 94% | 0.84 | 0.93 | 0.38 | 0.69 | 0.72
XGB with BERT | 93% | 0.78 | 0.93 | 0.47 | 0.65 | 0.74
XGB with DistilBERT | 94% | 0.85 | 0.94 | 0.62 | 0.65 | 0.74
Multimodal Transformer | 93% | 0.80 | 0.92 | 0.55 | 0.57 | 0.64
Figure 3: Precision-Recall (PR) curve comparison

As shown in Table 3 and Figure 3, empirically, we found that DistilBert often
performed better than BERT; this could be because the larger BERT models
require more samples to train or because they would overfit the train set. The
train set performance was significantly better than the validation and test
set. To avoid overfitting, we leveraged different training
parameters—specifically, changing the weight decay (L2 regularization) [34]
found that some of the higher regularization values do improve test set
scores.

On an AWS SageMaker GPU (ml.p3.8xlarge), prediction latency for each student
question is roughly 100 ms for the BERT model and even slower for the
multimodal architecture. The prediction for the DistilBERT model takes half
the time with the same settings. The final ensemble XGB_DistilBERT model
predicts the question-type almost as fast as the DistilBERT as the added
XGBoost architecture takes only 0.01 ms to generate predictions.

In addition to the slow inference time, our experiments with the multimodal
architecture were dissatisfying, and we ran into memory issues using the
available library. However, the NN architecture for numerical and categorical
features is not customizable to support different architectures. We only had
control over how to combine scores from the BERT model and the added NN nodes.

####  5.2.3 Error Analysis on MCQ Classifier

To determine the reasons for low _Recall_ in our experiments, we manually
analyzed the false-negative samples in the test set. We discovered that the
question text is noisy for roughly 50% of false-negative errors, meaning that
either the question text is incomplete or the answer option symbol is missing.
In addition, for around 70% of false negatives, the question text does not
contain any MCQ-related keyword. Here are two examples where our model could
not predict as MCQ:

Example 1: non-MCQ predicted

> If a country has a birthrate of 13 and a death rate of 9, what is the %
> increase?
> Options: .2 .4 2 4

Example 2: non-MCQ predicted

> Systems of equations with different slopes and different y-intercepts have
> more than one solution. (5 points) Always Sometimes Never

###  5.3 Multilabel Classifier

We trained a BERT-based ensemble model in a multilabel classification scheme
for all twelve question-types. Figure 4 depicts the distribution of question-
types asked by students on our online tutoring platform, which is imbalanced
and makes classification difficult. The figure shows that they ask more CALC,
ES, and MCQ than ORD, MAT, and LDGR questions.

Figure 4: Distribution of question-types asked by students in our online
tutoring platform

####  5.3.1 Model Training

Similar to the previous experiments, we trained the BERT model from
HuggingFace BERT Sequence Classification pre-trained models [32] on a training
set consisting of a golden set of 500 samples, answer-type, and TREC datasets
[22]. Later, we fine-tuned the trained BERT model on the same gold set, the
silver set, and the augmented gold set.

To build the handcraft and numeric features for this experiment, we utilized
the question subject and text because some question-types are more prevalent
in specific educational domains. For example, most questions asked in the
”Python programming” subject were identified as COD.

We trained the XGBoost model on the handcrafted features extracted from the
same gold, answer-type, silver, and augmented sets. The TREC dataset is not
used to train XGBoost since all samples are categorized as SA and this
question-type is hardly identified with structural patterns or keywords for
the handcrafted features.

To get a multilabel prediction from the XGBoost model, we ran a one-vs-all
experiment using the scikit-learn Python package [35]. If no label was
selected, we picked the label with the highest probability for the multilabel
prediction. Similar to the previous experiments, we performed hyper-parameter
tuning for the XGBoost model with F1-score as the optimizer loss function
using _Hyperopt_ library [33]. The final inference output was obtained by
averaging the probabilities of the BERT and XGBoost models.

Training the BERT model on an AWS SageMaker GPU (ml.p3.2xlarge) took around 14
hours for the first fine-tuning and around 1.5 hours for the final retuning,
with batch-size equals 8 per device. The prediction latency for the multilabel
classifier with the ensembled XGBoost and retuned BERT model took roughly 100
ms on an AWS SageMaker GPU (ml.p3.8xlarge).

####  5.3.2 Results

Table 4: Test results of multilabel classification for twelve question types Models | WF0.5 | WF1 | F1_CALC | F1_ES | F1_MCQ
---|---|---|---|---|---
Snorkel Generator | 0.67 | 0.48 | 0.76 | 0.26 | 0.58
BERT 1 (Gold, Answer-type, TREC) | 0.66 | 0.51 | 0.67 | 0.41 | 0.79
BERT 2 (Gold, Silver, Augmented) | 0.69 | 0.63 | 0.80 | 0.75 | 0.54
XGB | 0.66 | 0.61 | 0.67 | 0.71 | 0.57
Ensembled BERT 2 and XGB | 0.71 | 0.67 | 0.84 | 0.81 | 0.54

Table 4 reports the experimental results in terms of weighted F0.5 and
weighted F1 on the test set. We compared the final ensemble model against each
individual model and the Snorkel Generator as a baseline model. We used the
same Snorkel generator to create the silver set for our training data. As
shown in the table, the final combined model improves the performance of each
individual model. Furthermore, Table 4 lists F1-scores for three top-frequent
question-types. We have found that question types that are frequent in our
dataset (CALC, ES, and MCQ) show high classification scores; but results are
not yet satisfactory for other question-types.

BERT 1 in Table 4 refers to the first fine-tuning on gold, answer-type, and
TREC sets. BERT 2 refers to the second fine-tuning on gold, silver, and
augmented sets. Since our answer-type dataset contains only 10 classes
(including MCQ), the overall performance of BERT 2 is higher than BERT 1.
However, BERT 1 performed significantly better in MCQ.

####  5.3.3 Error Analysis on Multilabel Classifier

By analyzing the results, we have found that our multilabel classifier has
difficulties classifying ES and SA, which is a complex case even for humans.
For example, the following question, which is defined as ES, is predicted as
SA.

> How do we ensure we use the correct tRNA during translation?

Another example of misclassification is the following question that is asked
in the MySQL course and is labeled as COD but is predicted as SA.

> Return the id, host name, and host id of the accommodation that has the
> highest "host_total_listings_count". Return only the first result.

We believe that another reason for the lower performance in some classes is
because of the limited manual annotations and the imbalance in question-types.
Although the results are promising as a first step in building the multilabel
classifier, we hope to continue improving the model for all 12 types.

##  6 Conclusions

We hypothesized that both the semantics and structure of a question are vital
to gaining a deeper understanding of its type which makes it a multimodal
classification problem. We investigated multiple ensemble models for MCQ
identification and multilabel classification across 12 question types. We
examined BERT and DistilBERT models for question-type classification on the
question text embeddings and an XGBoost classifier to decide the numeric and
handcrafted features extracted from the question text and subject. For a
binary classification of MCQ versus non-MCQ, we also compared our results with
multimodal transformers. To train the models, we used both internal and public
datasets that were labeled manually or using weak supervision techniques. Our
ensemble model improved the performance of each classifier independently. On a
manually annotated test set, MCQ and 12-label classifiers showed an F1-score
of 0.94 and 0.67, respectively.

As an emerging feature, we deployed the question type identifier in our
tutoring platform to assist students across numerous academic domains. The
initial analysis and A/B testing convinced us of the usefulness of the
identifier in improving content understanding and optimizing the routing of
questions to the proper experts who can best support students when they are
stuck. The smart content understanding enables a personalized learning engine
to leverage MCQ, FIB, and TCQ questions to assist students with better
practice. It can also utilize ES, SA, CALC, and other long-form question types
to help students better understand critical concepts and topics.

These approaches for data generation and classification using both the
semantics and structure of text can be used in various multimodal
classification tasks, including query intent classification, document type
classification, and other similar text classification problems.

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## Authors

Azam Rabiee received her PhD in Computer Engineering from Islamic Azad
University, and she worked on Emotional Chatbots at KAIST, South Korea as a
postdoc researcher. Currently, she is a Senior Machine Learning Engineer in
Course Hero Inc. and her area of interests include Natural Language
Processing, Speech Processing, and Time-series Predictions.

Alok Goel received his Masters in Computer Science from North Carolina State
University. He worked at IBM Watson as an Advisory Software Engineer on
Knowledge Graphs and Discovery systems. Currently, he is a Senior Machine
Learning Engineer at Course Hero. His interests include Natural Language
Processing, Content Understanding, and Question Answering.

Johnson D’Souza has over 16 years of industry experience and specializes in
Natural Language Processing and Machine Learning for EdTech, Spoken Language
Processing Systems and Consumer Electronics domains. He currently leads NLP
initiatives at Course Hero with the goal of building delightful student
experiences. Prior to Course Hero, Johnson worked at Verizon, DirecTV and
TiVo.

Saurabh Khanwalkar leads Course Hero’s Machine Learning, Natural Language
Processing, Search, and Recommendations teams and is responsible for shipping
intelligent and personalized learning experiences to millions of students and
educators. He has over 18 years of R&D and leadership experience in machine
learning products in diverse industries such as DARPA research, social media
analytics, consumer electronics, healthtech, and edtech. Saurabh has technical
publications and patents in Speech Processing, Natural Language Processing,
and Information Retrieval and is on the review committee for NAACL, COLING,
and other academic conferences offering industry track Machine Learning
papers.


# Geometric invariants of $K3$ surfaces with purely non-symplectic
automorphisms

Dominik Burek  Instytut Matematyki
Wydział Matematyki ul. Łojasiewicza 6
30-348 Kraków
Poland<EMAIL_ADDRESS>
###### Abstract.

We give new relations between geometric invariants of $K3$ surfaces with
purely non-symplectic automorphisms of order 4 and 6. Our approach is based on
a comparison of two methods of computation of formulas for the Euler
characteristic of higher dimensional generalized Borcea-Vosin Calabi-Yau
manifolds constructed in [Bur21].

###### Key words and phrases:

Calabi–Yau manifolds, Borcea-Voisin construction, Chen-Ruan cohomology.

###### 2010 Mathematics Subject Classification:

Primary 14J32; Secondary 14J40, 14E15

##  1\. Introduction

Automorphisms of a $K3$ surfaces over $\mathbb{C}$ were intensively studied in
the recent years. Especially important are automorphisms of order $d$ which
acts on the period as a multiplication by a primitive root of unity of degree
$d$ i.e. (purely) non-symplectic automorphisms of order $d$.

Non-symplectic automorphisms have been investigated in several papers. A non-
symplectic automorphisms of prime order have been classified by M. Artebani,
A. Sarti, S. Taki in [AST11]. Some authors have started to investigate non-
symplectic automorphisms of composite order: D. Al Tabbaa and A. Sarti in
[TS17] (order 8), J. Dillies in [Dil12] (order 6), A. Sarti and A. Garbagnati
in [GS13] (order $2p,$ where $p$ is a prime) and J. Keum in [Keu15] (order
66). Also, S. Brandhorst in [Bra19] pursued the question when an automorphism
determines a $K3$ surface up to an isomorphism.

The study of purely non-symplectic automorphism on $K3$ surfaces involves,
among other things, the description of the fixed points locus of this
automorphism, this set is a disjoint union of finitely many isolated points
and curves. This leads to several very important numerical invariants. Some of
them are related i.e. topological or holomorphic Leftchetz’s formulas give
precise relation.

$K3$ surfaces with purely non-symplectic automorphism have been used in
several constructions of Calabi–Yau 3-folds. One of them was the construction
given independently by C. Borcea ([Bor97]) and C. Voisin ([Voi93]). They
constructed families of Calabi-Yau 3-folds using a non-symplectic involutions
of $K3$ surfaces and elliptic curves. A. Cattaneo and A. Garbagnati in [CG16]
observed that the Borcea-Voisin construction can be generalized by allowing
non-symplectic automorphism of orders $3,$ $4$ or $6.$ On the other hand S.
Cynk and K. Hulek in [CH07] studied where one can extend the classical
construction of Borcea-Voisin as far as taking the product of elliptic curves
and resolving the quotient in a crepant way.

In [Bur21] we investigated higher dimensional construction of Calabi-Yau
manifolds which generalizes the classical Borcea-Vosin construction by merging
ideas of Cynk-Hulek and Cattaneo-Garbagnati. More precisely: Let $E_{d}$ be an
elliptic curve with a not period preserving automorphism of order
$d\in\\{2,3,4,6\\}$ and $S_{d}$ be a $K3$ surface with purely non-symplectic
automorphism of order $d.$ Consider the following group

$G_{d,n}:=\\{(m_{1},m_{2},\ldots,m_{n})\in\mathbb{Z}_{d}^{n}\colon
m_{1}+m_{2}+\ldots+m_{n}=0\\}\simeq\mathbb{Z}_{d}^{n-1}$

acting naturally on $S_{d}\times E_{d}^{n-1}$. Then the quotient variety
$(S_{d}\times E_{d}^{n-1})/G_{d,n}$ admits a crepant resolution of
singularities $Y_{d,n}$. Consequently $Y_{d,n}$ is an $(n+1)$-dimensional
Calabi-Yau variety.

In this note we shall derive new relations among invariants of $K3$ surfaces
with purely non-symplectic automorphisms of order 4 and 6. Our approach is
based on varieties $Y_{d,n}$ constructed in [Bur21]. We shall compute Euler
characteristic of $Y_{d,n}$ in two different ways: by using Hodge numbers of
$Y_{d,n}$ computed in [Bur21] and by using orbifold Eulera Characteristic.
Since our approach is indirect, the relations have no clear geometrical
origin.

The paper is organised as follows. In the sections 2 we review ”stringy” Euler
characteristic formula and both topological and holomorphic Lefschetz’s
numbers. In the section 3 we recall classical Borcea-Voisin construction and
its generalizations. The section 4 is devoted to collect invariant of $K3$
surfaces with purely non-symplectic automorphisms. In the section 5 we shall
compare two methods of computation of formulas for the Euler characteristic of
$Y_{d,n}.$

### Acknowledgments

I would like to thank Sławomir Cynk for helpful suggestions and comments. The
author is supported by the National Science Center of Poland grant no.
2019/33/N/ST1/01502 and National Science Center of Poland grant no.
2020/36/T/ST1/00265.

##  2\. Preliminaries

###  2.1. Stringy Euler Characteristic

One of the most important cohomological invariant of the finite quotient of
compact manifolds conjectured by physicists is physicists (stringy) Euler
characteristic.

Let $G$ be a finite group acting on a compact, smooth differentiable manifold
$X$. For any $g\in G$ let

$X^{g}:=\left\\{x\in X\colon g(x)=x\right\\}.$

In [Dix+85, Dix+86] L. Dixon, J. Harvey, C. Vafa and E. Witten proposed the
following _orbifold Euler number_ :

(2.1) $e_{\operatorname{orb}}\left({{}^{\textstyle X}\big{/}_{\textstyle
G}}\right):=\dfrac{1}{\\#G}\sum_{\begin{subarray}{c}(g,h)\in G\times G\\\
gh=hg\end{subarray}}e(X^{g}\cap X^{h}).$

According to [HH90] the formula 2.1 can be rewritten as

(2.2) $e_{\operatorname{orb}}\left({{}^{\textstyle X}\big{/}_{\textstyle
G}}\right):=\sum_{[g]\in\textrm{Conj}(G)}e\left({{}^{\textstyle
X^{g}}\big{/}_{\textstyle\textup{C}(g)}}\right),$

where $\textrm{Conj}(G)$ is the set of conjugacy classes of $G$ (we choose a
representative $g$ of each conjugacy class) and $\textup{C}(g)$ is the
centralizer of $g$.

It is expected that orbifold Euler characteristic
$e_{\operatorname{orb}}\left({{}^{\textstyle X}\big{/}_{\textstyle G}}\right)$
coincides with the topological Euler characteristic
$e\Big{(}\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle{{}^{\textstyle
X}\big{/}_{\textstyle G}}$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}\Big{)}$ of any crepant resolution of
${{}^{\textstyle X}\big{/}_{\textstyle G}}.$ In fact:

######  2.1 Theorem ([Roa89] (abelian case), [Bat99]).

Let $G\subset\operatorname{SL}_{n}(\mathbb{C})$ be a finite group acting on
smooth algebraic variety $X$. If
$\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle{{}^{\textstyle
X}\big{/}_{\textstyle G}}$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}$ is a crepant resolution of variety
$X/G,$ then the following equality holds

$e\Big{(}\ThisStyle{\stackengine{-.1\LMpt}{\SavedStyle{{}^{\textstyle
X}\big{/}_{\textstyle G}}}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}\Big{)}=e_{\operatorname{orb}}\left({{}^{\textstyle
X}\big{/}_{\textstyle G}}\right).$

Some explicit examples of a group $G$ and possible calculations were studied
by T. Höfer and F. Hirzebruch in [HH90].

###  2.2. Topological and holomorphic Lefchetz’s numbers

In the following section we recall basic information about topological and
holomorphic Lefschetz numbers. We refer to [GH94] and [Pet86].

Let $X$ be a compact oriented manifold and $f\colon X\to X$ any continuous
map. The intersection number of the graph $\alpha_{f}$ of $f$ and diagonal
$\Delta_{X}$ in $X\times X$ is equal to the so called topological Lefschetz
number given by the following formula:

$\mathcal{L}_{\operatorname{top}}(f):=\sum_{q\geq
0}(-1)^{q}\operatorname{tr}\left(f^{*}\colon H^{q}(X)\to H^{q}(X)\right).$

######  2.2 Theorem ([Uen76]).

Let $X$ be a compact complex manifold and $f\colon X\to X$ a holomorphic map
with non-degenerate fixed locus then

(2.3)
$\mathcal{L}_{\operatorname{top}}(f)=\sum_{C\in\operatorname{Fix}(f)}e(C),$

where the sum is taken over connected components of $\operatorname{Fix}(f).$

Formula 2.3 holds in particular when $f^{n}=\operatorname{id}$ for some $n\geq
1$ (see [Car57]).

The map $f$ acts not only on the de Rham cohomology of $X$ but on the
Dolbeault cohomology too. Therefore there is a hope that the action of $f$ on
$H^{*,*}(X)$ will be reflected in local properties of $f$ around the fixed
point locus.

######  2.3 Definition.

Let $X$ be a compact complex manifold and $f\colon X\to X$ a holomorphic map.
The number

$\mathcal{L}_{\operatorname{hol}}(f):=\sum_{q\geq
0}(-1)^{q}\operatorname{tr}\left(f^{*}|H^{0,q}(X)\right)$

is called holomorphic Lefschetz number.

According to [AS68a] and [AS68] the holomorphic Lefschetz number can be
computed in different way. For our purpose assume that $X$ has dimension $2$
and let $G$ be a finite group of automorphisms of $X.$

######  2.4 Theorem ([AS68a], [AS68]).

For any $g\in G$ the following formula holds

$\mathcal{L}_{\operatorname{hol}}(g)=\sum_{j\in J}a(P_{j})+\sum_{k\in
K}b(C_{k}),$

where sets $\\{P_{j}\\}_{j\in J}$ and $\\{C_{k}\\}_{k\in K}$ denote fixed
points and fixed curves in $\operatorname{Fix}(g)$ and

$a(P):=\frac{1}{\det({1-g}|\mathcal{T}_{g})},\quad\textup{where
}\mathcal{T}_{P}\textup{ is a tangent space at $P$,}$
$b(C):=\frac{1-g(C)}{1-\zeta}-\frac{\zeta
C^{2}}{(1-\zeta)^{2}},\quad\textup{where }\zeta\textup{ is an eigenvalue of
$g$ on the normal bundle of $\operatorname{Fix}(g).$}$

##  3\. Borcea-Voisin construction

###  3.1. Classical Borcea-Voisin construction

One of the many reasons behind the interest in non-symplectic automorphisms of
$K3$ surfaces is the mirror symmetry construction of C. Borcea ([Bor97]) and
C. Voisin ([Voi93]). They independently constructed a family of Calabi-Yau
threefolds using a non-symplectic involutions of $K3$ surfaces and elliptic
curves. Moreover C. Voisin gave a construction of explicit mirror maps.

######  3.1 Theorem ([Bor97, Voi93]).

Let $E$ be an elliptic curve with an involution $\alpha_{E}$ which does not
preserve $\omega_{E}$. Let $S$ be a $K3$ surface with a non-symplectic
involution $\alpha_{S_{6}}$. Then any crepant resolution of the variety
${{}^{\textstyle E\times
S}\big{/}_{\textstyle\alpha_{E}\times\alpha_{S_{6}}}}$ is a Calabi-Yau
manifold.

###  3.2. Generalized Borcea-Voisin construction

In [CG16] A. Cattaneo and A. Garbagnati generalized the Borcea-Voisin
construction allowing a non-symplectic automorphisms of a $K3$ surfaces of
higher degrees i.e. $3,$ $4$ and $6.$

######  3.2 Theorem ([CG16]).

Let $S_{d}$ be a $K3$ surface admitting a purely non-symplectic automorphism
$\alpha_{S_{d}}$ of order $d=3,4,6.$ Let $E_{d}$ be an elliptic curve
admitting an automorphism $\alpha_{E_{d}}$ of order $d.$ Then ${{}^{\textstyle
S_{d}\times
E_{d}}\big{/}_{\textstyle\alpha_{S_{d}}\times\alpha_{E_{d}}^{n-1}}},$ is a
singular variety which admits a crepant resolution of singularities
$\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle{{}^{\textstyle S_{d}\times
E_{d}}\big{/}_{\textstyle\alpha_{S_{d}}\times\alpha_{E_{d}}^{d-1}}}$}{\stretchto{\scaleto{\SavedStyle\mkern-1.5mu\AC}{.5150}}{.5}}{O}{c}{F}{T}{S}}$.
In particular $\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle{{}^{\textstyle
S_{d}\times
E_{d}}\big{/}_{\textstyle\alpha_{S_{d}}\times\alpha_{E_{d}}^{d-1}}}$}{\stretchto{\scaleto{\SavedStyle\mkern-1.5mu\AC}{.5150}}{.5}}{O}{c}{F}{T}{S}}$
is a Calabi-Yau threefold.

######  3.3 Definition.

A crepant resolution of ${{}^{\textstyle S_{d}\times
E_{d}}\big{/}_{\textstyle\alpha_{S_{d}}\times\alpha_{E_{d}}^{d-1}}}\simeq{{}^{\textstyle
S_{d}\times E_{d}}\big{/}_{\textstyle\mathbb{Z}_{d}}}$ is called a Calabi-Yau
3-fold of Borcea-Voisin type.

In [Bur20] we investigated higher dimensional construction of Calabi-Yau
manifolds which generalizes the classical Borcea-Vosin construction by merging
ideas of Cynk-Hulek ([CH07]) and Cattaneo-Garbagnati. Precisely, we proved the
following:

######  3.4 Theorem ([Bur20]).

Let $E_{d}$ be an elliptic curve with an automorphism of order $d$ and $S_{d}$
be a $K3$ surface with purely non-symplectic automorphism of order
$d\in\\{2,3,4,6\\}.$ Consider the following group

$G_{d,n}:=\\{(m_{1},m_{2},\ldots,m_{n})\in\mathbb{Z}_{d}^{n}\colon
m_{1}+m_{2}+\ldots+m_{n}=0\\}\simeq\mathbb{Z}_{d}^{n-1}$

acting naturally on $S_{d}\times E_{d}^{n-1}$. Then the quotient variety
${{}^{\textstyle S_{d}\times E_{d}^{n-1}}\big{/}_{\textstyle G_{d,n}}}$ admits
a crepant resolution of singularities $Y_{d,n}$. Consequently $Y_{d,n}$ is an
$(n+1)$-dimensional Calabi-Yau variety.

######  3.5 Definition.

We call the resulting variety $Y_{d,n}$ an $(n+1)$-dimensional Calabi-Yau
manifold of Borcea-Voisin type.

##  4\. Invariants of $K3$ surfaces with non-symplectic automorphism of order
4 and 6

###  4.1. $K3$ surfaces with non-symplectic automorphism of order 6

We shall keep the following notation from [CG16]:

  * $r:=$

$\dim H^{2}(S,\mathbb{C})^{\alpha_{S_{6}}}$,

  * $m:=$

$\dim H^{2}(S,\mathbb{C})_{\zeta_{6}^{i}}$ for $1\leq i\leq 5$,

  * $l:=$

number of curves fixed by $\alpha_{S_{6}},$

  * $k:=$

number of curves fixed by $\alpha_{S_{6}}^{2},$

  * $N:=$

number of curves fixed by $\alpha_{S_{6}}^{3},$

  * $p_{(2,5)}+p_{(3,4)}:=$

number of isolated points fixed by $\alpha_{S_{6}}$ of type $(2,5)$ and
$(3,4)$ i.e. the action of $\alpha_{S_{6}}$ near these point linearises to
respectively $\operatorname{diag}(\zeta_{6}^{2},\zeta_{6}^{5})$ and
$\operatorname{diag}(\zeta_{6}^{3},\zeta_{6}^{4}),$

  * $n:=$

number of isolated points fixed by $\alpha_{S_{6}}^{2},$

  * $2n^{\prime}:=$

number of isolated points fixed by $\alpha_{S_{6}}^{2}$ and switched by
$\alpha_{S_{6}},$

  * $F_{1},F_{2}:=$

the curves with the highest genus in the fixed locus of $\alpha_{S_{6}}^{3}.$

  * $a:=$

number of triples $(A,A^{\prime},A^{\prime\prime})$ of curves fixed by
$\alpha_{S_{6}}^{3}$ such that $\alpha_{S_{6}}(A)=A^{\prime}$ and
$\alpha_{S_{6}}(A^{\prime})=A^{\prime\prime},$

  * $b:=$

number of pairs $(B,B^{\prime})$ of curves fixed by $\alpha_{S_{6}}^{2}$ such
that $\alpha_{S_{6}}(B)=B^{\prime},$

  * $D:=$

the curve with the highest genus in the fixed locus of $\alpha_{S_{6}},$

  * $G:=$

the curve with the highest genus in the fixed locus of $\alpha_{S_{6}}^{2}.$

######  4.1 Theorem ([Bur20]).

The Hodge number $h^{p,q}$ of the manifold $Y_{6,n}$ is equal to the
coefficient in $X^{p}Y^{q}$ of the following polynomial

$\displaystyle\Bigg{(}(XY)^{2}+r\cdot
XY+1+\sqrt[6]{XY}\cdot\Big{(}\ell+p_{(2,5)}\cdot XY+p_{(3,4)}\cdot
XY+g(D)\cdot(X+Y)+\ell\cdot XY\Big{)}+$
$\displaystyle+\sqrt[6]{(XY)^{2}}\cdot\Big{(}k-b+n^{\prime}\cdot
XY+p_{(2,5)}\cdot XY+g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)(X+Y)+(k-b)\cdot XY\Big{)}+$
$\displaystyle+\sqrt[6]{(XY)^{3}}\cdot\Big{(}N-2a+\left(g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\right)\cdot(X+Y)+(N-2a)\cdot
XY\Big{)}+$
$\displaystyle+\sqrt[6]{(XY)^{4}}\cdot\Big{(}k-b+n^{\prime}+p_{(2,5)}+g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)(X+Y)+(k-b)\cdot XY\Big{)}+$
$\displaystyle+\sqrt[6]{(XY)^{5}}\cdot\Big{(}\ell+p_{(2,5)}+p_{(3,4)}+g(D)\cdot(X+Y)+\ell\cdot
XY\Big{)}\Bigg{)}\cdot$
$\displaystyle\cdot\Big{(}1+XY+\sqrt[6]{XY}+2\sqrt[6]{(XY)^{2}}+2\sqrt[6]{(XY)^{3}}+2\sqrt[6]{(XY)^{4}}+\sqrt[6]{(XY)^{5}}\Big{)}^{n-1}+$
$\displaystyle+\Big{(}X^{2}+(m-1)\cdot XY\Big{)}\cdot
X^{n-1}+\Bigg{(}\alpha\cdot
XY+\sqrt[6]{(XY)^{3}}\cdot\Big{(}a+\frac{1}{2}\Big{(}g(F_{1})+g(F_{2})-$
$\displaystyle-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\Big{)}\cdot(X+Y)+a\cdot
XY\Big{)}\Bigg{)}\cdot\Big{(}\sqrt[6]{(XY)^{3}}\Big{)}^{n-1}+$
$\displaystyle+\Bigg{(}\beta\cdot
XY+\sqrt[6]{(XY)^{2}}\cdot\Big{(}b+n^{\prime}\cdot
XY+\Big{(}g(G)-g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\Big{)}\cdot(X+Y)+b\cdot
XY\Big{)}+$
$\displaystyle+\sqrt[6]{(XY)^{4}}\cdot\Big{(}b+n^{\prime}+\Big{(}g(G)-g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\Big{)}\cdot(X+Y)+b\cdot
XY\Big{)}\Bigg{)}\cdot\Big{(}\sqrt[6]{(XY)^{2}}+\sqrt[6]{(XY)^{4}}\Big{)}^{n-1}+$
$\displaystyle+\Bigg{(}\alpha\cdot
XY+\sqrt[6]{(XY)^{3}}\cdot\Big{(}a+\frac{1}{2}\Big{(}g(F_{1})+g(F_{2})-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\Big{)}\cdot(X+Y)+$
$\displaystyle+a\cdot
XY\Big{)}\Bigg{)}\cdot\Big{(}\sqrt[6]{(XY)^{3}}\Big{)}^{n-1}+\Big{(}Y^{2}+(m-1)\cdot
XY\Big{)}\cdot Y^{n-1}.$

From this formula we can compute the Euler characteristic by evaluating the
above formula at $6$-th roots of unity. Therefore we can compute this as a sum
of six geometric sequences, hence we expect that there exists recurrence of
degree at most 6. In fact two of them coincides and finally we obtain
recurrence of order 4.

######  4.2 Corollary.

The Euler characteristic of $Y_{6,n}$ is equal to $a_{n-1},$ where

$\begin{cases}a_{n}=12a_{n-1}-19a_{n-2}-12a_{n-3}+20a_{n-4},\\\
a_{0}=e(Y_{6,1})=24,\\\ \begin{aligned}
a_{1}=e(Y_{6,2})&=4+2r-2m+4l+6p_{(2,5)}+2p_{(3,4)}-4g(D)+8k-4b+6w-\\\
&-4g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-4g(G)+4N-4a-2g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-2g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-2g(F_{1})-2g(F_{2}),\end{aligned}\\\
\begin{aligned}
a_{2}=e(Y_{6,3})&=80+16r-2m-2\alpha+64l+66p_{(2,5)}+32p_{(3,4)}-64g(D)+68k-64b+36w-\\\
&-64g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-4g(G)+32N-64a-32g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-32g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right),\end{aligned}\\\
\begin{aligned}
a_{3}=e(Y_{6,4})&=380+166r-6m-4\alpha+660l+666p_{(2,5)}+330p_{(3,4)}-660g(D)+672k-660b+\\\
&+342w-660g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-12g(G)+332N-660a-330g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-330g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-\\\
&-2g(F_{1})-2g(F_{2}).\end{aligned}\end{cases}$

Therefore

$\displaystyle e(Y_{6,n})$
$\displaystyle=\frac{1}{3}\Bigg{(}46-r+2m-\alpha+2l+p_{(3,4)}-2g(D)-2k-2b-3w-2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+4g(G)-2N-2a-$
$\displaystyle-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+3g(F_{1})+3g(F_{2})\Bigg{)}\cdot(-1)^{n-1}-$
$\displaystyle-\frac{1}{3}\Bigg{(}-23+\frac{r}{2}+2m+2\alpha+2l+p_{(3,4)}-2g(D)-2k-2b-3w-2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+4g(G)+N-$
$\displaystyle-2a-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\Bigg{)}\cdot 2^{n-1}+$
$\displaystyle+\frac{1}{3}\Bigg{(}2+r+3\alpha-2l-2p_{(2,5)}-p_{(3,4)}+2g(D)-2k+2b-w+2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+2N+2a+$
$\displaystyle+g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-3g(F_{1})-3g(F_{2})\Bigg{)}-$
$\displaystyle-\frac{1}{3}\Bigg{(}-\frac{r}{2}-2l-2p_{(2,5)}-p_{(3,4)}+2g(D)-2k+2b-w+2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-N+2a+$
$\displaystyle+g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-1\Bigg{)}.$

###  4.2. $K3$ surfaces with non-symplectic automorphism of order 4

We shall keep the following notation from [CG16]:

  * $r:=$

$\dim H^{2}(S_{4},\mathbb{C})^{\alpha_{{S_{4}}}}$,

  * $m:=$

$\dim H^{2}(S_{4},\mathbb{C})_{\zeta_{4}^{i}}$ for $1\leq i\leq 3$,

  * $N:=$

number of curves which are fixed by $\alpha_{S_{4}}^{2},$

  * $k:=$

number of curves which are fixed by $\alpha_{S_{4}}$ (curves of the first
type),

  * $b:=$

number of curves which are fixed by $\alpha_{S_{4}}^{2}$ and are invariant by
$\alpha_{S_{4}}$ (curves of the second type),

  * $a:=$

number of pairs $(A,A^{\prime})$ of curves which are fixed by
$\alpha_{S_{4}}^{2}$ and $\alpha_{S_{4}}(A)=A^{\prime}$ (curves of the third
type),

  * $D:=$

the curve of the highest genus in $S^{\alpha_{S_{4}}^{2}},$

  * $n_{1}:=$

number of points which are fixed by $\alpha_{S_{4}}$ not laying on the curve
$D,$

  * $n_{2}:=$

number of points which are fixed by $\alpha_{S_{4}}$ laying on the curve $D.$

######  4.3 Theorem ([Bur20]).

The Hodge number $h^{p,q}$ of the manifold $Y_{4,n}$ is equal to the
coefficient in $X^{p}Y^{q}$ of the following polynomial

$\displaystyle\Bigg{(}(XY)^{2}+r\cdot XY+1+\Big{(}k+(n_{1}+n_{2})\cdot
XY+g(G)\cdot(X+Y)+k\cdot XY\Big{)}\cdot\sqrt[4]{XY}+$
$\displaystyle+\Big{(}N-a+g\left({{}^{\textstyle
D}\big{/}_{\textstyle\alpha_{S_{4}}}}\right)\cdot(X+Y)+(N-a)\cdot
XY\Big{)}\cdot\sqrt[4]{(XY)^{2}}+\Big{(}k+n_{1}+n_{2}+$
$\displaystyle+g(G)\cdot(X+Y)+k\cdot
XY\Big{)}\cdot\sqrt[4]{(XY)^{3}}\Bigg{)}\cdot\Big{(}1+XY+2\sqrt[4]{XY}+3\sqrt[4]{(XY)^{3}}+2\sqrt[4]{(XY)^{3}}\Big{)}^{n-1}+$
$\displaystyle+\Big{(}X^{2}+(m-1)\cdot XY\Big{)}\cdot
X^{n-1}+\Big{(}(22-r-2m)\cdot XY+\Big{(}a+\left(g(D)-g\left({{}^{\textstyle
D}\big{/}_{\textstyle\alpha_{S_{4}}}}\right)\right)(X+Y)+$
$\displaystyle+a\cdot
XY\Big{)}\cdot\sqrt[4]{(XY)^{2}}\Big{)}\cdot\Big{(}\sqrt[4]{(XY)^{2}}\Big{)}^{n-1}+\Big{(}Y^{2}+(m-1)\cdot
XY\Big{)}\cdot Y^{n-1}.$

Similarly as before we can obtain the following corollary

######  4.4 Corollary.

The Euler characteristic of $Y_{4,n}$ is equal to $a_{n-1},$ where

$\begin{cases}a_{n}=9a_{n-1}+a_{n-2}-9a_{n-3},\\\ a_{0}=e(Y_{4,1})=24,\\\
a_{1}=e(Y_{4,2})=2r+8k+4n_{1}+4n_{2}+6N-4a+4-14g(D)-2m,\\\
a_{2}=e(Y_{4,3})=64+20r+80k+40n_{1}+40n_{2}-120g(D)+40N-40a.\end{cases}$

Therefore

$\displaystyle e(Y_{4,n})$
$\displaystyle=\Big{(}-N+g(D)+m+12\Big{)}\cdot(-1)^{n-1}-\frac{1}{2}\Bigg{(}-N+a+3g(D)-2k-n_{1}-n_{2}-\frac{r}{2}-1\Bigg{)}\cdot
9^{n-1}+$
$\displaystyle+\frac{1}{2}\Bigg{(}N+a+g(D)-2k-2m-n_{1}-n_{2}-\frac{r}{2}+23\Bigg{)}.$

###  4.3. $K3$ surfaces with non-symplectic automorphism of order 3

We shall keep the following notation from [CG16]:

  * $r:=$

$\dim H^{2}(S_{3},\mathbb{C})^{\alpha_{S_{3}}}$,

  * $m:=$

$\dim H^{2}(S_{3},\mathbb{C})_{\zeta_{3}^{i}}$ for $i=1,2$,

  * $h:=$

number of points which are fixed by $\alpha_{S_{3}},$

  * $k:=$

number of curves which are fixed by $\alpha_{S_{3}}$.

######  4.5 Theorem ([Bur20]).

The Hodge number $h^{p,q}$ of the manifold $Y_{3,n}$ is equal to the
coefficient in $X^{p}Y^{q}$ of the following polynomial

$\displaystyle\Bigg{(}(XY)^{2}+r\cdot XY+1+\Big{(}k+h\cdot
XY+g(C)\cdot(X+Y)+k\cdot XY\Big{)}\cdot\sqrt[3]{XY}+\Big{(}k+h+$
$\displaystyle+g(C)\cdot(X+Y)+k\cdot
XY\Big{)}\cdot\sqrt[3]{(XY)^{2}}\Bigg{)}\cdot\Big{(}1+XY+3\sqrt[3]{XY}+3\sqrt[3]{(XY)^{2}}\Big{)}^{n-1}+$
$\displaystyle+\Big{(}X^{2}+(m-1)\cdot XY\Big{)}\cdot
X^{n-1}+\Big{(}Y^{2}+(m-1)\cdot XY\Big{)}\cdot Y^{n-1}.$

######  4.6 Corollary.

The Euler characteristic of $Y_{3,n}$ is equal to $a_{n-1},$ where

$\begin{cases}a_{n}=7a_{n-1}+8a_{n-2},\\\ a_{0}=e(Y_{3,1})=24,\\\
a_{1}=e(Y_{3,2})=2r+6h+12k+4-12g(C)-2m.\\\ \end{cases}$

Therefore

$e(Y_{3,n})=\dfrac{1}{9}\Big{(}188-2r-6h-12k+12g(C)+2m\Big{)}\cdot(-1)^{n-1}+\dfrac{1}{9}\Big{(}28+2r+6h+12k-12g(C)-2m\Big{)}\cdot
8^{n-1}.$

###  4.4. $K3$ surfaces with non-symplectic automorphism of order 2

  * $r:=$

$\dim H^{2}(S_{2},\mathbb{C})^{\alpha_{S_{2}}}$,

  * $m:=$

$\dim H^{2}(S_{2},\mathbb{C})_{\zeta_{2}^{i}}$ for $i=1,2$,

  * $N:=$

number of curves which are fixed by $\alpha_{S_{2}}$ with sum of genera equals
$N^{\prime}.$

######  4.7 Theorem ([Bur20]).

The Hodge number $h^{p,q}$ of the manifold $Y_{2,n}$ is equal to the
coefficient in $X^{p}Y^{q}$ of the following polynomial

$\displaystyle\Bigg{(}(XY)^{2}+r\cdot
XY+1+\Big{(}N+N^{\prime}\cdot(X+Y)+N\cdot(XY)^{2}\Big{)}\cdot\sqrt{XY}\Bigg{)}\cdot\Big{(}1+XY+4\sqrt{XY}\Big{)}^{n-1}+$
$\displaystyle+\Big{(}X^{2}+Y^{2}+(m-2)\cdot XY\Big{)}\cdot(X+Y)^{n-1}.$

######  4.8 Corollary.

The Euler characteristic of $Y_{2,n}$ is equal to $a_{n-1},$ where

$\begin{cases}a_{n}=4a_{n-1}+12a_{n-2},\\\ a_{0}=e(Y_{3,1})=24,\\\
a_{1}=e(Y_{3,2})=12N-12N^{\prime}.\\\ \end{cases}$

Therefore

$e(Y_{2,n})=\frac{1}{2}(12+3N-3N^{\prime})\cdot
6^{n-1}+\frac{1}{2}(36-3N+3N^{\prime})\cdot(-2)^{n-1}.$

##  5\. Relations

###  5.1. $K3$ surfaces with non-symplectic automorphism of order 6

There are many relations among numerical invariants attached to $S_{6}.$ Some
of them were pointed out in [CG16] and [Bur18]. Most of them follow from
Riemann-Hurwitz formula (see [CG16]). In [Bur18] we got new relation by
comparing Stringy Euler Characteristic of $Y_{6,2}$ with the Euler
characteristic computed from orbifold Hodge numbers.

In the present section we shall examine another formulas i.e. Hurwitz formula
and holomorphic and topological Lefschetz numbers.

###  5.2. Riemann-Hurwitz formula

Some isolated fixed points of $\alpha_{S_{6}}$ lie on curves in
$\operatorname{Fix}\left(\alpha_{S_{6}}^{2}\right)$ and all of them are of
type $(3,4).$ The canonical map $\pi_{G}\colon G\to{{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}$ can be considered as a covering of $G$
of degree two ramified at

$p_{(3,4)}-2\cdot\underbrace{(k-2b-\ell-1)}_{\begin{subarray}{c}\textup{number
of rational curves}\\\ \textup{invariant by }\alpha_{S_{6}}\end{subarray}}$

points lying on $G,$ which have ramification index $2,$ therefore by the
Reimann-Hurwitz formula we get:

$2-2g(G)=2\left(2-2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\right)-\left(p_{(3,4)}-2(k-2b-\ell-1)\right),$

thus

(5.1) $g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)=\frac{1}{4}\left(2g(G)-p_{(3,4)}+2k-4b-2\ell\right).$

Assuming $g(D)=0.$ Let $F:=F_{1}\cup F_{2}.$ The canonical map $\pi_{F}\colon
F\to{{}^{\textstyle F}\big{/}_{\textstyle\alpha_{S_{6}}}}$ is triple covering
of $F$, ramified at

$p_{(3,4)}+p_{(2,5)}-2\cdot\underbrace{(N-3a-1-\ell)}_{\begin{subarray}{c}\textup{number
of rational curves}\\\ \textup{invariant by }\alpha_{S_{6}}\end{subarray}}$

points, with ramification index equal to $3,$ so

$\displaystyle 2-2g(F_{1})+2-2g(F_{2})=3\left(2-2g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+2-2g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)\right)-$
$\displaystyle-2(p_{(3,4)}+p_{(2,5)}-2(N-3a-1-\ell)).$

Consequently

(5.2) $\displaystyle g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)=\frac{1}{6}\left(2g(F_{1})+2g(F_{2})-2p_{(2,5)}-2p_{(3,4)}+4N-12a-4l\right).$

###  5.3. Topological Lefschetz number

Using 2.2 we deduce:

$\mathcal{L}_{\operatorname{top}}\left(\alpha_{S_{6}}\right)=1+1+r+\zeta_{6}\cdot(1+m-1)+\zeta_{6}^{2}\cdot\alpha+\zeta_{6}^{3}\cdot\beta+\zeta_{6}^{4}\cdot\alpha+\zeta_{6}^{5}\cdot(m-1+1)=2+r+m-\alpha-\beta,$

therefore

$2+r+m-\alpha-\beta=e\left(\operatorname{Fix}\left(\alpha_{S_{6}}\right)\right)=2\ell-2g(D)+p_{(3,4)}+p_{(2,5)},$

which gives relation

(5.3) $2+r+m-\alpha-\beta-2\ell+2g(D)-p_{(3,4)}-p_{(2,5)}=0.$

And similarly

$\displaystyle\mathcal{L}_{\operatorname{top}}\left(\alpha_{S_{6}}^{2}\right)$
$\displaystyle=(1+r+\beta+1)+\zeta_{3}(1+m-1+\alpha)+\zeta_{3}^{2}(\alpha+m-1+1)=$
$\displaystyle=(1+r+\beta+1)-1-m-\alpha+1=-\alpha+\beta+r+2-m,$

so

$-\alpha+\beta+r+2-m=e\left(\operatorname{Fix}\left(\alpha_{S_{6}}^{2}\right)\right)=2k-2g(G),$

giving relation

(5.4) $-\alpha+\beta+r+2-m-2k+2g(G)=0.$

In the same way:

$\mathcal{L}_{\operatorname{top}}\left(\alpha_{S_{6}}^{3}\right)=(1+r+2\alpha+1)-(1+\beta+2m-2+1)=2+r+2\alpha-\beta-2m,$

so

$2+r+2\alpha-\beta-2m=e\left(\operatorname{Fix}\left(\alpha_{S_{6}}^{3}\right)\right)=2N-2g(F_{1})-2g(F_{2}),$

giving relation

(5.5) $2+r+2\alpha-\beta-2m-2N+2g(F_{1})+2g(F_{2})=0.$

###  5.4. Holomorphic Lefschetz number

We shall deduce another relations using holomorphic Lefschetz formulas.

On the one hand

$\mathcal{L}_{\operatorname{hol}}\left(\alpha_{S_{6}}\right)=\sum_{i=0}^{2}(-1)^{i}\operatorname{tr}\left(\left(\alpha_{S_{6}}\right)^{*}|H^{i}(S,\mathcal{O}_{S})\right)=1+\zeta_{6}^{5}.$

Let us compute numbers $a(P)$ and $b(C)$ from theorem 2.4:

$\displaystyle
a(P_{j})=\frac{1}{\det\left(1-\alpha_{S_{6}}^{*}\raise-2.15277pt\hbox{$|$}_{\mathcal{T}_{P_{j}}}\right)}=\frac{1}{\det\left(\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\begin{pmatrix}\zeta_{6}^{3}&0\\\
0&\zeta_{6}^{4}\end{pmatrix}\right)}=\frac{1}{\left(1-\zeta_{6}^{3}\right)\left(1-\zeta_{6}^{4}\right)},$
$\displaystyle
a(Q_{j})=\frac{1}{\det\left(1-\alpha_{S_{6}}^{*}\raise-2.15277pt\hbox{$|$}_{\mathcal{T}_{Q_{j}}}\right)}=\frac{1}{\det\left(\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\begin{pmatrix}\zeta_{6}^{2}&0\\\
0&\zeta_{6}^{5}\end{pmatrix}\right)}=\frac{1}{\left(1-\zeta_{6}^{2}\right)\left(1-\zeta_{6}^{5}\right)}$

and similarly

$\displaystyle b(K_{i})=\frac{1-g(K_{i})}{1-\zeta_{6}}-\frac{\zeta_{6}\cdot
K_{i}^{2}}{\left(1-\zeta_{6}\right)^{2}}=\frac{1}{1-\zeta_{6}}-\frac{\zeta_{6}\cdot(2g(K_{i})-2)}{\left(1-\zeta_{6}\right)^{2}}=\frac{1}{1-\zeta_{6}}-\frac{-2\cdot\zeta_{6}}{\left(1-\zeta_{6}\right)^{2}}=\frac{1+\zeta_{6}}{\left(1-\zeta_{6}\right)^{2}},$
$\displaystyle b(D)=\frac{1-g(D)}{1-\zeta_{6}}-\frac{\zeta_{6}\cdot
D^{2}}{\left(1-\zeta_{6}\right)^{2}}=\frac{1-g(D)}{1-\zeta_{6}}-\frac{\zeta_{6}\cdot(2-2g(D))}{\left(1-\zeta_{6}\right)^{2}}=\frac{(1+\zeta_{6})\cdot(1-g(D))}{\left(1-\zeta_{6}\right)^{2}}.$

Therefore

$\displaystyle\mathcal{L}_{\operatorname{hol}}\left(\alpha_{S_{6}}\right)$
$\displaystyle=\sum_{i=1}^{\ell-1}b(K_{i})+b(D)+\sum_{j=1}^{p_{(2,5)}}a(P_{j})+\sum_{j=1}^{p_{(3,4)}}a(Q_{j})=$
$\displaystyle=(\ell-1)\cdot\frac{1+\zeta_{6}}{\left(1-\zeta_{6}\right)^{2}}+\frac{(1+\zeta_{6})\cdot(1-g(D))}{\left(1-\zeta_{6}\right)^{2}}+\frac{p_{(3,4)}}{\left(1-\zeta_{6}^{3}\right)\left(1-\zeta_{6}^{4}\right)}+\frac{p_{(2,5)}}{\left(1-\zeta_{6}^{2}\right)\left(1-\zeta_{6}^{5}\right)}$

which is equivalent to

$1+\zeta_{6}^{5}=(\ell-1)\cdot\frac{1+\zeta_{6}}{\left(1-\zeta_{6}\right)^{2}}+\frac{(1+\zeta_{6})\cdot(1-g(D))}{\left(1-\zeta_{6}\right)^{2}}+\frac{p_{(3,4)}}{\left(1-\zeta_{6}^{3}\right)\left(1-\zeta_{6}^{4}\right)}+\frac{p_{(2,5)}}{\left(1-\zeta_{6}^{2}\right)\left(1-\zeta_{6}^{5}\right)}.$

Multiplying this equality by denominators and after some manipulations we get
the following relation:

(5.6) $3+3\ell-3g(D)-\frac{p_{(3,4)}}{2}-p_{(2,5)}=0,$

which agrees with [Dil12a].

###  5.5. Stringy Euler Characteristic

Using stringy Euler characteristic for $n=1,\ldots,6$ we have

$e_{s}(Y_{6,n})=\begin{cases}24&\textup{for }n=1,\\\ \begin{aligned}
&8l-8g(D)+8p_{(2,5)}+4p_{(3,4)}+k-8g(G)+8n^{\prime}+4N-4g(F_{1})-4g(F_{2})\end{aligned}&\textup{for
}n=2,\\\ \begin{aligned}
&96+128l-128g(D)+88p_{(2,5)}+64p_{(3,4)}+48k-48g(G)+\\\
&+48n^{\prime}+16N-16g(F_{1})-16g(F_{2})\end{aligned}&\textup{for }n=3,\\\
\begin{aligned}
&672+1320l-1320g(D)+888p_{(2,5)}+660p_{(3,4)}+456k-456g(G)+456n^{\prime}+\\\
&+168N-168g(F_{1})-168g(F_{2})\end{aligned}&\textup{for }n=4,\\\
\begin{aligned} &6720+13312l-13312g(D)+8888p_{(2,5)}+6656p_{(3,4)}+4464k+\\\
&+4464n^{\prime}-4464g(G)+1664N-1664g(F_{1})-1664g(F_{2})\end{aligned}&\textup{for
}n=5,\\\ \begin{aligned}
&66720+133288l-133288g(D)+88888p_{(2,5)}+66644p_{(3,4)}+44488k+\\\
&+44488n^{\prime}-44488g(G)+16664N-16664g(F_{1})-16664g(F_{1})\end{aligned}&\textup{for
}n=6.\\\ \end{cases}$

Comparing $e_{s}(Y_{6,n})$ and 4.2 for $n=1\ldots 6$ we obtain three new
relations:

(5.7) $-2\alpha+10+N-r-g(F_{1})-g(F_{2})=0$ (5.8)
$\displaystyle-m+2+r-2l-p_{(2,5)}-p_{(3,4)}+2g(D)-2b-w-2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+2g(G)$
$\displaystyle-2a-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g(F_{1})+g(F_{2})=0.$ (5.9)
$\displaystyle-n^{\prime}-3+\frac{3}{2}\cdot
r-6l-2p_{(2,5)}-3p_{(3,4)}+6g(D)+2k-6b-6g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+$
$\displaystyle+4g(G)+\frac{3}{2}\cdot N-6a-3g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-3g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+\frac{3}{2}\cdot
g(F_{1})+\frac{3}{2}\cdot g(F_{2})=0.$

To summarize we obtained the following relations:

######  5.1 Theorem.

The following relation holds among parameters attached to $S_{6}:$

$\displaystyle\textup{1)\;\;}0=2m+r+\alpha+\beta-20,$
$\displaystyle\textup{2)\;\;}0=n-p_{(2,5)}-2n^{\prime},$
$\displaystyle\textup{3)\;\;}0=2+r+m-\alpha-\beta-2\ell+2g(D)-p_{(2,5)}-p_{(3,4)},$
$\displaystyle\textup{4)\;\;}0=-\alpha+\beta+r+2-m-2k+2g(G),$
$\displaystyle\textup{5)\;\;}0=2+r+2\alpha-\beta-2m-2N+2g(F_{1})+2g(F_{2}),$
$\displaystyle\textup{6)\;\;}0=-2\alpha+10+N-r-g(F_{1})-g(F_{2}),$
$\displaystyle\textup{7)\;\;}0=3+3\ell-3g(D)-\frac{p_{(3,4)}}{2}-p_{(2,5)},$
$\displaystyle\textup{8)\;\;}0=-g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+\frac{1}{4}\left(2g(G)-p_{(3,4)}+2k-4b-2\ell\right),$
$\displaystyle\textup{9)\;\;}0=-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+\frac{1}{6}\big{(}2g(F_{1})+2g(F_{2})-2p_{(2,5)}-2p_{(3,4)}+$
$\displaystyle+4N-12a-4l\big{)}\;\;\;\textup{(assuming $g(D)=0$)},$
$\displaystyle\textup{10)\;\;}0=-m+2+r-2l-p_{(2,5)}-p_{(3,4)}+2g(D)-2b-w-2g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+2g(G)$
$\displaystyle-2a-g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+g(F_{1})+g(F_{2}),$
$\displaystyle\textup{11)\;\;}0=-n^{\prime}-3+\frac{3}{2}\cdot
r-6l-2p_{(2,5)}-3p_{(3,4)}+6g(D)+2k-6b-6g\left({{}^{\textstyle
G}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+$
$\displaystyle+4g(G)+\frac{3}{2}\cdot N-6a-3g\left({{}^{\textstyle
F_{1}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)-3g\left({{}^{\textstyle
F_{2}}\big{/}_{\textstyle\alpha_{S_{6}}}}\right)+\frac{3}{2}\cdot
g(F_{1})+\frac{3}{2}\cdot g(F_{2}).$

###  5.6. $K3$ surfaces with a non-symplectic automorphism of order 4

Some of geometric parameters attached to $S_{4}$ are related in the obvious
way i.e.

(5.10) $-N+k+b+2a=0.$

###  5.7. Topological Lefschetz number

Using 2.2 we deduce:

$\mathcal{L}_{\operatorname{top}}\left(\alpha_{S_{4}}\right)=1+1+r+\zeta_{4}\cdot(1+m-1)+\zeta_{4}^{2}\cdot(22-r-2m)+\zeta_{4}^{3}\cdot(m-1+1)=-20+2r+2m,$

therefore

$-20+2r+2m=e\left(\operatorname{Fix}\left(\alpha_{S_{4}}\right)\right)=n_{1}+n_{2}+2k-2g(D),$

which gives relation

(5.11) $-20+2r+2m-n_{1}-n_{2}-2k+2g(D)=0.$

And similarly

$\mathcal{L}_{\operatorname{top}}\left(\alpha_{S_{4}}^{2}\right)=(1+22-2m+1)+\zeta_{4}^{2}\cdot(1+1+2m-2)=24-4m,$

so

$2N-2g(D)=e\left(\operatorname{Fix}\left(\alpha_{S_{4}}^{2}\right)\right)=24-4m,$

giving relation

(5.12) $N-g(D)-12+2m=0.$

###  5.8. Holomorphic Lefschetz number

Computation of holomorphic Lefschetz number in two ways, leads to the
equation:

$1+\zeta_{4}^{3}=\frac{(k-1)\cdot(1+\zeta_{4})}{(1-\zeta_{4})^{2}}+\frac{(1-g(D))\cdot(1+\zeta_{4})}{(1-\zeta_{4})^{2}}+\frac{n_{1}+n_{2}}{(1-\zeta_{4}^{2})(1-\zeta_{4}^{3})}.$

After some simplifications we get

(5.13) $4+2k-2g(D)-n_{1}-n_{2}=0,$

which agrees with [AS15].

###  5.9. Stringy Euler Characteristic

Using stringy Euler characteristic one can see that

$e_{s}(Y_{4,n})=\begin{cases}24&\textup{for }n=1,\\\
18k-18g(D)+6n_{1}+6n_{2}+6b+12a&\textup{for }n=2,\\\
144+150k-150g(D)+60n_{1}+60n_{2}+30b+60a&\textup{for }n=3,\\\
1080+1368k-1368g(D)+546n_{1}+546n_{2}+276b+552a&\textup{for }n=4,\\\
9864+12300k-12300g(D)+4920n_{1}+4920n_{2}+2460b+4920a&\textup{for }n=5,\\\
88560+110718k-110718g(D)+44286n_{1}+44286n_{2}+22146b+44292a&\textup{for
}n=6.\\\ \end{cases}$

Comparing $e_{s}(Y_{4,n})$ and 4.4 for $n=1\ldots 6$ we obtain two new
relations:

(5.14) $-b+8+3k-3g(D)+2n_{1}+2n_{2}+2a-2r=0$ (5.15)
$-m-2k+2g(D)-n_{1}-n_{2}-2a+r+2=0.$

To summarize we get the following relations:

######  5.2 Theorem.

The following relation holds among parameters attached to $S_{4}:$

$\displaystyle 0=-N+k+b+2a,$ $\displaystyle 0=-20+2r+2m-n_{1}-n_{2}-2k+2g(D),$
$\displaystyle 0=N-g(D)-12+2m,$ $\displaystyle 0=4+2k-2g(D)-n_{1}-n_{2},$
$\displaystyle 0=-b+8+3k-3g(D)+2n_{1}+2n_{2}+2a-2r,$ $\displaystyle
0=-m-2k+2g(D)-n_{1}-n_{2}-2a+r+2.$

Therefore, for computing all parameters it is sufficient to know only the
following:

$k,\;\;g(D),\;\;r\textup{ or }m,\;\;n_{1}\textup{ or }n_{2},\;\;a\textup{ or
}b.$

######  5.3 Remark.

Last two relations in Thm. 5.2 and last four relations in Thm. 5.1 seems to be
new. In the case of purely non-symplectic automorphism of order 2 and 3 we did
not get any new relations.

We have found relations by a careful analysis of two ways of computing Euler
characteristics. In this approach they follow from an enumeration of orbits of
the group action of components of the fixed locus, in particular they have no
clear geometric explanation. This leads to the following question:

######  5.4 Question.

Is there any geometrical argument proving last two relations from Thm. 5.2 and
last four relations from Thm. 5.1?

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