# Generative Entity Typing with Curriculum Learning

Siyu Yuan$\heartsuit$1, Deqing Yang$\heartsuit$2, Jiaqing Liang$\heartsuit$2,
Zhixu Li$\spadesuit$2 Jinxi Liu$\heartsuit$1,
Jingyue Huang$\heartsuit$2, Yanghua Xiao$\spadesuit\clubsuit$211footnotemark:
1
$\heartsuit$School of Data Science, Fudan University, Shanghai, China
$\spadesuit$Shanghai Key Laboratory of Data Science, School of Computer
Science, Fudan University
$\clubsuit$Fudan-Aishu Cognitive Intelligence Joint Research Center
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Corresponding authors.

###### Abstract

Entity typing aims to assign types to the entity mentions in given texts. The
traditional classification-based entity typing paradigm has two unignorable
drawbacks: 1) it fails to assign an entity to the types beyond the predefined
type set, and 2) it can hardly handle few-shot and zero-shot situations where
many long-tail types only have few or even no training instances. To overcome
these drawbacks, we propose a novel generative entity typing (GET) paradigm:
given a text with an entity mention, the multiple types for the role that the
entity plays in the text are generated with a pre-trained language model
(PLM). However, PLMs tend to generate coarse-grained types after fine-tuning
upon the entity typing dataset. In addition, only the heterogeneous training
data consisting of a small portion of human-annotated data and a large portion
of auto-generated but low-quality data are provided for model training. To
tackle these problems, we employ curriculum learning (CL) to train our GET
model on heterogeneous data, where the curriculum could be self-adjusted with
the self-paced learning according to its comprehension of the type granularity
and data heterogeneity. Our extensive experiments upon the datasets of
different languages and downstream tasks justify the superiority of our GET
model over the state-of-the-art entity typing models. The code has been
released on https://github.com/siyuyuan/GET.

##  1 Introduction

Entity typing aims to assign types to mentions of entities from a predefined
type set, which enables machines to better understand natural languages and
benefit many downstream tasks, such as entity linking Yang et al. (2019) and
text classification Chen et al. (2019). Traditional entity typing approaches
follow the classification paradigm to classify (assign) the entity into a
predefined set of types, which have the following two unignorable drawbacks.
1) Closed Type Set: The classification-based approaches fail to assign the
entity to the types out of the predefined set. 2) Few-shot Dilemma for Long-
tail Types: Although fine-grained entity typing (FET) and ultra-fine entity
typing approaches can classify entities into fine-grained types, they can
hardly handle few-shot and zero-shot issues. In fact, there are many long-tail
types only having few or even no training instances in the datasets. For
example, more than 80% types have less than 5 instances and 25% types even
never appear in the training data from the ultra-fine dataset Choi et al.
(2018).

Figure 1: A toy example of entity typing through generation and classification
paradigm, respectively.

To address these drawbacks, in this paper, we propose a novel generative
entity typing (GET) paradigm: given a text with an entity mention, the
multiple types for the role that the entity plays in the text are generated by
a pre-trained language model (PLM). Compared to traditional classification-
based entity typing methods, PLM-based GET has two advantages. First, instead
of a predefined closed type set, PLMs can generate more open types for entity
mentions due to their strong generation capabilities. For example, in Figure
1, fine-grained types such as “large detergent company" and “large detergent
manufacturer" can be generated by PLMs for entity P&G, which contain richer
semantics but are seldom included by a predefined type set. Second, PLMs are
capable of conceptual reasoning and handling the few-shot and zero-shot
dilemma Hwang et al. (2021), since massive knowledge has been learned during
their pre-training.

However, it is nontrivial to realize PLM-based GET due to the following
challenges: 1) Entity typing usually requires generating fine-grained types
with more semantics, which are more beneficial to downstream tasks. However,
PLMs are biased to generate high-frequency vocabulary in the corpus due to
their primary learning principle based on statistical associations. As a
result, a typical PLM tends to generate high-frequent but coarse-grained types
even if we carefully finetune the PLM on the fine-grained entity typing
dataset (refer to Figure 5 in Section 4). Therefore, how to guide a PLM to
generate high-quality and fine-grained types for entities is crucial. 2) It is
costly for humans to annotate a great number of samples with fine-grained
types. Therefore, most existing works adopt heterogeneous data consisting of a
small portion (less than 10%) of human-annotated data and a large portion
(more than 90%) of auto-generated low-quality data (e.g., by distant
supervision), which greatly hurts the performance of entity typing models Gong
et al. (2021). How to train a PLM to generate desirable types on these low-
quality heterogeneous data is also challenging.

The difficulty of using PLMs to generate high-quality fine-grained types based
on the low-quality heterogeneous training data motivates us to leverage the
idea from curriculum learning (CL) Bengio et al. (2009), which better learns
heterogeneous data by ordering the training samples based on their quality and
difficulty Kumar et al. (2019). In this paper, we propose a CL-based strategy
to train our GET model. Specifically, we first define a fixed curriculum
instruction and partition the training data into several subsets according to
the granularity and heterogeneity of samples for model training. Based on the
curriculum instruction, CL can control the order of using these training
subsets from coarse-grained and lower-quality ones to fine-grained and higher-
quality ones. However, a fixed curriculum ignores the feedback from the
training process. Thus, we combine the predetermined curriculum with self-
paced learning (SPL) Kumar et al. (2010), which can enforce the model
dynamically self-adjusting to the actual learning order according to the
training loss. In this way, our CL-based GET model can make the learning
process move towards a better global optimum upon the heterogeneous data to
generate high-quality and fine-grained types.  Our contributions in this paper
are summarized as follows:

  * •

To the best of our knowledge, our work is the first to propose the paradigm of
generative entity typing (GET).

  * •

We propose to leverage curriculum learning to train our GET model upon
heterogeneous data, where the curriculum can be self-adjusted with self-paced
learning.

  * •

Our extensive experiments on the data of different languages and downstream
tasks justify the superiority of our GET model.

##  2 Related Work

#### Classification-based Entity Typing

The traditional classification-based entity typing methods can be categorized
into three classes. 1) Coarse-grained entity typing methods Weischedel and
Brunstein (2005); Tokarchuk et al. (2021) assign mentions to a small set of
coarse types; 2) Fine-grained entity typing (FET) methods Yuan and Downey
(2018); Onoe et al. (2021) classify mentions into more diverse and
semantically richer ontologies; 3) Ultra-fine entity typing methods Choi et
al. (2018); Ding et al. (2021); Dai et al. (2021) use a large open type
vocabulary to predict a set of natural-language phrases as entity types based
on texts. However, FET and ultra-fine entity typing methods hardly perform
satisfactorily due to the huge predefined type set. They also hardly handle
few-shot and zero-shot issues. Comparatively, our GET model can generate high-
quality multi-granularity types even beyond the predefined set for the given
entity mentions.

#### Concept Acquisition

Concept acquisition is very related to entity typing which also aims to obtain
the types for the given entities, since entity types are often recognized as
concepts. Concept acquisition can be categorized into the extraction-based or
generation-based scheme. The extraction scheme cannot acquire concepts not
existing in the given text Yang et al. (2020). The existing approaches of
concept generation Zeng et al. (2021) focus on utilizing the existing concept
taxonomy or knowledge bases to generate concepts but neglect to utilize the
large corpus. Our GET model can also achieve text-based concept generation.

#### Curriculum Learning

According to the curriculum learning (CL) paradigm, a model is first trained
with the easier subsets or subtasks, and then the training difficulty is
gradually increased Bengio et al. (2009) to improve model performance in
difficult target tasks, such as domain adaption Wang et al. (2021) and
training generalization Huang and Du (2019). The existing CL methods can be
divided into predefined CL (PCL) Bengio et al. (2009) and automatic CL (ACL)
Kumar et al. (2010). PCL divides the training data by the difficulty level
with prior knowledge, while ACL, such as self-paced learning (SPL), measures
the difficulty according to its losses or other models.

##  3 Methodology

Figure 2: Our PLM-based GET framework trained with curriculum learning.

In this section, we first formalize our task in this paper and overview the
framework of our GET model. Then, we introduce the details of model
implementation.

###  3.1 Task Formalization

Given a piece of text $X$ and an entity mention $M$ within it, the task of
generative entity typing (GET) is to generate multiple types
$TS=\\{T_{1},T_{2},...,T_{K}\\}$, where each $T_{k}(1\leq k\leq K)$ is a type
for $M$ w.r.t. the context of $X$.

###  3.2 Framework

As most of the previous entity typing models Choi et al. (2018); Lee et al.
(2020); Gong et al. (2021), our GET model is also trained upon the
heterogeneous data consisting of a small portion of human-annotated data and a
large portion of auto-generated data, due to the difficulty and high cost of
human annotation. We will introduce how to obtain our auto-generated data in
Section 4.1. The framework of our model learning includes the following three
steps, as shown in Figure 2.

  1. 1.

Prompt Construction: To better leverage the knowledge obtained from the pre-
training of PLM, we employ the prompt mechanism Liu et al. (2021a) to guide
the learning of our PLM-based GET model;

  2. 2.

Curriculum Instruction: As a key component of CL, the curriculum instruction
is responsible for measuring the difficulty of each sample in the
heterogeneous training data, and then designing a suitable curriculum for the
model training process;

  3. 3.

CL-based Learning: In this step, our PLM-based GET model is trained with the
designed curriculum, which is capable of adjusting its learning progress
dynamically through self-paced learning (SPL).

###  3.3 Prompt Construction

To generate the types of given entities by a PLM, we construct the prompts in
cloze format from the Hearst patterns listed in Table 1. Specifically, each
input text $X$ including an entity mention $M$ is concatenated with a cloze
prompt constructed with $M$, and the PLM is asked to fill the blank within the
cloze prompt. Recall the example in Figure 1, the original text “In the early
1980s, P & G tried to launch here a concentrated detergent under the Ariel
brand name that it markets in Europe” can be concatenated with a cloze prompt
such as “P & G is a  ” to construct an input prompt for the PLM, which
predicts “large detergent company", “large detergent manufacturer" and
“company" as the types for P & G to fill the blank.

M is a  |  such as M
---|---
M is one of  |  especially M
M refers to  |  , including M
M is a member of  |
Table 1: Prompts constructed from Hearst patterns.

###  3.4 Curriculum Instruction

Curriculum instruction is the core issue of CL, which requires estimating the
difficulty of samples in terms of model learning, to decide the order of using
samples for model training.

For our specific PLM-based GET model, we argue that the difficulty of a sample
in terms of model learning greatly depends on the granularity of its type,
which could be roughly measured by the length of the type term. To prove this,
we generate two subsets of training samples in the same size from the auto-
generated data according to their type length: one subset with type length=1,
and the other subset with type length$\geq$2111Here we chose length=1 and
length$\geq$2 to divide the training data since most of the types are no
longer than 2. Then, we train a typical PLM, T5 Raffel et al. (2019) for one
epoch in the two subsets and record the landscapes of loss. As observed in
Figure 3, the length$\geq$2 subset has lower converge and higher cross-entropy
(CE) losses than the length=1 subset when training converges, which shows that
it is more difficult for the PLM to fit the training samples of fine-grained
types.

Based on this observation, we partition the auto-generated data into two
subsets, i.e., the subset with one-word types (denoted as $D_{A}$), and the
subset with types of more than one word (denoted as $D_{B}$), and $D_{A}$ is
used for model training earlier than $D_{B}$. The human-annotated data
(denoted as $D_{C}$) is finally used, as it usually contains many ultra fine-
grained types annotated by human annotators, which is harder for model
learning.  For easier presentation later, we denote the whole training data as
$D=D_{A}\bigcup D_{B}\bigcup
D_{C}=\\{<X^{(i)},M^{(i)},TS^{(i)}>\\}_{i=1}^{N}$, where $N$ is the training
sample size and a sample is denoted as $D^{(i)}_{k}=$
$<X^{(i)},M^{(i)},T^{(i)}_{k}>$. Based on the fixed curriculum instruction, CL
can control the order in which data are used for model training, i.e., from
$D_{A}$ to $D_{B}$ to $D_{C}$.

Figure 3: The landscapes of CE loss comparison for the two training subsets
with type length=1 and type length$\geq$2.

###  3.5 CL-based Learning

#### T5 Backbone

To meet the cloze format and generate more fine-grained types, we choose T5
Raffel et al. (2019) as the backbone PLM of our GET model. T5 is an encoder-
decoder pre-trained model, which can focus on the entire text and allow
multiple words to fill in the blanks. To train the T5, in our settings, we
define the loss function of sample $D^{(i)}_{k}$ as,

$\displaystyle
L\big{(}D^{(i)}_{k}\big{)}=L_{CE}\big{(}T_{k}^{(i)},f(X^{(i)},\bm{\theta},M^{(i)})\big{)}$
(1)

where $L_{CE}$ is a CE loss function that calculates the cost between the
ground truth type $T_{k}^{(i)}$ and the predicted type
$f(X^{(i)},\bm{\theta},M^{(i)})$. $\bm{\theta}$ denote the model parameters.

#### SPL-based Training Process

Our model training can be performed according to the above predefined
curriculum, but the feedback from the learning process is inevitably ignored,
which may lead to divergent solutions Jiang et al. (2015). As an alternative,
we adopt self-paced learning (SPL) to enforce the model to self-adjust the
curriculum according to the feedback from the training loss. Formally, we
define the objective of our CL as,

$\displaystyle\min\limits_{\bm{\theta},\bm{v}}E(\bm{\theta},\bm{v};\lambda)$
$\displaystyle=\sum_{i=1}^{N}\sum_{k=1}^{K^{(i)}}v^{(i)}_{k}L\big{(}D^{(i)}_{k}\big{)}+g(\bm{v};\lambda)$
(2)

where the binary variable $v_{k}^{(i)}\in[0,1]$ indicates whether sample
$D_{k}^{(i)}$ should be incorporated into the calculation of the objective.
Specifically,

$v^{(i)}_{k}=\left\\{\begin{aligned}
1&&{L\big{(}D^{(i)}_{k}\big{)}<\lambda}\\\
0&&{L\big{(}D^{(i)}_{k}\big{)}\geq\lambda}\end{aligned}\right.$ (3)

and

$g(\bm{v};\lambda)=-\lambda\sum_{i=1}^{N}\sum_{k=1}^{K^{(i)}}v^{(i)}_{k},$ (4)

where $K^{(i)}$ is the number of types for $<X^{(i)},M^{(i)},TS^{(i)}>$, and
$\lambda$ is the “age" of SPL to control the learning pace. The regularizer
$g(;)$ is the binary self-paced function used to avoid over-fitting Jiang et
al. (2014).

In the training process, “easy" samples with small losses are used first for
training. We update $\lambda=\mu\lambda$ to increase $\lambda$ gradually,
where $\mu>1$ is the step size. With the growth of $\lambda$, more samples
with larger losses are gradually incorporated into model training to obtain a
more “mature" model.

#### Prior Knowledge to Optimize SPL

As mentioned in Section 3.4, we expect that the model is trained orderly by
the three subsets according to the predetermined curriculum and generates more
fine-grained types. However, the order of using the data for model training
totally depends on the loss function (Eq. 1) in SPL. Thus, SPL is limited in
incorporating predetermined curriculum and the type granularity into learning.
Therefore, we treat the predetermined curriculum and the type granularity as
prior knowledge to optimize SPL. Specifically, we increase the weight of the
samples with fine-grained types to let the model pay more attention to these
data and assign each subset with different weights to ensure that the training
process is executed according to the predetermined curriculum. In particular,
given the sample $D^{(i)}_{k}$, we define its weight as

$w(D^{(i)}_{k})=length(T_{k}^{(i)})*\gamma(D^{(i)}_{k})$ (5)

where

$\gamma(D^{(i)}_{k})=\left\\{\begin{aligned} 1&&\text{if }D^{(i)}_{k}\in
D_{A},\\\ 2&&\text{if }D^{(i)}_{k}\in D_{B},\\\ 3&&\text{if }D^{(i)}_{k}\in
D_{C}.\end{aligned}\right.$ (6)

Then, the loss function $L$ in Eq. 1 is updated as

$\displaystyle
L_{CE}\big{(}T_{k}^{(i)},f(X^{(i)},\bm{\theta},M^{(i)}))*w\big{(}D^{(i)}_{k}\big{)},$
(7)

which indicates that a sample with a large weight (more difficult) can be
incorporated later into the training process since its $v_{k}^{(i)}$ is more
likely to be 0 according to Eq. 3. We adopt an alternative convex search (ACS)
to realize SPL, of which the algorithm is shown in Appendix A. We use Adam
Kingma and Ba (2015) to update the model parameters.

###  3.6 Type Generation

When the PLM in our GET model has been trained through the aforementioned CL-
based learning process, we use it to generate the types for the given entity
mention. Specifically, we let the PLM fill in the blank of the input text. To
obtain more diverse candidate types, we apply the beam search Reddy et al.
(1977) with beam size as $b$, and select the $b$ most probable candidates.
Then, we reserve the types with confidence scores bigger than 0.5.

##  4 Experiments

Dataset | Type | Language | Size of D3 | Size of test set
---|---|---|---|---
BNN Weischedel and Brunstein (2005) | Coarse-grained | English | 10,000 | 500
FIGER Shimaoka et al. (2016) | Fine-grained | English | 10,000 | 278
Ultra-Fine Choi et al. (2018) | Ultra fine-grained | English | 5,500 | 500
GT Lee et al. (2020) | Multilingual | English | 4,750 | 250
Chinese | 4,750 | 250
Table 2: The statistic of different entity typing datasets. Model | BNN | FIGER
---|---|---
CT # | Prec. | R-Recall | R-F1 | CT # | Prec. | R-Recall | R-F1
Zhang et al. (2018) | 555 | 58.10% | 50.49% | 54.03% | 348 | 62.00% | 49.85% | 55.26%
Lin and Ji (2019) | 534 | 55.90% | 48.58% | 51.98% | 353 | 62.90% | 50.57% | 56.07%
Xiong et al. (2019) | 558 | 58.40% | 50.75% | 54.31% | 350 | 62.30% | 50.09% | 55.53%
Ali et al. (2020) | 697 | 73.00% | 63.43% | 67.88% | 399 | 71.00% | 57.08% | 63.29%
Chen et al. (2020) | 718 | 75.20% | 65.35% | 69.93% | 388 | 69.10% | 55.56% | 61.59%
Zhang et al. (2021) | 732 | 76.70% | 66.65% | 71.32% | 394 | 70.10% | 56.36% | 62.48%
Li et al. (2021) | 668 | 69.90% | 60.74% | 65.00% | 397 | 70.60% | 56.76% | 62.93%
Ours | 875 | 82.30% | 79.62% | 80.94% | 444 | 66.20% | 63.52% | 64.83%
Table 3: Comparison results of different approaches on the sample test set in coarse-grained and fine-grained entity typing dataset. Model | Ultra-Fine
---|---
CT # | Prec. | R-Recall | R-F1
Xiong et al. (2019) | 782 | 50.30% | 24.28% | 32.75%
Onoe and Durrett (2019) ELMo | 884 | 51.50% | 27.44% | 35.81%
Onoe and Durrett (2019) BERT | 884 | 51.60% | 27.44% | 35.83%
López and Strube (2020) | 915 | 43.40% | 28.41% | 34.34%
Onoe et al. (2021) | 1039 | 52.80% | 32.26% | 40.05%
Liu et al. (2021b) | 1042 | 54.50% | 32.35% | 40.60%
Dai et al. (2021) | 1213 | 53.60% | 37.66% | 44.24%
Ours | 1275 | 87.10% | 39.58% | 54.43%
Table 4: Comparison results of different approaches on the sample test set in Ultra-fine entity typing dataset. Dataset | MaNew | MiNew | R.New
---|---|---|---
BNN | 4 | 100 | 11.61%
FIGER | 25 | 137 | 26.81%
Ultra-Fine | 73 | 543 | 42.14%
Table 5: The number and ratio of new types generated by our model on different datasets. Model | Dataset | Chinese | English
---|---|---|---
CT # | Prec. | R-F1 | Len. | CT # | Prec. | R-F1 | Len.
FT |  Auto- generated data | 690 | 84.46% | 70.81% | 2.80 | 870 | 75.85% | 52.87% | 1.48
PCL | 646 | 91.76% | 70.37% | 2.75 | 864 | 85.97% | 54.87% | 1.32
SPL w/o PK | 672 | 92.18% | 72.22% | 2.75 | 900 | 87.12% | 56.66% | 1.54
Ours | 714 | 90.04% | 74.18% | 2.86 | 928 | 87.14% | 57.84% | 1.62
FT |  Human- annotated data | 383 | 72.54% | 53.98% | 2.65 | 352 | 84.82% | 48.82% | 1.72
PCL | 370 | 77.24% | 54.01% | 2.64 | 375 | 88.03% | 51.62% | 1.69
SPL w/o PK | 383 | 78.64% | 55.59% | 2.61 | 370 | 90.46% | 51.53% | 1.74
Ours | 409 | 83.64% | 59.28% | 2.63 | 373 | 90.75% | 51.88% | 1.82
Table 6: Performance comparisons of our model and its variants on the auto-
generated and human-annotated test set.

In this section, we verify the advantages of our GET model over the
classification-based entity typing models through our experiments. We also
explore the role of CL in guiding PLM-based type generation with different
language data. We further display the effectiveness of our generated entity
types in two downstream tasks.

###  4.1 Datasets

As we mentioned before, due to the expensive manual labeling of fine-grained
types, the training dataset consists of a large portion of auto-generated data
and a small portion of human-annotated data. Since PLMs have more difficulty
fitting the training samples of fine-grained types (elaborated in Figure 3 in
Section 3.4), we partition the auto-generated data into two subsets, and
denote the subset with one-word types as $D_{A}$ while the other as $D_{B}$.
Furthermore, human-annotated data with ultra fine-grained types is denoted as
$D_{C}$.

#### Auto-generated Data

The auto-generated data used in our model is obtained from the abstracts of
entities on Wikipedia Vrandečić and Krötzsch (2014). Specifically, we collect
the abstract texts and their hyperlinks pointing to the web pages of mentioned
entities, from which the type labels of these mentioned entities can be
obtained. In this way, the obtained type labels are more consistent with the
contexts of entities, and thus of much higher quality than those auto-
generated with distant supervision Gong et al. (2021).

To construct $D_{A}$ and $D_{B}$ from the auto-generated data, we collect
100,000 Chinese and English abstracts from Wikipedia, from which we randomly
select 500 samples as our test set, and the rest are used as the training set.
Then we split the training set into two subsets $D_{A}$ and $D_{B}$, according
to the length of the types as mentioned in Section 3.4.

#### Human-annotated Data

To demonstrate that GET is superior to the classification-based approaches, we
collect the human-annotated data from four different entity type datasets. The
statistics of them are listed in Table 2. We compare our model with baselines
on BNN, FIGER and Ultra-fine to demonstrate the superior performance of GET on
entity typing. GT dataset is used to evaluate the effectiveness of CL upon the
texts of different languages and heterogeneous data. Please note that we do
sample the test set to reduce the cost of manual evaluation on assessing
whether the newly-generated types are correct. However, the results from the
baselines and our model are evaluated in the same test instances.

###  4.2 Evaluation Metrics

The detailed information about the baselines and some experiment settings is
shown in Appendix B. Please note that the baselines on BNN and FIGER are
different from those on Ultra-Fine.

For BNN and FIGER, we only reserve the type in the predefined type set with
the highest probability predicted by the model since there is only one golden
label in the dataset. Ultra-fine entity typing aims to predict a set of
natural-language phrases that describe the types of entity mentions based on
texts. Therefore, we reserve the types with the probability bigger than 0.5
for all models. Please note that previous work adopts a classification
paradigm, while our GET model can generate new types not existing in the
predefined type set. Therefore, annotators are invited to assess whether the
generated types are correct. The annotation detail is shown in Appendix B.4.

We record the number of correct types assigned to the entity mentions (CT #)
and strict macro-averaged precision (Prec.). Obviously, it is impossible to
know all correct types generated based on the input text in advance due to
incalculable search space. Therefore, we report the relative recall.
Specifically, suppose CTs # is the total number of new types obtained by all
models. Then, the relative recall (R-Recall) is calculated as CT # divided by
CTs #. Accordingly, the relative F1 (R-F1) can also be calculated with Prec.
and R-Recall. In addition, we also record the average length of types (Len.)
under different training strategies to investigate the effectiveness of CL and
prior knowledge.

###  4.3 Overall Comparison Results

The comparison results on traditional entity typing and Ultra-fine entity
typing are shown in Table 3 and Table 4 222 The statistical significance test
and data size analysis are provided in Appendix B.5. The tables show that our
model (Ours) achieves consistent performance improvements on these datasets.
For BNN, our model significantly improves Prec. and covers more entity types.
For FIGER, our model generates more types than other baselines. For Ultra-
Fine, the existing models based on the classification paradigm are extremely
difficult to select the appropriate types from the large predefined type set.
Comparatively, our GET model has no classification constraint since it
transforms multi-classification into a generation paradigm that is more
suitable for PLMs. Therefore, our model greatly improves the precision of
Ultra-Fine and covers more types of entities.

We further display the capability of our model to generate new types beyond
the predefined type set, as shown in Table 5. In the table, MaNew (Macro-New)
is the total number of generated types beyond the predefined type set, MiNew
(Micro-New) is the total number of generated types beyond the human-annotated
type set (i.e., golden labeled set $TS$) of each instance, and R.New is the
ratio of new generated types per sample. The listed results are counted upon
the test sets in different datasets, from which we find that our model can
generate abundant types that are not in the golden label set, thereby
increasing the diversity of entity types.

Figure 4: The ratio of training samples of different learning strategies in
each epoch. Figure 5: The ratio of the types of length=1 and length $\geq$ 2
in the original training dataset, and the types generated by our GET model
with fine-tuning (FT) or curriculum learning (CL).

###  4.4 Effectiveness of CL

We further compare our model with the following ablated variants to verify the
effectiveness of CL. FT is fine-tuned directly with training data without CL;
PCL adopts Baby Step Bengio et al. (2009) instead of SPL, which inputs the
subsets into the model in turn according to a fixed curriculum, but ignores
the feedback from the learning process; SPL w/o PK adopts SPL but ignores
prior knowledge in training.

In order to demonstrate the performance of the compared models on Chinese and
English data, we only consider $D_{C}$ of GT in our ablation studies. Since
the models are designed toward heterogeneous training data, we investigate
their performance on both the auto-generated test set and human-annotated test
set (in GT), which is displayed in Table 6. Please note that we only report
the Prec. and R-F1 due to the limited space. Based on the results, the
superiority of PCL and SPL over FT verifies the advantage of CL over the
general training strategy, while the superiority of SPL w/o PK over PCL
verifies the effectiveness of SPL. Furthermore, our GET model performs well on
type generation upon abstract texts (auto-generated data) and common free
texts (human-annotated data).

To explore the reason for the advantage of SPL, we record the ratio of
incorporated training samples in each epoch. As shown in Figure 4, SPL
gradually incorporates the whole training data to train the model. The
training on the former subsets can be regarded as a pre-training process that
helps model optimization and regularizes the training on the latter subsets.
Thus, SPL can better guide the model to find a global minimum loss and make it
more generalizable.

###  4.5 Effectiveness of Prior Knowledge

From the Table 6, we also find that Ours can generate more fine-grained types
than SPL w/o PK. Without the prior knowledge, SPL only relies on the self-
judgment of the model and treats all the selected samples equally, which
ignores the data heterogeneity and type granularity during training and harms
the model performance.

As shown in Figure 5, compared to the original training dataset, there are
more coarse-grained types (length=1) than fine-grained types (length$\geq$2)
generated by the directly fine-tuned GET model (Generation with FT), while the
GET model with CL can generate more fine-grained types of the almost same
ratio as the coarse-grained types. It is because that the prior knowledge
about the type length is considered to re-weight the importance of samples,
making the model pay more attention to fine-grained types. Thus, more fine-
grained and high-quality types are generated. Based on these results, we
believe that combining prior knowledge with SPL is an excellent way to
optimize CL.

We also explore the influence of different $\lambda$ and $\mu$ which is shown
in Appendix B.6.

Method | Prec. | Recall | F1
---|---|---|---
No type | 72.92% | 72.70% | 72.47%
types (KG) | 73.99% | 73.17% | 73.30%
types (Gen.) | 74.51% | 73.41% | 73.53%
Table 7: Performance of short text classification based on Bi-LSTM without/with different external knowledge on NLPCC2017 dataset. Dataset | Method | F1
---|---|---
AIDA CoNLL-YAGO | triples (KG.) | 94.58%
triples (Gen.) | 94.92%
ACE 2014 | triples (KG) | 89.74%
triples (Gen.) | 90.54%
Table 8: Performance of entity linking model DCA-SL with different external
knowledge.

###  4.6 Applications

We further conduct experiments on the task of short text classification and
entity linking to prove that the types generated by our model can promote the
downstream tasks.

#### Short Text Classification.

Existing short text classification approaches Chen et al. (2019) directly use
KG as external knowledge to improve model performance. However, how to choose
the context-consistent types for the roles that the entities play in the text
relation is still a problem, which may lead to unsatisfactory results. GET can
generate context-consistent types for entities, and thus it can be adopted to
promote the classification performance. We conduct our experiments in the
NLPCC2017 dataset 333http://tcci.ccf.org.cn/conference/2017/taskdata.php, the
Chinese news title dataset with 18 classes (e.g., entertainment, game, food).
We first use an NER model to identify entities in the text and directly apply
our model upon NLPCC2017 dataset to generate types for the entities. Then we
choose Bi-LSTM to achieve classification. we also collect corresponding types
of entities in the representative KG CN-DBpedia for comparison. The results in
Table 7 show that external knowledge enhances the classification performance,
and the types generated by our GET model are more effective than those
directly obtained from KG.

#### Entity Linking.

The representative entity linking model DCA-SL Yang et al. (2019) adopts the
entity description and triples in KG as an external knowledge to enhance the
performance of the model in the entity linking task. To prove that the types
generated by our model are of high quality, we first adopt our model to
generate types for entities based on texts. Then we replace the types in the
original triples in KGs with the types we generated. From Table 8 we find that
the generated types by our model can improve the entity lining performance of
DCA-SL effectively, indicating that the generated types are of high quality
and meaningful.

##  5 Conclusion

In this paper, we propose a novel generative paradigm for entity typing, which
employs a generative PLM trained with curriculum learning. Compared with the
traditional classification-based entity typing methods, our generative entity
typing (GET) model can generate new types beyond the predefined type set for
given entity mentions. Our extensive experiments on several benchmark datasets
justify that the curriculum learning with SPL and the prior knowledge of type
length and subset order help our model generate more high-quality fine-grained
types.

Input Text | Golden | Generated
---|---|---
| He was capped 42
---
times and scored 8
goals for Sweden,
and he played at the
2002 FIFA World Cup
nation |  | nation,
---
nordic country,
scandinavian
country
| The Audit Bureau of
---
Circulations
was formed in 1914
to verify publication
circulation figures and
track media rates
| administration,
---
organization
| organization,
---
government
agency,
investigative
service
| Chapman retired from
---
playing hockey after
the 1943 to 1944
hockey season
| tournament,
---
event,contest,
game,activity,
sport,hockey
| sport,
---
contact sport,
team sport,
winter sport
Table 9: Ultra-fine entity typing examples with the corresponding golden
labels and generated types. Entity mentions are in bold and underlined.

##  6 Limitations

Although we have proven that our GET model can generate high-quality and new
types beyond the predefined type set for given entity mentions, it also has
some limitations. In this section, we analyze these limitations and hopefully
advance future work in GET.

###  6.1 Uncontrolled Generation

To delve into the model performance, we compare the types generated by our
approach with the golden type labels in Ultra-fine entity typing datasets.
Table 9 lists three examples with the correct types generated by our model and
the golden labeled type set of the entities in the Ultra-Fine dataset. The
first example shows that our model can generate more fine-grained types which
may not appear in the golden labeled set. The second and third examples
demonstrate that although our model can generate new concepts, it may ignore
some correct types in the golden label set, e.g., “administration". However,
enforcing the model by constraint decoding to generate the types in the
predefined type set may compromise the flexibility of our model to generate
new concepts. Therefore, we hope that future work can handle this dilemma with
better methods.

###  6.2 Type Selection

As mentioned in Sec. 3.6, T5 adopts beam search to generate the $b$ most
probable types with confidence scores. Then we reserve the types with
confidence scores larger than the selection threshold. However, it can hardly
achieve a satisfactory balance to reserve the types by choosing a specific
threshold to directly truncate the output of T5. If we select a relatively big
threshold, we can get more accurate types but may lose some correct types. If
the recall is preferred, precision might be hurt. Therefore, we suggest that
future work consider how to achieve a better trade-off between precision and
recall.

## Acknowledgement

We would like to thank the anonymous reviewers for their valuable comments and
suggestions for this work. This work is supported by the National Natural
Science Foundation of China (No.62072323), Shanghai Science, the Science and
Technology Commission of Shanghai Municipality Grant (No. 22511105902) and
Technology Innovation Action Plan (No.21511100401& 22511104700).

## References

  * Ali et al. (2020) Muhammad Asif Ali, Yifang Sun, Bing Li, and Wei Wang. 2020.  Fine-grained named entity typing over distantly supervised data based on refined representations.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 7391–7398.
  * Bengio et al. (2009) Yoshua Bengio, Jérôme Louradour, Ronan Collobert, and Jason Weston. 2009\.  Curriculum learning.  In _Proceedings of the 26th annual international conference on machine learning_ , pages 41–48.
  * Chen et al. (2019) Jindong Chen, Yizhou Hu, Jingping Liu, Yanghua Xiao, and Haiyun Jiang. 2019.  Deep short text classification with knowledge powered attention.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 33, pages 6252–6259.
  * Chen et al. (2020) Tongfei Chen, Yunmo Chen, and Benjamin Van Durme. 2020.  Hierarchical entity typing via multi-level learning to rank.  _arXiv preprint arXiv:2004.02286_.
  * Choi et al. (2018) Eunsol Choi, Omer Levy, Yejin Choi, and Luke Zettlemoyer. 2018.  Ultra-fine entity typing.  _arXiv preprint arXiv:1807.04905_.
  * Dai et al. (2021) Hongliang Dai, Yangqiu Song, and Haixun Wang. 2021.  Ultra-fine entity typing with weak supervision from a masked language model.  _arXiv preprint arXiv:2106.04098_.
  * Ding et al. (2021) Ning Ding, Yulin Chen, Xu Han, Guangwei Xu, Pengjun Xie, Hai-Tao Zheng, Zhiyuan Liu, Juanzi Li, and Hong-Gee Kim. 2021.  Prompt-learning for fine-grained entity typing.  _arXiv preprint arXiv:2108.10604_.
  * Dror et al. (2018) Rotem Dror, Gili Baumer, Segev Shlomov, and Roi Reichart. 2018.  The hitchhiker’s guide to testing statistical significance in natural language processing.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1383–1392, Melbourne, Australia. Association for Computational Linguistics.
  * Gong et al. (2021) Hongyu Gong, Alberto Valido, Katherine M Ingram, Giulia Fanti, Suma Bhat, and Dorothy L Espelage. 2021.  Abusive language detection in heterogeneous contexts: Dataset collection and the role of supervised attention.  _arXiv preprint arXiv:2105.11119_.
  * Huang and Du (2019) Yuyun Huang and Jinhua Du. 2019.  Self-attention enhanced cnns and collaborative curriculum learning for distantly supervised relation extraction.  In _Proceedings of the 2019 conference on empirical methods in natural language processing and the 9th international joint conference on natural language processing (EMNLP-IJCNLP)_ , pages 389–398.
  * Hwang et al. (2021) Jena D. Hwang, Chandra Bhagavatula, Ronan Le Bras, Jeff Da, Keisuke Sakaguchi, Antoine Bosselut, and Yejin Choi. 2021.  Comet-atomic 2020: On symbolic and neural commonsense knowledge graphs.  In _AAAI_.
  * Jiang et al. (2014) Lu Jiang, Deyu Meng, Teruko Mitamura, and Alexander G Hauptmann. 2014.  Easy samples first: Self-paced reranking for zero-example multimedia search.  In _Proceedings of the 22nd ACM international conference on Multimedia_ , pages 547–556.
  * Jiang et al. (2015) Lu Jiang, Deyu Meng, Qian Zhao, Shiguang Shan, and Alexander G Hauptmann. 2015.  Self-paced curriculum learning.  In _Twenty-Ninth AAAI Conference on Artificial Intelligence_.
  * Kingma and Ba (2015) Diederik P. Kingma and Jimmy Ba. 2015.  Adam: A method for stochastic optimization.  In _3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings_.
  * Kumar et al. (2019) Gaurav Kumar, George Foster, Colin Cherry, and Maxim Krikun. 2019.  Reinforcement learning based curriculum optimization for neural machine translation.  _arXiv preprint arXiv:1903.00041_.
  * Kumar et al. (2010) M Pawan Kumar, Benjamin Packer, and Daphne Koller. 2010.  Self-paced learning for latent variable models.  In _NIPS_ , volume 1, page 2.
  * Lee et al. (2020) Chin Lee, Hongliang Dai, Yangqiu Song, and Xin Li. 2020.  A chinese corpus for fine-grained entity typing.  _arXiv preprint arXiv:2004.08825_.
  * Li et al. (2021) Jinqing Li, Xiaojun Chen, Dakui Wang, and Yuwei Li. 2021.  Enhancing label representations with relational inductive bias constraint for fine-grained entity typing.
  * Lin and Ji (2019) Ying Lin and Heng Ji. 2019.  An attentive fine-grained entity typing model with latent type representation.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 6197–6202.
  * Liu et al. (2021a) Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang, Hiroaki Hayashi, and Graham Neubig. 2021a.  Pre-train, prompt, and predict: A systematic survey of prompting methods in natural language processing.  _arXiv preprint arXiv:2107.13586_.
  * Liu et al. (2021b) Qing Liu, Hongyu Lin, Xinyan Xiao, Xianpei Han, Le Sun, and Hua Wu. 2021b.  Fine-grained entity typing via label reasoning.  _arXiv preprint arXiv:2109.05744_.
  * López and Strube (2020) F López and M. Strube. 2020.  A fully hyperbolic neural model for hierarchical multi-class classification.
  * Onoe et al. (2021) Yasumasa Onoe, Michael Boratko, Andrew McCallum, and Greg Durrett. 2021.  Modeling fine-grained entity types with box embeddings.  _arXiv preprint arXiv:2101.00345_.
  * Onoe and Durrett (2019) Yasumasa Onoe and Greg Durrett. 2019.  Learning to denoise distantly-labeled data for entity typing.  _arXiv preprint arXiv:1905.01566_.
  * Raffel et al. (2019) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2019.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _arXiv preprint arXiv:1910.10683_.
  * Reddy et al. (1977) D Raj Reddy et al. 1977.  Speech understanding systems: A summary of results of the five-year research effort. department of computer science.
  * Shimaoka et al. (2016) Sonse Shimaoka, Pontus Stenetorp, Kentaro Inui, and Sebastian Riedel. 2016.  Neural architectures for fine-grained entity type classification.  _arXiv preprint arXiv:1606.01341_.
  * Tokarchuk et al. (2021) Evgeniia Tokarchuk, David Thulke, Weiyue Wang, Christian Dugast, and Hermann Ney. 2021.  Investigation on data adaptation techniques for neural named entity recognition.  _arXiv preprint arXiv:2110.05892_.
  * Vrandečić and Krötzsch (2014) Denny Vrandečić and Markus Krötzsch. 2014.  Wikidata: a free collaborative knowledgebase.  _Communications of the ACM_ , 57(10):78–85.
  * Wang et al. (2021) X. Wang, Y. Chen, and W. Zhu. 2021.  A survey on curriculum learning.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , PP(99).
  * Weischedel and Brunstein (2005) Ralph Weischedel and Ada Brunstein. 2005.  Bbn pronoun coreference and entity type corpus.  _Linguistic Data Consortium, Philadelphia_ , 112.
  * Xiong et al. (2019) Wenhan Xiong, Jiawei Wu, Deren Lei, Mo Yu, Shiyu Chang, Xiaoxiao Guo, and William Yang Wang. 2019.  Imposing label-relational inductive bias for extremely fine-grained entity typing.  _arXiv preprint arXiv:1903.02591_.
  * Yang et al. (2020) Xi Yang, Jiang Bian, William R Hogan, and Yonghui Wu. 2020.  Clinical concept extraction using transformers.  _Journal of the American Medical Informatics Association_ , 27(12):1935–1942.
  * Yang et al. (2019) Xiyuan Yang, Xiaotao Gu, Sheng Lin, Siliang Tang, Yueting Zhuang, Fei Wu, Zhigang Chen, Guoping Hu, and Xiang Ren. 2019.  Learning dynamic context augmentation for global entity linking.  _arXiv preprint arXiv:1909.02117_.
  * Yuan and Downey (2018) Zheng Yuan and Doug Downey. 2018.  Otyper: A neural architecture for open named entity typing.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 32.
  * Zeng et al. (2021) Qingkai Zeng, Jinfeng Lin, Wenhao Yu, Jane Cleland-Huang, and Meng Jiang. 2021.  Enhancing taxonomy completion with concept generation via fusing relational representations.  _arXiv preprint arXiv:2106.02974_.
  * Zhang et al. (2021) Haoyu Zhang, Dingkun Long, Guangwei Xu, Muhua Zhu, Pengjun Xie, Fei Huang, and Ji Wang. 2021.  Learning with noise: improving distantly-supervised fine-grained entity typing via automatic relabeling.  In _Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence_ , pages 3808–3815.
  * Zhang et al. (2018) Sheng Zhang, Kevin Duh, and Benjamin Van Durme. 2018.  Fine-grained entity typing through increased discourse context and adaptive classification thresholds.  _arXiv preprint arXiv:1804.08000_.

##  Appendix A CL Algorithm

We adopt an alternative convex search (ACS) to realize the SPL of our model
training. As shown in Algorithm 1, we use Adam to update the model parameters.

Algorithm 1 Self-paced Learning with prior knowledge for types generation.

Input: Input training data D, predetermined curriculum $\gamma$, self-paced
function $g$, step size $\mu$, pre-training learning parameters $\bm{\theta}$
, maximum number of iterations $T$ and the number of training epoch $EP$
Output: Model parameters $\hat{\bm{\theta}}$

1: Initialize $\bm{v}$ and $\lambda$;

2: Derive prior knowledge according to $\gamma$ and type length;

3: while $ep<EP$ do

4: while $t<T$ do

5: Compute the objective function $E(\bm{\theta_{t}},\bm{v};\lambda)$ with
prior knowledge;

6: Update $\bm{\theta_{t+1}}=Adam(E(\bm{\theta_{t}},\bm{v};\lambda))$;

7: $t=t+1$;

8: end while

9: Record $\hat{\bm{\theta}}$ = $\bm{\theta_{T}}$

10: Record $\bm{v}$ = $argmin_{\bm{v}}(E(\hat{\bm{\theta}},\bm{v};\lambda))$

11: $\lambda=\mu\lambda$;

12: $ep=ep+1$;

13: end while

14: return $\hat{\bm{\theta}}$

##  Appendix B Experiment Detail

###  B.1 Baselines of Traditional Entity Typing

Upon traditional entity typing dataset, namely FIGER and BNN, we compare our
model with following baselines:

  * •

Zhang et al. (2018): This approach uses a neural architecture to learn a
distributional semantic representation to classify.

  * •

Lin and Ji (2019): This approach proposes a two-step mention-aware attention
mechanism to enable the model to focus on the important words in mentions and
contexts to improve type classification performance.

  * •

Xiong et al. (2019): This approach utilizes a graph propagation layer to
capture label correlations for type classification.

  * •

Ali et al. (2020): This method adopts edge-weighted attentive graph
convolution network to refine the noisy mention representations.

  * •

Chen et al. (2020): Under the undefined case, this approach does not modify
the labels in the dataset.

  * •

Zhang et al. (2021): This approach utilizes a probabilistic automatic
relabeling method that treats all training samples uniformly to handle noisy
samples.

  * •

Li et al. (2021): This approach proposes a novel method based on a two-phase
graph network for the Fine-Grained Entity Typing task to enhance the label
representations via imposing the relational inductive biases of instance-to-
label and label-to-label.

###  B.2 Baselines of Ultra-Fine Entity Typing

For Ultra-Fine dataset, we compare our model with the following baselines:

  * •

Onoe and Durrett (2019): This approach adopts ELMo and BERT as the encoder to
fine-tune on the crowdsourced train split or raw and denoised distantly-
labeled data.

  * •

López and Strube (2020): This approach proposes a fully hyperbolic model for
multi-class multi-label classification, which performs all operations in
hyperbolic space.

  * •

Onoe et al. (2021): This approach adopts a BERT-based model with box
embeddings to capture latent type hierarchies for type classification.

  * •

Liu et al. (2021b): This approach discovers and exploits label dependencies
knowledge entailed in the data to sequentially reason fine-grained entity
labels for type classification.

  * •

Dai et al. (2021): This approach uses a BERT Masked Language Model to generate
weak labels for ultra-fine entity typing to improve the performance of type
classification.

###  B.3 Experiment Settings

Our experiments are conducted on a workstation of dual GeForce GTX 1080 Ti
with 32G memory, and the environment of torch 1.7.1. We adopted a T5-base with
12 layers and 12 self-attention heads for the English dataset and mT5-small
with 8 layers and 6 self-attention heads for the Chinese dataset. The
hyperparameter settings of training our PLM-based GET are: $\lambda=0.5$,
$\mu=2$. The beam size $b$ is 8. The coefficient weight $\alpha$ in the loss
function is 4.

###  B.4 Human Assessment

It is impossible to know all newly-generated types apriori. Thus human
annotators are needed to assess whether the generated types are correct. We
employ two annotators to ensure the quality of the assessment. Each predicated
type is labeled with 0 or 1 by two annotators, where 0 means a wrong type for
the given entity and 1 represents the right type for the given entity. If the
results from the two annotators are different, the third annotator will be
hired for a final check.

Data Size | CT #. | Prec. | Recall | F1
---|---|---|---|---
50% | 1040 | 81.95% | 32.29% | 46.32%
100% | 1275 | 87.10% | 39.58% | 54.43%
Table 10: Model Performance with different sizes of training data in Ultra-fine entity typing task. 50% and 100% are the proportions of auto-generated data used to train the model. $\mu$ | $\lambda$ | CT # | Prec. | Recall | F1
---|---|---|---|---|---
2 | 0.5 | 373 | 90.75% | 30.52% | 45.68%
2 | 0.1 | 330 | 80.10% | 27.00% | 40.39%
2 | 1 | 359 | 86.51% | 29.38% | 43.86%
4 | 0.5 | 328 | 82.21% | 26.84% | 40.47%
4 | 0.1 | 361 | 82.42% | 29.54% | 43.49%
4 | 1 | 348 | 81.88% | 28.48% | 42.26%
Table 11: Our model’s performance with different hyper-parameter settings.

###  B.5 Result Confidence

We also conduct a statistical significance test Dror et al. (2018) to show our
experiment results are convincing. Specifically, we run our method on the test
set of the Ultra-fine entity typing dataset twice with different random seeds.
Then we implement a t-test on the two results with a 0.05 significance level.
The result is not significant (p-value: 0.208) and thus we can not reject the
null hypothesis (H0: result1-result2=0, where resulti=(CT#, Prec., R-recall,
R-F1)). Based on the above hypothesis test, we believe that our experiment
results are confident and reproducible.

Besides, we do a run with 50% auto-generated training data for the Ultra-fine
entity typing task and the results are shown in Table 10. We find that our
method suffers from a slight performance drop but still outperforms the
baselines, which shows the effectiveness of auto-generated data.

###  B.6 Parameter Tuning Results

We explore the influence of different $\lambda$ and $\mu$ on the performance
of our model, as shown in Table 11.


# On the automorphism group of a toral variety

Anton Shafarevich and Anton Trushin  Moscow Center of Fundamental and Applied
Mathematics, Moscow, Russia; and Faculty of Computer Science, HSE University,
Pokrovsky Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>Moscow
Center of Fundamental and Applied Mathematics, Moscow, Russia; and Faculty of
Computer Science, HSE University, Pokrovsky Boulevard 11, Moscow, 109028
Russia<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero. An
affine algebraic variety $X$ over ${\mathbb{K}}$ is toral if it is isomorphic
to a closed subvariety of a torus $({\mathbb{K}}^{*})^{d}$. We study the group
$\operatorname{Aut}(X)$ of regular automorpshims of a toral variety $X$. We
prove that if $T$ is a maximal torus in $\operatorname{Aut}(X)$, then $X$ is a
direct product $Y\times T$, where $Y$ is a toral variety with a trivial
maximal torus in the automorphism group. We show that knowing
$\operatorname{Aut}(Y)$, one can compute $\operatorname{Aut}(X)$. In the case
when the rank of the group ${\mathbb{K}}[Y]^{*}/{\mathbb{K}}^{*}$ is $\dim
Y+1$, the group $\operatorname{Aut}(Y)$ can be described explicitly.

###### Key words and phrases:

Affine variety, invertible function, algebraic torus, automorphism, rigid
variety

###### 2010 Mathematics Subject Classification:

Primary 14M25, 14L30; Secondary 14R20, 13N15, 14M17

The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”

##  1\. Introduction

Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero.
The set of solutions of a system of polynomial equations in affine space has
been studied for a very long time. But some interesting properties may appear
when we consider the set of solutions inside a torus $({\mathbb{K}}^{*})^{d}$.
In other words we consider only solutions with non-zero coordinates. One of
examples of this approach is the Bernstein-Kushnirenko Theorem; see [2, 6].

Vladimir Popov in [9] proposed the following definition.

######  Definition 1.

An irreducible affine algebraic variety $X$ is called _toral_ if it is
isomorphic to a closed subvariety of a torus $({\mathbb{K}}^{*})^{d}$.

Some authors also use the term a ”very affine variety”; see [11, 3]. It can be
seen that $X$ is toral if and only if the algebra of regular functions on $X$
is generated by invertible functions; see [9, Lemma 1.14]. One of the reasons
why toral varieties are interesting is that they are rigid varieties; see [9,
Lemma 1.14].

######  Definition 2.

An affine algebraic variety $X$ is called _rigid_ if there is no non-trivial
action of the additive group $({\mathbb{K}},+)$ on $X.$

Despite the fact that the automorphism group of an affine algebraic variety
has a complicated structure sometimes it is possible to describe it for rigid
varieties. It was proven in [1] that the group of regular automorphisms
$\operatorname{Aut}(X)$ of a rigid variety $X$ contains a unique maximal torus
$T$. Moreover if we denote by $C(T)$ the centralizer of $T$ in
$\operatorname{Aut}(X)$ then the quotient group $\operatorname{Aut}(X)/C(T)$
is a discrete group. One can find examples of computation of
$\operatorname{Aut}(X)$ for rigid varieties in [1, 7, 8].

In this paper we study the automorphism group $\operatorname{Aut}(X)$ of a
toral variety $X$. We denote by ${\mathbb{K}}[X]$ the algebra of regular
functions on $X$ and by ${\mathbb{K}}[X]^{*}$ the multiplicative group of
invertible regular functions on $X$. Let $E(X)$ be the quotient group
${\mathbb{K}}[X]^{*}/{\mathbb{K}}^{*}.$ By [10], the group $E(X)$ is a free
finitely generated abelian group. For a toral variety $X$ the rank of $E(X)$
is not less than $\dim X.$

Any automorphism of $X$ induces an automorphism of $E(X).$ So we obtain a
homomorphism from $\operatorname{Aut}(X)$ to $E(X).$ We denote by $H(X)$ the
kernel of this homomorphism.

Suppose that $X$ is a closed subvariety of a torus
$T_{d}=({\mathbb{K}}^{*})^{d}$. In Proposition 1 we show that the group $H(X)$
is naturally isomorphic to a subgroup in $T_{d}$ which consists of elements
that preserve $X$ under the action by multiplication. In Proposition 2 we
propose a way to compute the subgroup $H(X)$.

In Theorem 1 we show that if $T$ is a maximal torus in $\operatorname{Aut}(X)$
then $X$ is isomorphic to a direct product $T\times Y$ where $Y$ is a toral
variety with a discrete automorphism group. Theorem 3 gives a way to find
$\operatorname{Aut}(X)$ knowing $\operatorname{Aut}(Y).$ If the rank of $E(Y)$
is $\dim Y+1$ then one can describe $\operatorname{Aut}(Y)$ (Theorem 3).

Also we consider the case when the rank of $E(X)$ is equal to $\dim X.$ By
Proposition 3 in this case $X$ is a torus. Moreover, it is the only case when
$\operatorname{Aut}(X)$ acts on $X$ with an open orbit.

We use the following notation. If $\varphi$ is a regular automorphism of an
affine variety $X$ then by $\varphi^{*}$ we mean an automorphism of
${\mathbb{K}}[X]$ dual to $\varphi$. If $A$ is a group and $B$ is a normal
subgroup in $A$ then by $[a]$ we denote the image of an element $a\in A$ in
the quotient group $A/B.$ If $X$ is a closed subvariety of an affine variety
$Z$ then by $I(X)$ we mean the ideal of regular functions on $Z$ which are
equal to zero on $X$.

The authors are grateful to Segrey Gaifullin for useful discussions. Also we
would like to thank Ivan Arzhantsev for helpful remarks and comments.

##  2\. General facts on toral varieties

Here we prove some initial properties of toral varieties and propose a way to
compute the group $H(X)$ for a toral variety $X$

Let $T_{d}$ be a torus of dimension $d$. We recall that the group
$\operatorname{Aut}(T_{d})$ is isomorphic to
$T_{d}\rtimes\mathrm{GL}_{d}({\mathbb{Z}})$; see[1, Example 2.3]. Here the
left factor $T_{d}$ acts on itself by multiplications and a matrix
$(a_{ij})\in\mathrm{GL}_{d}({\mathbb{Z}})$ defines an automorphism of $T_{d}$
which is given by the formula

$t_{i}\to t_{1}^{a_{i1}}\ldots t_{d}^{a_{id}},$

where $t_{1},\ldots,t_{d}$ are coordinate functions on $T_{d}.$

######  Proposition 1.

  1. (1)

Let $X$ be a closed irreducible subvariety of the torus $T_{d}.$ Then $H(X)$
is a quasitorus isomorphic to a closed subgroup in $T_{d}$ which consists of
all elements that preserve $X$.

  2. (2)

Let $X$ be a toral variety. Suppose that the rank of the group $E(X)$ is equal
to $r$. Then $X$ is isomorphic to a closed subvariety of
$T_{r}=({\mathbb{K}}^{*})^{r}$ and the group $\operatorname{Aut}(X)$ is
isomorphic to a subgroup of
$\operatorname{Aut}(T_{r})=T_{r}\rtimes\mathrm{GL}_{r}({\mathbb{Z}})$ which
consists of all automorphisms of $T_{r}$ that preserve $X.$

###### Proof.

1) Let $H$ be a subgroup in $T_{d}$ which consists of all elements which
preserve $X$. Any element of $H$ defines an automorphism of $X$. So we obtain
a homomorphism

$\psi:H\to\operatorname{Aut}(X).$

We will show that $\psi$ is injective and the image of $\psi$ is $H(X)$.

Let $t_{1},\ldots,t_{d}$ be coordinate functions on $T_{d}.$ Then their
restrictions to $X$ define invertible functions $f_{1},\ldots,f_{d}$ which
generate ${\mathbb{K}}[X].$ Let $\varphi$ be an automorphism in $H(X)$. Then
the automorphism $\varphi^{*}$ defines a trivial automorphism of $E(X).$ So
$\varphi^{*}(f_{i})=\lambda_{i}f_{i},$ for some
$\lambda_{1},\ldots,\lambda_{d}\in{\mathbb{K}}^{*}.$ Then the element
$(\lambda_{1},\ldots,\lambda_{d})\in T_{d}$ preserves $X$ and its restriction
to $X$ is $\varphi.$ So the image of $\psi$ contains $H(X).$

Conversely, consider an element $t=(\lambda_{1},\ldots,\lambda_{d})\in H$.
Then $(\psi(t))^{*}$ maps $f_{i}$ to $\lambda_{i}f_{i}.$ So it acts trivially
on $E(X).$ Therefore, the image of $\psi$ is contained in $H(X).$

The group $T_{d}$ acts on itself freely. So only the identity can act on $X$
trivially. Hence $\psi$ is injective.

2) Since rank $E(X)$ is equal to $r$ one can choose invertible functions
$f_{1},\ldots,f_{r}\in{\mathbb{K}}[X]^{*}$ such that $[f_{1}],\ldots,[f_{r}]$
form a basis of the group $E(X)$. Then $f_{1},\ldots,f_{r}$ generate the
algebra ${\mathbb{K}}[X]$ and there is a closed embedding of $X$ into $T_{r}.$
We denote by $t_{1},\ldots,t_{r}$ coordinate functions on $T_{r}$ which
corresponds to $f_{1},\ldots,f_{r}.$

Let $P$ be the subgroup in $T_{r}\rtimes\mathrm{GL}_{r}({\mathbb{Z}})$ of all
automorphisms that preserve $X$. Again, there is a homomorphism
$\psi:P\to\operatorname{Aut}(X)$ and we will show that $\psi$ is a bijection.

Each automorphism $\varphi\in\operatorname{Aut}(X)$ defines an automorphism of
the group $E(X).$ Since $E(X)$ is a free abelian group of the rank $r$ the
group $\operatorname{Aut}E(X)$ is isomorphic to
$\mathrm{GL}_{r}({\mathbb{Z}}).$

So the automorphism $\varphi^{*}$ has a form
$\varphi^{*}(f_{i})=\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}},$ where
$(a_{ij})\in\mathrm{GL}_{r}({\mathbb{Z}})$ and
$\lambda_{i}\in{\mathbb{K}}^{*}.$ But then the automorphism $\varphi$ is the
restriction of an automorphism
$\widetilde{\varphi}\in\operatorname{Aut}(T_{r})$ which is given by the
formula

$\widetilde{\varphi}(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}}.$

So $\psi$ is surjective.

Now let $\phi$ be an automorphism of $T_{r}$ which acts trivially on $X$. Then
$\phi$ is given by the formula

$\phi(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}}$

where $\lambda_{i}\in{\mathbb{K}}^{*}$ and the matrix $(a_{ij})$ belongs to
$\mathrm{GL}_{r}({\mathbb{Z}}).$ The corresponding automorphism of
${\mathbb{K}}[X]$ is given by the formula

$f_{i}\to\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}}$

but it is a trivial automorphism so we have

$f_{i}=\lambda_{i}f_{1}^{a_{i1}}\ldots f_{r}^{a_{ir}}.$

But the elements $[f_{1}],\ldots[f_{r}]\in E(X)$ are linearly independent. The
matrix $(a_{ij})$ is the identity matrix. So $\phi(t_{i})=\lambda_{i}t_{i}$
and $\phi$ belongs to $T_{r}\subseteq\operatorname{Aut}(T_{r})$. But the group
$T_{r}$ acts freely on itself. Hence $\phi$ is a trivial automorphism. So
$\psi$ is injective.

∎

######  Remark 1.

It follows from Proposition 1 that a toral variety $X$ can be embedded in a
torus $T_{r}$ in such a way that any automorphism $X$ can be extended to an
automorphism of $T_{r}.$ If $X$ is a subvariety of $Z$ it is always natural to
ask whether an automorphism of $X$ can be extended to an automorphism of $Z$.
Some results concerning this problem can be found in [4, 5].

Now let $X$ be a closed irreducible subvariety in $T_{d}.$ Then one can
describe the subgroup $H(X).$ Let $M\simeq{\mathbb{Z}}^{d}$ be the lattice of
characters of $T_{d}$. For $m=(m_{1},\ldots,m_{d})\in M$ by $\chi^{m}$ we mean
the character $t\to t_{1}^{m_{1}}\ldots t_{d}^{m_{d}}.$ Then each function in
${\mathbb{K}}[t_{1}^{\pm 1},\ldots,t_{d}^{\pm 1}]$ is a linear combinations of
characters. For a function
$f=\sum_{i}\alpha_{m_{i}}\chi^{m_{i}}\in{\mathbb{K}}[t_{1}^{\pm
1},\ldots,t_{d}^{\pm 1}]$ by _support of $f$_ we mean the subset

$\operatorname{Supp}f=\\{m_{i}\in M|\ \alpha_{m_{i}}\neq 0\\}\subseteq M.$

Let $I(X)$ be the ideal of functions in ${\mathbb{K}}[t_{1},\ldots,t_{d}]$
which are equal to zero on $X$. We say that $f\in I(X)$ is _minimal_ if there
is no non-zero $g\in I(X)$ such that
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f.$

######  Lemma 1.

The ideal $I(X)$ is generated by minimal polynomials.

###### Proof.

If $f\in I(X)$ is not minimal then there is a $g\in I(X)$ with
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f$. One can choose a
constant $\alpha$ such that $\operatorname{Supp}(f-\alpha
g)\subsetneq\operatorname{Supp}f.$ Applying induction by cardinality of
$\operatorname{Supp}f$ we see that $g$ and $f-\alpha g$ can be represented as
a sum of minimal polynomials. Then $f$ is also a sum of minimal polynomials. ∎

######  Remark 2.

We denote by $M(X)$ the subgroup of $M$ which is generated by all elements of
the form $m_{a}-m_{b}$ such that there is a minimal polynomial
$f=\sum\alpha_{m_{i}}\chi^{m_{i}}\in I(X)$ with non-zero $\alpha_{m_{a}}$ and
$\alpha_{m_{b}}$.

######  Proposition 2.

The subgroup $H(X)$ is isomorphic to a subgroup in $T_{d}$ which is given by
equations $\chi^{m}(t)=1$ for all $m\in M(X).$

###### Proof.

Denote by $H$ the subgroup in $T_{d}$ consisting of all elements which
preserve $X$. By Proposition 1 the group $H$ is isomorphic to $H(X).$ We will
show that $H$ is given in $T_{d}$ by equations $\chi^{m}(t)=1$ for all $m\in
M(X).$

Let $h\in H$ and $f=\sum_{i}\alpha_{m_{i}}\chi^{m_{i}}$ be a minimal
polynomial in $I(X).$ Then $h\circ
f=\sum_{i}\alpha_{i}\chi^{m_{i}}(h)\chi^{m_{i}}.$ The ideal $I(X)$ is
invariant under the action of $H$. So $h\circ f\in I(X)$. Suppose that there
are $a,b$ such that $\alpha_{a},\alpha_{b}\neq 0$ and
$\chi^{m_{a}}(h)\neq\chi^{m_{b}}(h).$ Then $g=\chi^{m_{a}}(h)f-h\circ f$ is a
non-zero function in $I(X)$ and
$\operatorname{Supp}g\subsetneq\operatorname{Supp}f.$ But $f$ is minimal. So
$\chi^{m_{a}}(h)=\chi^{m_{b}}(h)$. Therefore, $\chi^{m_{a}-m_{b}}(h)=1$ and
this implies that $\chi^{m}(h)=1$ for all $m\in M(X).$

Now consider the subgroup $H^{\prime}=\\{t\in T_{d}|\chi^{m}(t)=1,\ \forall\
m\in M(X)\\}.$ Then every minimal polynomial in $I(X)$ is semi-invariant with
respect to $H^{\prime}$. But $I(X)$ is generated by minimal polynomials. So
$I(X)$ is invariant under the action of $H^{\prime}$. Therefore,
$H^{\prime}\subseteq H.$

∎

At the end of this section, we note that toral varieties satisfy the following
conjecture formulated by Alexander Perepechko and Mikhail Zaidenberg.

######  Conjecture 1 (Conjecture 1.0.1 in [8]).

If $Y$ is a rigid affine algebraic variety over ${\mathbb{K}}$, then the
connected component $\operatorname{Aut}^{0}(Y)$ is an algebraic torus of rank
not greater than $\dim Y$.

Indeed, if $Y$ is a toral variety then the group $\operatorname{Aut}(Y)/H(Y)$
is isomorphic to a subgroup in
$\operatorname{Aut}(E(Y))\simeq\mathrm{GL}_{r}({\mathbb{Z}})$, where $r$ is a
rank of $E(Y).$ Then $\operatorname{Aut}(Y)/H(Y)$ is a discrete group. So
$\operatorname{Aut}^{0}(Y)$ is contained in $H(Y).$ But $H(Y)$ is a
quasitorus. Therefore, $\operatorname{Aut}^{0}(Y)$ is a torus.

##  3\. The structure of the automorphism group

It follows from [9, Lemma 1.14] that toral varieties are rigid. By [1, Theorem
2.1], there is a unique maximal torus in the automorphism group of an
irreducible rigid variety.

######  Theorem 1.

Let $X$ be a toral variety and $T$ be the maximal torus in
$\operatorname{Aut}(X).$ Then $X\simeq Y\times T$ where $Y$ is a toral variety
with a discrete automorphism group.

###### Proof.

Let $r$ by the rank of the group $E(X).$ By Proposition 1, the variety $X$ can
be embedded into the torus $T_{r}.$ We denote by $M$ the lattice of characters
of $T_{r}$ and by $M(X)$ the sublattice in $M$ which corresponds to $X$; see
Remark 2. One can choose a basis $e_{1},\ldots,e_{r}\in M$ such that
$b_{1}e_{1},\ldots b_{l}e_{l}$ is a basis of $M(X)$ for some
$b_{1},\ldots,b_{l}\in{\mathbb{N}}$ and $l\leq r$. Denote by
${t_{1}},\ldots{t_{r}}$ coordinates on $T_{r}$ corresponding to
$e_{1},\ldots,e_{r}$.

Then the equations $\chi^{m}(t)=1$ for all $m\in M(X)$ define a subgroup $H$
in $T_{r}$ which consists of elements of the form

$(\epsilon_{1},\ldots,\epsilon_{l},t_{l+1},\ldots,t_{r}),$

where $\epsilon_{1},\ldots,\epsilon_{l}$ are the roots of unity of degrees
$b_{1},\ldots,b_{l}$ respectively and $t_{l+1},\ldots
t_{r}\in{\mathbb{K}}^{*}.$

Then all minimal polynomials in $I(X)$ are semi-invariant with respect to $H$.
This means that minimal polynomials in $I(X)$ are homogeneous with respect to
variables ${t}_{l+1},\ldots,{t}_{r}.$ Since functions ${t}_{i}$ are invertible
one can choose a set of minimal generators of $I(X)$ which do not depend on
${t}_{l+1},\ldots,{t}_{r}.$ It implies that $X\simeq Y\times T_{r-l}$ where
$Y$ is a subvariety of $T_{l}.$

We see that $T_{r-l}$ coincides with the maximal torus in $H\simeq H(X)$. The
variety $Y$ is also a toral variety which is given by the ideal
$I(X)\cap{\mathbb{K}}[t_{1},\ldots t_{l}].$ Since the maximal torus in
$\operatorname{Aut}(X)$ is isomorphic to $T_{l}$ the maximal torus in
$\operatorname{Aut}(Y)$ is trivial. It follows from Proposition 1 that in this
case $\operatorname{Aut}(Y)$ is a discrete group.

∎

Let $X$ be a toral variety and suppose that $X\simeq T_{s}\times Y$ where $Y$
is a toral variety with a discrete automorphism group and $T_{s}$ is the torus
$({\mathbb{K}}^{*})^{s}.$ One can see that $\operatorname{Aut}(X)$ contains
the following subgroups.

There is a subgroup which is isomorphic to $\operatorname{Aut}(Y)$. This
subgroup acts in a natural way on $Y$ and acts trivially on $T_{s}.$ The
subgroup $\mathrm{GL_{s}}({\mathbb{Z}})$ acts naturally on $T_{s}$ and
trivially on $Y$. Moreover there is a subgroup which is isomorphic to
$({\mathbb{K}}[Y]^{*})^{s}\simeq(E(Y)\times{\mathbb{K}}^{*})^{s}.$ This
subgroup acts in the following way. If
$f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}$ then we can define an automorphism
of $T_{s}\times Y$ as follows

$(t_{1},\ldots,t_{s},y)\to(f_{1}(y)t_{1},\ldots,f_{s}(y)t_{s},y).$

The following theorem was proposed to the authors by Sergey Gaifullin.

######  Theorem 2.

Let $X\simeq T_{s}\times Y$ be a toral variety, where $Y$ is a toral variety
with a discrete automorphism. Then

$\operatorname{Aut}(X)\cong\operatorname{Aut}(Y)\ltimes(\mathrm{GL}_{s}({\mathbb{Z}})\ltimes(E(Y)\times{\mathbb{K}}^{*})^{s}).$

###### Proof.

There is a natural action of $T_{s}$ on $X$. We see that ${\mathbb{K}}[Y]$ is
the algebra of invariants of this action. Since $T_{s}$ is a maximal torus in
$\operatorname{Aut}(X)$ each automorphism of $T_{s}\times Y$ preserves
${\mathbb{K}}[Y].$ So we obtain a homomorphism
$\Phi:\operatorname{Aut}(X)\to\operatorname{Aut}(Y).$ Let $B$ be the kernel of
$\Phi$. The group $\operatorname{Aut}(Y)$ is naturally embedded into
$\operatorname{Aut}(T_{s}\times Y)$ and intersects trivially with $B$. At the
same time $\operatorname{Aut}(Y)$ maps isomorphically to the image of $\Phi$.
It implies that

$\operatorname{Aut}(T_{s}\times Y)=\operatorname{Aut}(Y)\ltimes B.$

We denote by $t_{1},\ldots,t_{s}$ coordinate functions on $T_{s}.$ Then

${\mathbb{K}}[T_{s}\times
Y]\simeq{\mathbb{K}}[T_{s}]\otimes{\mathbb{K}}[Y]={\mathbb{K}}[Y][t_{1}^{\pm
1},\ldots t_{s}^{\pm 1}].$

Let $\phi\in B.$ The algebra ${\mathbb{K}}[Y]$ is invariant with respect to
$\phi^{*}$. So for all $t\in T_{s}$ and $y\in Y$ we have

$\phi((t,y))=(t^{\prime},y),$

for some $t^{\prime}\in T_{s}.$ Therefore, for each $y\in Y$ the automorphism
$\phi$ defines an automorphism $\phi_{y}:T_{s}\to T_{s}.$ Hence for each $y\in
Y$ we have

$\phi^{*}(t_{i})(t,y)=t_{i}((\phi(t),\phi(y)))=t_{i}((\phi_{y}(t),y))=f_{i}(y)t_{1}^{a_{i1}(y)}\ldots
t_{s}^{a_{is}(y)}$

for some non-zero constant $f_{i}(y)$ and a matrix
$A(y)=(a_{ij}(y))\in\mathrm{GL}_{s}({\mathbb{Z}}).$ In reasons of continuity
the matrix $A(y)$ is the same for all $y\in Y$ and $f_{i}:Y\to Y$ are regular
functions on $Y$. Since $f_{i}(y)\neq 0$ for all $y\in Y$ the functions
$f_{i}$ are invertible. So we have

$\phi^{*}(t_{i})=f_{i}t_{1}^{a_{i1}}\ldots t_{s}^{a_{is}}$

for some $f_{i}\in{\mathbb{K}}[Y]^{*}$ and
$A\in\mathrm{GL}_{s}({\mathbb{Z}}).$

Then we have a homomorphism
$\overline{\Phi}:B\to\mathrm{GL}_{s}({\mathbb{Z}}),\ \phi\to A.$ Again, it is
easy to see that $\mathrm{GL}_{s}({\mathbb{Z}})$ is naturally embedded into
$B$ and maps isomorphically to $\mathrm{GL}_{s}({\mathbb{Z}})$ with respect to
$\overline{\Phi}.$ So

$B=\mathrm{GL}_{s}({\mathbb{Z}})\ltimes\mathrm{Ker}\ \overline{\Phi}.$

The kernel of $\overline{\Phi}$ consists of automorphisms
$\varphi\in\operatorname{Aut}(T_{s}\times Y)$ which have a form

$\varphi(t_{1},\ldots,t_{s},y)=(f_{1}(y)t_{1},\ldots,f_{s}(y)t_{s},y).$

for some $f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}.$ We see that for all
$f_{1},\ldots,f_{s}\in{\mathbb{K}}[Y]^{*}$ this formula defines an
automorphism of $T_{s}\times Y,$ so $\mathrm{{\rm Ker}\,}\
\overline{\Phi}=({\mathbb{K}}[Y]^{*})^{s}=(E(Y)\times{\mathbb{K}}^{*})^{s}.$ ∎

##  4\. The case $\mathrm{rk}\ E(X)=\dim X$

Let $X$ be a toral variety. Then $\mathrm{rk}\ E(X)\geq\dim X.$ Indeed,
suppose that $f_{1},\ldots,f_{r}$ are invertible functions and
$[f_{1}],\ldots,[f_{r}]$ is a basis in $E(X).$ Then $f_{1},\ldots,f_{r}$
generate ${\mathbb{K}}[X].$ So $r\geq\dim X$.

The following result shows that if $\mathrm{rk}\ E(X)=\dim X$ then $X$ is a
torus. Moreover this is the only case when $\operatorname{Aut}(X)$ acts with
an open orbit on $X$.

######  Proposition 3.

Let $X$ be a toral variety. Then the following conditions are equivalent.

  1. (1)

$X$ is a torus;

  2. (2)

$\mathrm{rk}\ E(X)=\dim X.$

Suppose also that the field ${\mathbb{K}}$ is uncountable. Then $X$ is a torus
if and only if $\operatorname{Aut}(X)$ acts on $X$ with an open orbit.

###### Proof.

The implication $1)\Rightarrow 2)$ is trivial.

Suppose that $\mathrm{rk}\ E(X)=\dim X.$ Then one can choose invertible
functions $f_{1},\ldots,f_{n}$ such that $[f_{1}],\ldots,[f_{n}]$ is a basis
of $E(X)$. Then ${\mathbb{K}}[X]$ is generated by
$f_{1},f_{1}^{-1},\ldots,f_{n},f_{n}^{-1}$. But $f_{1},\ldots,f_{n}$ are
algebraically independent, otherwise $\dim X<\mathrm{rk}\ E(X)$. So
${\mathbb{K}}[X]$ is isomoprhic to the algebra of Laurent polynomials. So we
obtain implication $2)\Rightarrow 1).$

Now we prove the second statement. If $X$ is a torus then
$\operatorname{Aut}(X)$ acts on $X$ transitively. So suppose $X$ is a toral
variety over an uncountable field ${\mathbb{K}}$ and $\operatorname{Aut}(X)$
acts on $X$ with an open orbit $U$.

Let $T$ be the maximal torus in $\operatorname{Aut}(X).$ Sine the quotient
group $\operatorname{Aut}(X)/T$ is a discrete group the set $U$ is a countable
union of orbits of $T.$ Since ${\mathbb{K}}$ is uncountable it implies that
one of the orbits of $T$ is open in $X.$ Then $\dim X=\dim T.$ By Theorem 1 we
have $X\simeq T\times Y$ for some toral variety $Y$. But since $\dim T=\dim
T\times Y$ we obtain that $Y$ is a point and $X\simeq T.$

∎

##  5\. The case $\mathrm{rk}\ E(X)=\dim X+1$

By Theorem 1 any toral variety is a direct product $T\times Y$ where $T$ is a
torus and $Y$ is a toral variety with a discrete automorphism group. By
Theorem 2 one can find $\operatorname{Aut}(X)$ knowing
$\operatorname{Aut}(Y).$ In this section we provide a way to find
$\operatorname{Aut}(Y)$ when $\mathrm{rk}\ E(Y)=\dim\ Y+1.$

Let $Y$ be a toral variety with a trivial maximal torus in
$\operatorname{Aut}(Y)$. Let $r$ be the rank of $E(Y)$. We suppose that
$r=\dim(Y)+1.$

By Proposition 1 the variety $Y$ can be embedded into $T_{r}$ as a
hypersurface. Then there is an irreducible polynomial
$h\in{\mathbb{K}}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$ such that $I(Y)=(h).$
The polynomial $h$ has a form

$h=\sum_{m\ \in\ \mathrm{Supp}\ h}\alpha_{m}\chi^{m}.$

Let $M$ be the lattice of characters of $T_{r}$ and $M(Y)$ be a sublattice in
$M$ which corresponds to $Y$; see Definition 2. Since the maximal torus in
$\operatorname{Aut}(Y)$ is trivial, the rank of the lattice $M(Y)$ is equal to
$r$. It means that the elements $m_{a}-m_{b}$ with
$m_{a},m_{b}\in\mathrm{Supp}\ h$ generate a sublattice of full rank in $M$.

We denote by $\mathrm{GAff}(M,h)$ the group of all degenerate integer affine
transformations $\varphi$ of $M$ which preserve $\mathrm{Supp}\ h$ and for any
linear combination

$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h$

$\varphi$ satisfies

$\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}=\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{i}}.$
(1)

######  Remark 3.

Let us fix a point $m_{0}\in\mathrm{Supp}\ h.$ Then any equation

$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h$

can be written as

$\sum_{i}{b_{i}}(m_{i}-m_{0})=0,\ b_{i}\in{\mathbb{Z}},\
m_{i}\in\mathrm{Supp}\ h.$ (2)

Therefore, it is enough to check equations (1) for linear combinations of the
form (2). Let $N$ be the number of points in $\mathrm{Supp}\ h.$ Then the set

$\\{(b_{1},\ldots,b_{N-1})\in\mathbb{Z}^{N-1}|\sum_{i}{b_{i}}(m_{i}-m_{0})=0\\}$

is a sublattice in ${\mathbb{Z}}^{N-1}$ and it is enough to check the
equations in (1) for some basis in this sublattice.

######  Theorem 3.

Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Suppose that $\mathrm{rk}\ E(Y)=\dim Y+1.$ Then

$\operatorname{Aut}(Y)/H(Y)\simeq\mathrm{GAff}(M,h).$

###### Proof.

Let $\psi$ be an automorphism of $Y$. By Proposition 1 the automorphism $\psi$
can be extended to an automorphism of $T_{r}$. We denote by $\psi^{*}$ the
respective automorphism of ${\mathbb{K}}[t^{\pm 1}_{1},\ldots,t^{\pm 1}_{r}]$.
Then $\psi^{*}$ has the form

$\psi^{*}(t_{i})=\lambda_{i}t_{1}^{a_{i1}}\ldots t_{r}^{a_{ir}},$

where $\lambda_{i}\in{\mathbb{K}}^{*}$ and
$(a_{ij})\in\mathrm{GL}_{r}({\mathbb{Z}}).$ We denote by $\lambda$ the element
$(\lambda_{1},\ldots,\lambda_{r})\in T_{r}$ and by $\overline{\psi}$ the
automorphism of $M$ which respects to the matrix $(a_{ij}).$ Then

$\psi^{*}(\chi^{m})=\chi^{m}(\lambda)\chi^{\overline{\psi}(m)}$

for all $m\in M.$

The polynomial $\psi^{*}(h)$ also generates $I(Y).$ So it differs from $h$ by
an invertible element of ${\mathbb{K}}[t^{\pm 1}_{1},\ldots,t^{\pm 1}_{r}].$
Then

$\psi^{*}(h)=\alpha\chi^{v}h$

for some $\alpha\in{\mathbb{K}}^{*}$ and $v\in M.$ Therefore, we have an
equation

$\psi^{*}(h)=\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{m}\chi^{m}(\lambda)\chi^{\overline{\psi}(m)}=\alpha\sum_{m\ \in\
\mathrm{Supp}\ h}\alpha_{m}\chi^{m+v}.$ (3)

We define the map $\varphi:M\to M$ by the following formula:

$\varphi(m)=\overline{\psi}(m)-v.$

Then $\psi$ is an affine transformation of $M$ which preserves $\mathrm{Supp}\
h.$

We consider a linear combination

$\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})=0,\ a_{i}\in\mathbb{Z},\
m_{i,1},m_{i,2}\in\mathrm{Supp}\ h.$

Equation (3) can be written as

$\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{m}\chi^{m}(\lambda)\chi^{\varphi(m)}=\alpha\sum_{m\ \in\
\mathrm{Supp}\ h}\alpha_{m}\chi^{m}=\alpha\sum_{m\ \in\ \mathrm{Supp}\
h}\alpha_{\varphi(m)}\chi^{\varphi(m)}.$

and it implies

$\frac{\alpha_{m_{i,1}}\chi^{m_{i,1}}(\lambda)}{\alpha_{m_{i,2}}\chi^{m_{i,2}}(\lambda)}=\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\chi^{m_{i,1}-m_{i,2}}(\lambda)=\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}$

for all $i.$

Then we have

$\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\chi^{m_{i,1}-m_{i,2}}(\lambda)\right)^{a_{i}}=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}\left(\chi^{\sum_{i}{a_{i}}(m_{i,1}-m_{i,2})}(\lambda)\right)=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{a_{i}}$
$=\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{i}}.$

So $\varphi\in\mathrm{Gaff}(M,h).$ Then we obtain a homomorphism
$\eta:\operatorname{Aut}(Y)\to\mathrm{Gaff}(M,h).$ Moreover we see that the
kernel of $\eta$ is $H(Y).$ Now we will show that $\eta$ is surjective.

Let $\varphi\in\mathrm{Gaff}(M,h)$ and $f_{1},\ldots,f_{r}$ be a basis in
$M(Y).$ Then there are $m_{i,1},m_{i,2}\in\ \mathrm{Supp}(h),$ and
$a_{ij}\in{\mathbb{Z}}$ such that

$f_{j}=\sum_{i}a_{ij}(m_{i,1}-m_{i,2}).$

There is a $\lambda\in T_{r}$ such that

$\chi^{f_{j}}(\lambda)=\prod_{i}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-a_{ij}}\prod_{i}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{a_{ij}}.$

Let $d\varphi$ be the differential of $\varphi.$ We define an automorphism
$\psi^{*}$ of ${\mathbb{K}}[t_{1}^{\pm 1}\ldots t_{r}^{\pm 1}]$ by the
following rule:

$\psi^{*}(\chi^{m})=\chi^{m}(\lambda)\chi^{d\varphi(m)}.$

Let us check that $\psi^{*}$ preserves $I(Y).$ We have

$\psi^{*}(h)=\sum\alpha_{m}\chi^{m}(\lambda)\chi^{d\varphi(m)}.$

There is $v\in M$ such that

$\varphi(m)=d\varphi(m)+v.$

Then

$\chi^{v}\psi^{*}(h)=\sum\alpha_{m}\chi^{m}(\lambda)\chi^{\varphi(m)}.$

We see that $\mathrm{Supp}\ \chi^{v}\psi^{*}(h)=\mathrm{Supp}\ h.$ We will
show that there is $\alpha\in{\mathbb{K}}$ such that

$\chi^{v}\psi^{*}(h)=\alpha h.$

For any $b,c\in\mathrm{Supp}\ h$ there are numbers $d_{j}\in{\mathbb{Z}}$ such
that

$b-c=\sum_{j}d_{j}f_{j}.$

So

$\frac{\alpha_{b}\chi^{b}(\lambda)}{\alpha_{c}\chi^{c}(\lambda)}=\frac{\alpha_{c}}{\alpha_{b}}\chi^{b-c}(\lambda)=\frac{\alpha_{b}}{\alpha_{c}}\chi^{\sum_{j}d_{j}f_{j}}(\lambda)=$
$\frac{\alpha_{b}}{\alpha_{c}}(\prod_{j}\chi^{f_{j}}(\lambda))^{d_{j}}=\frac{\alpha_{b}}{\alpha_{c}}\prod_{i,j}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-d_{j}a_{ij}}\prod_{i,j}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{d_{j}a_{ij}}.$

We have a combination

$b-c-\sum_{j}d_{j}f_{j}=b-c-\sum_{j,i}d_{j}a_{ij}(m_{i,1}-m_{i,2})=0.$

Since $\varphi\in\mathrm{Gaff}(M,h)$ we obtain

$\frac{\alpha_{b}}{\alpha_{c}}\prod_{i,j}\left(\frac{\alpha_{m_{i,1}}}{\alpha_{m_{i,2}}}\right)^{-d_{j}a_{ij}}=\frac{\alpha_{\varphi(b)}}{\alpha_{\varphi(c)}}\prod_{i,j}\left(\frac{\alpha_{\varphi(m_{i,1})}}{\alpha_{\varphi(m_{i,2})}}\right)^{-d_{j}a_{ij}}.$

It implies that

$\frac{\alpha_{b}\chi^{b}(\lambda)}{\alpha_{c}\chi^{c}(\lambda)}=\frac{\alpha_{\varphi(b)}}{\alpha_{\varphi(c)}}.$

So the coefficients of the polynomials $\chi^{v}\psi^{*}(h)$ and $h$ are
proportional. Then there is an $\alpha\in{\mathbb{K}}$ such that
$\chi^{v}\psi^{*}(h)=\alpha h.$ Hence $\psi^{*}(h)=\alpha\chi^{-v}h\in I(Y)$.
Therefore, $\psi^{*}$ preserves $I(Y)$ and defines an automorphism $\psi$. It
is a direct check that $\eta(\psi)=\varphi$. So $\varphi$ is surjective.

∎

######  Corollary 1.

Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Suppose that $\mathrm{{\mathrm{rk}}}\ E(Y)=\dim Y+1.$
Then $\operatorname{Aut}(Y)$ is a finite group.

It is natural to formulate the following question.

######  Question 1.

Let $Y$ be a toral variety with trivial maximal torus in
$\operatorname{Aut}(Y).$ Is $\operatorname{Aut}(Y)$ a finite group?

Note that this is not true for a general rigid variety. One can find a
counterexample in [7].

At the end, we give two examples illustrating Theorem 3.

######  Example 1.

Let $Y$ be the affine line $\mathbb{A}^{1}$ without two points. The variety
$Y$ can be given in $({\mathbb{K}}^{*})^{2}$ as the set of solutions of the
equation

$h=x+y-1=0,\ (x,y)\in({\mathbb{K}}^{*})^{2}.$

So $Y$ is a toral variety. Let $M\simeq{\mathbb{Z}}^{2}$ be the lattice of
characetrs of $({\mathbb{K}}^{*})^{2}$. We see that the lattice $M(Y)$
contains elements $(1,0),(0,1),(1,-1)$, so $M(Y)=M.$ Therefore, $H(Y)$ is a
trivial group. The group $E(Y)$ is generated by $[x]$ and $[y]$. So we can
apply Theorem 3.

Now we find the group $\mathrm{Gaff}(M,h)$. The set $\mathrm{Supp}\ h$
consists of points $m_{0}=(0,0),\ m_{1}=(1,0),m_{2}=(0,1).$

$m_{2}$$m_{0}$$m_{1}$$M$ Figure 1. $\mathrm{Supp}(x+y-1)$

A linear combination

$a_{1}(m_{1}-m_{0})+a_{2}(m_{2}-m_{0})+a_{3}(m_{2}-m_{1})=a_{1}(1,0)+a_{2}(0,1)+a_{3}(-1,1)$

is equal to zero if and only if $a_{1}=a_{3}=-a_{2}.$ But then equations (1)
are trivial. By affine transformations of $M$ we can permute all points in
$\mathrm{Supp}\ h.$ Therefore,

$\operatorname{Aut}(Y)\simeq\mathrm{GAff}(M,h)\simeq S_{3}.$

The answer looks natural since the affine line without two points is the
projective line without three points.

######  Example 2.

Let $a,b,c\in{\mathbb{K}}^{*}$ be non-zero constants and $Y$ be the set of
solutions of the equation

$h=1+ax+by+cxy=0,\ (x,y)\in({\mathbb{K}}^{*})^{2}.$

Let $M\simeq{\mathbb{Z}}^{2}$ be the lattice of characters of
$({\mathbb{K}}^{*})^{2}$. The lattice $M(Y)$ contains $(1,0)$ and $(0,1)$ so
$M(Y)=M.$ Then $H(Y)$ is a trivial group. The group $E(Y)$ is generated by
$[x]$ and $[y]$. Again, we can apply Theorem 3.

The set $\mathrm{Supp}\ h$ consists of points $m_{0}=(0,0),\
m_{1}=(1,0),m_{2}=(0,1),m_{3}=(1,1).$

$m_{3}$$m_{2}$$m_{0}$$m_{1}$$M$ Figure 2. $\mathrm{Supp}(1+ax+by+cxy)$

The set of affine transformation which preserve $\mathrm{Supp}\ h$ is the
dihedral group $D_{4}.$

To understand what elements of $D_{4}$ belong to $\mathrm{GAff}(M,h)$ we need
to check the equations (3). To simplify notations we will denote $m_{0}$ by
$1,$ $m_{1}$ by $x,$ $m_{2}$ by $y$ and $m_{3}$ by $xy$. By $\alpha_{m}$ we
denote the coefficient in $h$ of monomial $\chi^{m}$. So we have

$\alpha_{1}=1,\ \alpha_{x}=a,\ \alpha_{y}=b,\ \alpha_{xy}=c.$

By Remark 3 it is enough to check the equations (3) only for linear
combinations of the form

$a_{1}(m_{1}-m_{0})+a_{2}(m_{2}-m_{0})+a_{3}(m_{3}-m_{0})=a_{1}(1,0)+a_{2}(0,1)+a_{3}(1,1)=0,$

where $a_{1},a_{2},a_{3}\in{\mathbb{Z}}.$ So we have $a_{1}=a_{2}=-a_{3}.$
Again, by Remark 3 it is enough to check the equation

$\left(\frac{\alpha_{x}}{\alpha_{1}}\right)^{1}\left(\frac{\alpha_{y}}{\alpha_{1}}\right)^{1}\left(\frac{\alpha_{xy}}{\alpha_{1}}\right)^{-1}=\left(\frac{\alpha_{\varphi(x)}}{\alpha_{\varphi(1)}}\right)^{1}\left(\frac{\alpha_{\varphi(y)}}{\alpha_{\varphi(1)}}\right)^{1}\left(\frac{\alpha_{\varphi(xy)}}{\alpha_{\varphi(1)}}\right)^{-1}.$

So an element $\varphi\in D_{4}$ belongs to $\mathrm{GAff}(M,h)$ if and only
if

$\frac{ab}{c}=\frac{\alpha_{\varphi(x)}\alpha_{\varphi(y)}}{\alpha_{\varphi(1)}\alpha_{\varphi(xy)}}.$

Substituting various elements of the dihedral group into this equality, we
obtain that the rotation by $\frac{\pi}{2}$ and the symmetries with respect to
the lines passing through the midpoints of the opposite sides belong to
$\mathrm{GAff}(M,h)$ if and only if

$\frac{ab}{c}=\frac{c}{ab}\iff(ab)^{2}=c^{2}.$

The rotation by $\pi$ and the symmetries with respect to the diagonals belong
to $\mathrm{GAff}(M,h)$ for any $a,b,c.$

So we obtain that if $(ab)^{2}=c^{2}$ then $\operatorname{Aut}(Y)\simeq D_{4}$
and if $(ab)^{2}\neq c^{2}$ then

$\operatorname{Aut}(Y)\simeq({\mathbb{Z}}/2{\mathbb{Z}})\oplus({\mathbb{Z}}/2{\mathbb{Z}}).$

## References

  * [1] Ivan Arzhantsev and Sergey Gaifullin. The automorphism group of a rigid affine variety. Math. Nachr., 290:5, 662–671, 2017
  * [2] David Bernstein. The number of roots of a system of equations. Funct. Anal. Appl., 9:3, 183–185, 1975
  * [3] June Huh. The maximum likelihood degree of a very affine variety. Compos. Math., 149:8, 1245–1266, 2013
  * [4] Shulim Kaliman. Extensions of isomorphisms between affine algebraic subvarieties of $k^{n}$ to automorphisms of $k^{n}$. Proc. Amer. Math. Soc. 113:2, 325–334, 1991
  * [5] Shulim Kaliman and David Udumyan. On automorphisms of flexible varieties. Adv. Math., 396, Article no. 108112, 2022
  * [6] Anatoly Koushnirenko. Newton polytopes and the Bezout theorem. Funct. Anal. Appl., 10:3, 82–83, 1976
  * [7] Alexander Perepechko. Automorphisms of surfaces of Markov type. Math. Notes, 110:5, 732–737, 2021
  * [8] Alexander Perepechko and Michael Zaidenberg. Automorphism group of affine rigid surfaces: the identity component. arXiv:2208.09738, 2022
  * [9] Vladimir Popov. On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties. CRM Proc. Lect. Notes, 54:17, 289–311, 2011
  * [10] Maxwell Rosenlicht. Some rationality questions on algebraic groups. Ann. Mat. Pura Appl., 43, 25–50, 1957
  * [11] Jenia Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129:4, 1087–1104, 2007