Ryo Masumura, Naoki Makishima, Taiga Yamane, Yoshihiko Yamazaki, Saki Mizuno,
Mana Ihori, Mihiro Uchida, Keita Suzuki, Hiroshi Sato, Tomohiro Tanaka,
Akihiko Takashima, Satoshi Suzuki, Takafumi Moriya, Nobukatsu Hojo, Atsushi
Ando

# End-to-End Joint Target and Non-Target Speakers ASR

###### Abstract

This paper proposes a novel automatic speech recognition (ASR) system that can
transcribe individual speaker's speech while identifying whether they are
target or non-target speakers from multi-talker overlapped speech. Target-
speaker ASR systems are a promising way to only transcribe a target speaker's
speech by enrolling the target speaker’s information. However, in
conversational ASR applications, transcribing both the target speaker’s speech
and non-target speakers’ ones is often required to understand interactive
information. To naturally consider both target and non-target speakers in a
single ASR model, our idea is to extend autoregressive modeling-based multi-
talker ASR systems to utilize the enrollment speech of the target speaker. Our
proposed ASR is performed by recursively generating both textual tokens and
tokens that represent target or non-target speakers. Our experiments
demonstrate the effectiveness of our proposed method.

Index Terms: automatic speech recognition, target-speaker, non-target
speakers, autoregressive model, enrollment speech

##  1 Introduction

Single-talker automatic speech recognition (ASR) systems limit their
applicability because multiple utterances are often overlapped in many
practical scenes. Therefore, recognizing multi-talker overlapped monaural
speech signals has been the focus of much attention [1, 2]. In particular,
target-speaker ASR (TS-ASR) systems that can transcribe only a target
speaker's speech from the overlapped ones is known to be an effective way when
the target speaker is specified. The systems are informed about the target
speaker using an enrollment speech of that speaker, and can extract the same
speaker's speech as the enrollment one. In this paper, we aim to improve ASR
systems that use the speaker enrollment to make it suitable for conversational
ASR applications such as meetings.

Toward a better performance of TS-ASR, several methods have been developed
with the progress of deep learning technology. Initial TS-ASR systems have
been built by cascading single-talker ASR with target-speaker speech
extraction [3, 4, 5, 6, 7], which is a process to separate a target speaker's
speech from a multi-talker overlapped one. The cascading systems are
comparatively feasible to build because individual components are separately
trained, but their performance is limited because overall optimization of TS-
ASR cannot be achieved. Therefore, previous studies challenge to directly
build TS-ASR in an end-to-end manner [8, 9, 10]. The main advantage of the
end-to-end TS-ASR systems is that overall optimization to transcribe a target
speaker's speech into text can be performed. In addition, since there is no
need to separate the target speaker's speech as a signal, computation
complexity can be reduced compared with the cascading systems.

TS-ASR systems are really effective to execute personalized speech commands in
a cocktail-party situation or record personalized life-logs; however, ignoring
non-target speakers' speech is unsuitable for understanding interactions
between the target speaker and non-target speakers (e.g., interactions between
a target salesperson and customers in a store). In conversational ASR
applications, both the target speaker's speech and non-target speakers’ ones
often need to be transcribed. In addition, we suspect that eliminating non-
target speakers' speech is ineffective to perform accurate TS-ASR. Considering
non-target speakers' information will explicitly enables us to model the
dependency among the speakers and lead to improving TS-ASR performance. In
fact, considering non-target interference speaker's information has achieved
better performance in senone-based acoustic modeling [11]. One challenge is
how we naturally consider both the target speaker and non-target speakers in
an end-to-end ASR framework.

In this paper, we propose end-to-end joint target and non-target speakers ASR
(TS-NTS-ASR) systems that jointly transcribe a target speaker's and non-target
speakers’ speech using a unified autoregressive model. Our key idea is to
extend autoregressive modeling-based multi-talker ASR (MT-ASR) systems
(detailed in Sections 2 and 3.2) [12, 13] to utilize the enrollment speech of
the target speaker. In existing MT-ASR systems, transcriptions of multiple
speakers are generated as a serialized token sequence, but it does not
determine whether each transcription is spoken by the target speaker or not.
Therefore, our proposed method utilizes one unified autoregressive model for
not only transcribing all speakers' speech but also identifying whether they
are the target speaker or non-target speakers using the enrollment speech. The
identification is achieved by handling both textual tokens and tokens that
represent the target or non-target speakers within the serialized token
sequence. Our proposed TS-NTS-ASR systems have two advantages: they can
transcribe both the target speaker's speech and non-target speakers' speech in
a simple beam search decoding, and they improve TS-ASR performance compared
with conventional TS-ASR systems that ignore non-target speakers' speech.

We detail our contributions as follows.

  * •

This paper is the first study of end-to-end joint target and non-target
speakers ASR systems. For the proposed autoregressive modeling, we show three
serialized token sequence patterns to take the target speaker and non-target
speakers' spoken text into consideration. We also show a network structure
that can effectively utilize the speaker enrollment in autoregressive
modeling.

  * •

This paper also shows a method to implement non-target speakers ASR (NTS-ASR)
systems that transcribe an individual speaker's speech except for the target
speaker's speech.

  * •

This paper demonstrates that our proposed method effectively transcribes both
the target speaker's speech and non-target speakers' speech in experiments
using a Japanese multi-talker overlapped speech dataset.

##  2 Related Work

Speech processing with target speaker enrollment: This paper is related to
speech processing that uses speaker enrollment. In previous studies, target-
speaker speech extraction [3, 4, 5, 6, 7], TS-ASR [8, 9, 10], and target-
speaker voice activity detection [14, 15, 16, 17, 18, 19] have been mainly
investigated. These studies use the enrollment speech of a target speaker to
extract the target speaker's information. In contrast, our proposed method
examines speech processing for both the target speaker and non-target speakers
using the target speaker enrollment.

Multi-talker ASR: This paper is related to end-to-end MT-ASR systems that can
transcribe an individual speakers' speech from a multi-talker overlapped one.
Initial end-to-end MT-ASR systems introduce modeling with multiple output
branches, in which each branch generates a transcription for one speaker by
considering all possible permutations of speakers [20, 21, 22, 23, 24, 25,
26]. A recent hopeful approach for end-to-end modeling is autoregressive
modeling in which transcriptions of multiple speakers are recursively
generated from one output branch [12, 13]. Our proposed method is regarded as
an extended modeling of the latter MT-ASR to identify a target speaker or non-
target speakers using the target speaker enrollment.

##  3 Preliminaries

This section describes target-speaker ASR (TS-ASR) and multi-talker ASR (MT-
ASR) systems based on autoregressive modeling. These two systems can handle
monaural multi-talker overlapped speech. The former generates only the target
speaker's spoken text and the latter generates all speakers' spoken text.

###  3.1 Target-speaker ASR

TS-ASR based on autoregressive modeling predicts a generation probability of a
target speaker's spoken text $\bm{W}=\\{w_{1},\cdots,w_{N}\\}$ from monaural
multi-talker overlapped speech $\bm{X}$ and the target speaker's enrollment
speech $\bm{E}$, where $w_{n}\in{\cal V}$ is the $n$-th token in the spoken
text, $N$ is the number of tokens in the spoken text, and ${\cal V}$ is the
vocabulary set. In autoregressive modeling, the generation probability of
$\bm{W}$ is defined as

$P(\bm{W}|\bm{X},\bm{E};\bm{\Theta}_{\tt
ts})=\prod_{n=1}^{N}P(w_{n}|w_{1:n-1},\bm{X},\bm{E};\bm{\Theta}_{\tt ts}),$
(1)

where $\bm{\Theta}_{\tt ts}$ represents the trainable model parameter sets and
$w_{1:n-1}=\\{w_{1},\cdots,w_{n-1}\\}$. The loss function to optimize the
model parameter sets is defined as

${\cal L}(\bm{\Theta}_{\tt ts})=-\sum_{(\bm{X},\bm{E},\bm{W})\in{\cal D}_{\tt
ts}}\log P(\bm{W}|\bm{X},\bm{E};\bm{\Theta}_{\tt ts}),$ (2)

where ${\cal D}_{\tt ts}$ represents a paired dataset of input multi-talker
speech, the target speaker's enrollment speech, and the target speaker's
spoken text. Note that spoken text becomes empty when the target speaker is
not included in the multi-talker overlapped speech.

###  3.2 Multi-talker ASR

MT-ASR based on autoregressive modeling predicts a generation probability of
all speakers' spoken text $\bm{W}^{1:T}=\\{\bm{W}^{1},\cdots,\bm{W}^{T}\\}$
from monaural multi-talker overlapped speech $\bm{X}$, where
$\bm{W}^{t}=\\{w_{1}^{t},\cdots,w_{N^{t}}^{t}\\}$ is the $t$-th speaker's
spoken text, $N^{t}$ is the number of tokens in the $t$-th speaker's spoken
text, and $T$ is the number of speakers in the multi-talker overlapped speech.
Multiple spoken texts in one autoregressive model are serialized into a single
token sequence. Thus, the generation probability of $\bm{W}^{1:T}$ is defined
as

$\begin{split}P(\bm{W}^{1:T}|\bm{X};\bm{\Theta}_{\tt
mt})&=P(\bm{S}|\bm{X};\bm{\Theta}_{\tt mt})\\\
&=\prod_{l=1}^{|\bm{S}|}P(s_{l}|s_{1:l-1},\bm{X};\bm{\Theta}_{\tt
mt}),\end{split}$ (3)

where $\bm{\Theta}_{\tt mt}$ represents the trainable model parameter sets,
$\bm{S}=\\{s_{1},\cdots,s_{|\bm{S}|}\\}$ is the serialized token sequence, and
$s_{l}\in\\{{\cal V}\cup{\cal O}\\}$ is the $l$-th token in the serialized
token sequence. $\cal O=\\{{\tt[sep]},{\tt[eos]}\\}$ represents the special
token set, where $\tt[sep]$ represents the speaker change and $\tt[eos]$
represents the end-of-sentence. There are multiple permutations in the order
of the multiple spoken texts $\bm{W}^{1:T}$, so they are sorted by their start
times, which is called first-in first-out. When the speaker index $t$ is
ordered by the start time, the serialized token sequence is represented as

$\bm{S}=\\{w_{1}^{1},\cdots,w_{N^{1}}^{1},{\tt[sep]},w_{1}^{2},\cdots,w_{N^{2}}^{2},\\\
\cdots,w_{N^{T-1}}^{T-1},{\tt[sep]},w_{1}^{T},\cdots,w_{N^{T}}^{T},{\tt[eos]}\\}.$
(4)

Thus, the serialized token sequence is composed by concatenating multiple
spoken texts while inserting $\tt[sep]$ between them and $\tt[eos]$ at the end
of the entire sequence.

The loss function to optimize the model parameter sets is defined as

${\cal L}(\bm{\Theta}_{\tt mt})=-\sum_{(\bm{X},\bm{S})\in{\cal D}_{\tt
mt}}\log P(\bm{S}|\bm{X};\bm{\Theta}_{\tt mt}),$ (5)

where ${\cal D}_{\tt mt}$ represents a paired dataset of the serialized token
sequence and the multi-talker overlapped speech.

##  4 Proposed Method

This paper proposes end-to-end joint target and non-target speaker ASR (TS-
NTS-ASR) systems that can transcribe an individual speaker's speech while
identifying whether they are the target speaker or non-target speakers from
multi-talker overlapped speech and the target speaker's enrollment speech.

###  4.1 Modeling

We construct a TS-NTS-ASR model by extending autoregressive modeling-based MT-
ASR (described in Section 3.2) to utilize the target speaker's enrollment
speech. The TS-NTS-ASR model predicts a joint generation probability of a
target speaker's spoken text $\bm{W}=\\{w_{1},\cdots,w_{N}\\}$ and non-target
speakers' spoken texts
$\bar{\bm{W}}^{1:T-1}=\\{\bar{\bm{W}}^{1},\cdots,\bar{\bm{W}}^{T-1}\\}$ from
monaural multi-talker overlapped speech $\bm{X}$ and the target speaker's
enrollment speech $\bm{E}$, where
$\bar{\bm{W}}^{t}=\\{\bar{w}_{1}^{t},\cdots,\bar{w}_{N^{t}}^{t}\\}$ is the
$t$-th non-target speaker's spoken text. To handle these outputs within an
autoregressive model, the target speaker's spoken text and non-target
speakers' spoken texts are serialized as a single token sequence
$\bm{Z}=\\{z_{1},\cdots,z_{|\bm{Z}|}\\}$. In this case, the joint generation
probability is defined as

$\begin{split}P(\bm{W},\bar{\bm{W}}^{1:T-1}|\bm{X},\bm{E};\bm{\Theta})&=P(\bm{Z}|\bm{X},\bm{E};\bm{\Theta})\\\
&=\prod_{l=1}^{|\bm{Z}|}P(z_{l}|z_{1:l-1},\bm{X},\bm{E};\bm{\Theta}),\end{split}$
(6)

where $\bm{\Theta}$ represents the trainable model parameter sets.
$z_{l}\in\\{{\cal V}\cup{\cal U}\\}$ is the $l$-th token in the serialized
token sequence. $\cal U=\\{{\tt[t]},{\tt[nt]},{\tt[eos]}\\}$ represents the
special token set, where $\tt[t]$ represents the target speaker's section and
$\tt[nt]$ represents non-target speaker's section.

TS-NTS-ASR systems can be associated with those described in Section 3. By
disregarding $\bm{E}$ from Eq. (6) to not identify the target or non-target
speakers, the system can be regarded as the same modeling as MT-ASR defined by
Eq. (3). In addition, by disregarding $\bar{\bm{W}}^{1:T-1}$ from Eq. (6), the
system can be regarded as the same modeling as TS-ASR defined by Eq. (1). On
the other hand, by disregarding $\bm{W}$ from Eq. (6), the system can be
regarded as a non-target speakers ASR (NTS-ASR) system that transcribes an
individual speaker's speech except for the target speaker's speech.

The loss function to optimize the model parameter sets is defined as

${\cal L}(\bm{\Theta})=-\sum_{(\bm{X},\bm{E},\bm{Z})\in{\cal D}}\log
P(\bm{Z}|\bm{X},\bm{E};\bm{\Theta}),$ (7)

where ${\cal D}$ represents a paired dataset of input multi-talker speech, the
target speaker's enrollment speech, and the serialized token sequence.

###  4.2 Serialization of tokens

For the proposed autoregressive modeling, we show three serialized token
sequence patterns. In each serialization, we can transcribe an individual
speaker’s speech while identifying whether they are the target or non-target
speaker.

Target-speaker first: We put the target speaker's spoken text first, then the
non-target speakers' spoken texts. The serialized token sequence is composed
by

$\bm{Z}=\\{{\tt[t]},w_{1},\cdots,w_{N},{\tt[nt]},\bar{w}_{1}^{1},\cdots,\bar{w}_{N^{1}}^{1},\\\
\cdots,{\tt[nt]},\bar{w}_{1}^{T-1},\cdots,\bar{w}_{N^{T-1}}^{T-1},{\tt[eos]}\\}.$
(8)

Note that we can perform TS-ASR by stopping when the first ${\tt[nt]}$ is
generated.

Non-target speaker first: We put the non-target speakers' spoken texts first,
then the target speaker's spoken text. The serialized token sequence is
composed by

$\bm{Z}=\\{{\tt[nt]},\bar{w}_{1}^{1},\cdots,\bar{w}_{N^{1}}^{1},\cdots,{\tt[nt]},\bar{w}_{1}^{T-1},\cdots,\bar{w}_{N^{T-1}}^{T-1},\\\
{\tt[t]},w_{1},\cdots,w_{N},{\tt[eos]}\\}.$ (9)

Note that we can perform NTS-ASR by stopping when the first ${\tt[t]}$ is
generated.

First-in first-out: We sort each transcription by their start times regardless
of whether the text is spoken by the target or non-target speaker. For
example, when the target speaker starts speaking a second earlier among all
speakers, the serialized token sequence is composed by

$\bm{S}=\\{{\tt[nt]},\bar{w}_{1}^{1},\cdots,\bar{w}_{N^{1}}^{1},{\tt[t]},w_{1},\cdots,w_{N},\\\
{\tt[nt]},\bar{w}_{1}^{2},\cdots,\bar{w}_{N^{2}}^{2},\cdots,{\tt[nt]},\bar{w}_{1}^{T-1},\cdots,\bar{w}_{N^{T-1}}^{T-1}\\}.$
(10)

Note that this is the same arrangement as that for MT-ASR.

Figure 1: Network structure of TS-NTS-ASR. Table 1: Experimental results in terms of character error rate (%). System | Speaker | Serialization | Single-talker speech | Multi-talker overlapped speech
---|---|---|---|---
| enrollment | of tokens | Target | Non-target | Target | Non-target | Multi-talker
MT-ASR | - | First-in first-out | - | - | - | - | 12.75
TS-ASR | $\checkmark$ | - | 6.95 | - | 12.54 | - | -
NTS-ASR | $\checkmark$ | - | - | 8.36 | - | 13.80 | -
TS-NTS-ASR (proposed) | $\checkmark$ | Target first | 6.15 | 7.87 | 12.36 | 13.05 | 12.60
TS-NTS-ASR (proposed) | $\checkmark$ | Non-target first | 6.52 | 7.40 | 13.18 | 13.24 | 13.57
TS-NTS-ASR (proposed) | $\checkmark$ | First-in first-out | 5.29 | 7.14 | 11.80 | 12.53 | 12.15

###  4.3 Network structure

We construct the TS-NTS-ASR system from two speaker encoders and a text
decoder. Figure 1 shows the network structure of TS-NTS-ASR.

Speaker encoder: The speaker encoder converts acoustic features of the target
speaker's enrollment speech into a speaker vector. This conversion is defined
as

$\bm{c}={\rm SpeakerEnc}(\bm{E};\bm{\theta}_{\rm e}),$ (11)

where $\bm{\theta}_{\rm e}\in\bm{\Theta}$ is its parameters. We implement this
function using convolution and pooling layers, a positional encoding layer,
transformer encoder blocks, an attentive pooling layer, and a linear layer.

Speech encoder: The speech encoder converts acoustic features of the multi-
talker overlapped speech and the speaker vector into hidden representations.
This conversion is defined as

$\bm{H}={\rm SpeechEnc}(\bm{X},\bm{c};\bm{\theta}_{\rm x}),$ (12)

where $\bm{\theta}_{\rm e}\in\bm{\Theta}$ is its parameters. We implement this
function using convolution and pooling layers to generate subsampled speech
representations, a positional encoding layer, an element-wise multiplication
layer with a linear layer, and transformer encoder blocks. The element-wise
multiplication is performed between the speaker vector and subsampled speech
representations to consider the target speaker's information [5].

Text decoder: The token decoder computes the generation probability of a token
given preceding tokens and the hidden representations produced in the speech
encoder. The generation probability is computed from

$P(z_{l}|z_{1:l-1},\bm{X},\bm{E},\bm{\Theta})={\rm
TextDec}(z_{1:l-1},\bm{H};\bm{\theta}_{\rm z}),$ (13)

where $\bm{\theta}_{\rm z}\in\bm{\Theta}$ is its parameters. We implement this
function using a linear embedding layer, a positional encoding layer,
transformer decoder blocks, and a softmax layer with a linear transformation.

##  5 Experiments

In the experiments, we used the Corpus of Spontaneous Japanese (CSJ) [27],
which is a single-talker speech dataset. We first split the dataset into
training set (518.4 h), validation set (1.3 h), and test set (1.9 h) and,
following procedures were conducted for each set. To generate multi-talker
overlapped speech dataset, we mixed multiple audio signals as a monaural
signal. To this end, we randomly chose multiple audio signals so as not to
select the same speakers. We set the number of speakers in the mixed signals
as two or three. When mixing the audio signals, the original volume of each
utterance was kept unchanged, resulting in an average signal-to-interference
ratio of about 0 dB. For the delay applied to each utterance, the start times
of the individual utterances differ by 0.5 s or longer. In addition, every
utterance in each mixed audio sample has at least one speaker-overlapped
region with other utterances. In addition, we prepared enrollment speech for
the single-talker speech and the multi-talker overlapped speech sets. For the
single-talker speech dataset, we prepared two cases: a different utterance of
the target speaker was used as the speaker enrollment, and an utterance of a
different speaker from the target speaker was used as the speaker enrollment.
For the multi-talker overlapped speech dataset, an utterance spoken by one of
the talkers was used as the enrollment. In training, all paired patterns
(about 2,500 h) were jointly used. In testing, each of paired patterns were
individually evaluated.

###  5.1 Setups

We constructed an MT-ASR, TS-ASR, NTS-ASR, and three TS-NTS-ASR systems.
Except for MT-ASR, which does not use speaker enrollment, we introduced the
network structure described in Section 4.3 for each system. MT-ASR used a
network structure that excludes a speaker encoder and its connection. For
these systems, the transformer blocks were composed under the following
conditions: the dimensions of the output continuous representations were set
to 512, the dimensions of the inner outputs were set to 2,048, and the number
of heads in the multi-head attentions was set to 4. In the nonlinear
transformational functions, the Swish activation was used. For both the
speaker and the speech encoders, we used 80 log mel-scale filterbank
coefficients as acoustic features. The frame shift was 10 ms. The acoustic
features passed two convolution and max pooling layers with a stride of 2, so
we down-sampled them to $1/4$ along with the time axis. After these layers, we
stacked 4-layer transformer encoder blocks. For the text decoder, we used
512-dimensional character embeddings where the vocabulary size was set to
3,262. We also stacked 3-layer transformer decoder blocks. For the training,
we used the RAdam optimizer [28]. For the systems that used speaker
enrollment, the speaker encoder was first trained using VoxCeleb2 dataset [29]
with ArcFace criterion [30], and then the speech encoder and text decoder were
trained while freezing the speaker encoder. We set the mini-batch size to 64
utterances and the dropout rate in the transformer blocks to 0.1. We
introduced label smoothing where its smoothing parameter was set to 0.1. In
addition, we applied SpecAugment [31] and enrollment-less training [17]. For
testing, we used a beam search algorithm in which the beam size was set to 4.

###  5.2 Results

Table 1 shows the results in terms of character error rate for each ASR
system. The ``Target'' column represents TS-ASR performance and the ``Non-
target'' column represents NTS-ASR performance, each of which was only
evaluated when using the speaker enrollment. The ``Multi-talker'' column
represents MT-ASR performance that evaluated all speakers' results. First, the
results show that each TS-NTS-ASR system outperformed the TS-ASR system. This
indicates that considering non-target speakers' information explicitly leads
to improving TS-ASR performance. In addition, each TS-NTS-ASR system
outperformed the NTS-ASR system. This also indicates that joint consideration
of the target and non-target speakers is important. Next, among the TS-NTS-ASR
systems, the best results were attained by ``first-in first-out''
serialization. It is considered that ``first-in first-out,'' which can attend
to the input of the multi-talker overlapped speech in left-to-right order, is
easier to train than ``target first'' and ``non-target first.'' Furthermore,
Table 2 shows the results of evaluating the target speaker or non-target
speaker detection error rate against single-talker speech to reveal the
performance of considering the speaker enrollment. The error was calculated
using the number of utterances that misdetects the target or non-target
speaker. The results show that TS-NTS-ASR could reduce the detection error
rate for both the target and non-target speakers. This induces TS-ASR and NTS-
ASR performance improvements. These results demonstrate that our proposed
method effectively transcribes not only the target speaker's speech but also
non-target speakers' speech.

Table 2: Experimental results in terms of target speaker or non-target speaker detection error rate (%). TS-NTS-ASR system introduced ``first-in first-out'' serialization. | Single-talker speech
---|---
| Target | Non-target
TS-ASR | 1.41 | 2.03
NTS-ASR | 2.10 | 2.35
TS-NTS-ASR (proposed) | 0.00 | 1.82

##  6 Conclusions

We presented end-to-end joint target and non-target speakers ASR systems that
jointly transcribe a target speaker's speech and non-target speakers’ speech
using the target speaker's enrollment speech. The key strength of the proposed
method is to effectively transcribe all speakers' speech while identifying
whether they are the target speaker or non-target speakers using the
enrollment speech. This is suitable to understand interactions between the
target speaker and non-target speakers. Our experimental results showed that
the proposed method effectively transcribes not only the target speaker's
speech but also non-target speakers' speech, and outperformed the target-
speaker ASR system and the non-target speaker ASR system.

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# On the continuity of Weil-Petersson volumes of the moduli space weighted
points on the projective line

Salvatore Tambasco<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we show that the Weil-Petersson volume (which coincides with the
CM degree) in the case of weighted points in the projective line is continuous
when approaching the Calabi-Yau geometry from the Fano geometry. More
specifically, the CM volume computed via localization converges to the
geometric volume, computed by McMullen with different techniques, when the sum
of the weights approaches the Calabi-Yau geometry.

##  1 Introduction

When considering a moduli space of weighted pairs it becomes interesting to
study how such moduli space changes towards the _geography_ of the weights. In
this direction, we study the behaviour of the _volume function_ of the moduli
space of points in $\mathbb{P}^{1}$. We denote the latter by $M_{d},$ where
$d$ denotes the weight vector $(d_{1},...,d_{n})$ of a configuration of
$n$-points in $\mathbb{P}^{1}.$ We assume, henceforth, that the components of
such vector are rational numbers in the open interval $(0,1).$ A point of
$M_{d}$ is a pair $(\mathbb{P}^{1},\sum_{i=1}^{n}d_{i}p_{i}),$ where $p_{i}$
are points in general position in $\mathbb{P}^{1},$ $\forall
i\\{1,2,...,n\\}.$ The sum of the weights determine the nature of the pair.
Namely, given the pair $(\mathbb{P}^{1},\sum_{i=1}^{n}d_{i}p_{i}),$ a choice
of the rational weights correspond to three distinguished _geometries_

$\begin{cases}\sum_{i=1}^{n}d_{i}<2&\ \text{log Fano,}\\\
\sum_{i=1}^{n}d_{i}=2&\ \text{log Calabi-Yau,}\\\ \sum_{i=1}^{n}d_{i}>2&\
\text{log General type.}\end{cases}$

We want to study how the Weil-Petersson volume (induced by _canonical constant
curvature metrics_) of the moduli space of $n$ points in the projective line
varies in the above geometries, henceforth denoted by $M_{d}.$ This study is
motivated by the relations among the first Chern class of the CM line bundle
and the Weil-Petersson metric. The aim of this work is to show that the CM
volume of $M_{d}$ _converges_ to the geometric volume computed by McMullen in
[33] (Theorem 8.1), when the sum of the weights approaches the log Calabi-Yau
geometry. To do so, we take the below explicit description of $M_{d}$ as GIT
quotient

$M_{d}=(\mathbb{P}^{1})^{n}//_{\mathcal{L}_{a}}\mathrm{SL}(2),$ (1)

where $d=(d_{1},...,d_{n})$ is a weight vector. This compact space
parametrizes configurations of points in $\mathbb{P}^{1}$ which carry
canonical metrics. Indeed, when $\sum_{i=1}^{n}d_{i}=2,$ it parametrizes flat
metrics (with cone angle) thanks to the work of [32], [29]. When
$\sum_{i=1}^{n}d_{i}<2,$ by works of Li in [73] and Fujita in [80], the latter
moduli space is actually a Fano K-moduli space, parametrizing K-polystable
Fano pairs (positive constant curvature metric). Finally, when
$\sum_{i=1}^{n}d_{i}=2+\epsilon,$ and generic then it is a Hasset/KSBA type
moduli space ([11, Theorem 5.5.2], [25, Theorem 1.5]) of constant negative
curvature metric.

Volumes of $M_{d}$ have been computed in many works, via localisation by [38],
[35], [37], [34], using the isomorphism with the moduli space of polygons. The
correspondence between the Weil-Petersson volume and the degree of the CM line
bundle it is shown in [77] in the absolute case, and the latter result is
extended in the log case in [94].

In this work we will prove the following

######  Theorem 1.1.

The volume of $M_{d}$ with $d_{i}\in(0,1)\cap\mathbb{Q}$, for all
$i\in\\{1,2,....,n\\}$ converges to the volume computed by McMullen in [33]
(Theorem 8.1) when the sum of the weights approaches 2.

The above proves the continuity of the _volume function_ when passing from the
Fano geometry to the Calabi-Yau geometry. The continuity result holds also
when the sum of the weights is $2+\epsilon,$ due to [11, Theorem 5.5.2], [25,
Theorem 1.5]. When the sum of the weights exceed significantly two, then we do
not have the above explicit description of the moduli space of weighted points
in $\mathbb{P}^{1}.$ Namely, the natural K-moduli compactification is not
simply given by the above GIT quotient. However, in this work, as an example,
we fully examine the simple case of four points in $\mathbb{P}^{1}$ giving an
explanation of the continuity of the volume function across the above
mentioned geometries through the picture of $K$-stability.

We organised this work as follows

  * •

In section 2 we recall the notion of CM line bundle for pairs and we relate it
to the moduli of log Fano hyperplane arrangements.

  * •

In section 3 we fully examine the case of four points on the projective line,
by calculating the volume of the moduli space of four points in
$\mathbb{P}^{1}.$ We prove Theorem 1.1 in this case. Later we discuss the
continuity of the volume function across the above mentioned three geometries.

  * •

In section 4 we give a proof of Theorem 1.1 by using a continuity argument
arising from the continuity of the construction of the Weil-Petersson metrics.

  * •

In the final remark, as a _miscellanea_ we show that the CM volume can be
expressed as the sum of Donaldson-Futaki invatiant of some _special_ test
configurations.

## Acknowledgments

I would like to express my most sincere gratitude towards my supervisors Prof.
Cristiano Spotti and Prof. Alessandro Ghigi for their patient guidance and
advice they have provided me. Last, but not least I wish to extend my thanks
to the Mathematics Department of Aarhus University for hosting me for two
years, to the AUFF starting grant 24285, and to the Villum Fonden 0019098 for
partially funding my research.

##  2 The log CM line bundle

###  2.1 The CM line bundle for pairs

Fix $f:\mathcal{X}\rightarrow B$ to be a proper flat morphism of scheme of
finite type over $\mathbb{C}.$ Let $\mathcal{L}$ be a relatively ample on
$\mathcal{X}$ and assume that $f$ has relative dimension $n\geq 1,$ namely
$\forall b\in B,\ (\mathcal{X}_{b},\mathcal{L}_{b})$ has constant dimension
$n.$

######  Definition 2.1.

[91] Let $\mathcal{D}_{i},\forall i=1,2,..,k$ be a closed subscheme of
$\mathcal{X}$ such that $f|_{\mathcal{D}_{i}}:\mathcal{D}_{i}\rightarrow B$
has relarive dimension $n-1,$ and $f|_{\mathcal{D}_{i}}$ is proper and flat.
Let $d_{i}\in[0,1]\cap\mathbb{Q}$ we define the log CM $\mathbb{Q}-$line
bundle of the data
$(f,\mathcal{D}:=\sum_{i=1}^{k}d_{i}\mathcal{D}_{i}),\mathcal{L})$ to be

$\lambda_{CM,\mathcal{D}}:=\lambda_{CM}-\frac{n\mathcal{L}_{b}^{n-1}\cdot\mathcal{D}_{b}}{(\mathcal{L}_{b}^{n})}\lambda_{\mathrm{CH}}+(n+1)\lambda_{\mathrm{CH},\mathcal{D}},$

Where $\lambda_{CM}$ is the CM line bundle defined in Chapter two and
$\lambda_{CH}$ is the Chow line bundle, defined as the leading order term of
the Hilbert Mumford expansion, namely $\lambda_{CH}=\lambda_{n+1},$ and
$\lambda_{CH,\mathcal{D}}=\bigotimes_{i=1}^{k}\lambda_{CH}^{d_{i}}.$

We outline some easy and well known properties of the log CM line bundle, for
a full description of the below facts see [91].

######  Proposition 2.1.

Let $f:(\mathcal{X},\mathcal{D})\rightarrow B$ be a $\mathbb{Q}-$Gorenstein
flat family of $n$ dimensional pairs over a normal proper variety $B.$ Then
for any $\mathcal{L}$ relatively $f-$ample line bundle we have

$c_{1}(\lambda_{CM,\mathcal{D}})=n\frac{(-K_{\mathcal{X}_{b}}-\mathcal{D}_{b})\cdot\mathcal{L}_{b}^{n-1}}{(\mathcal{L}_{b}^{n})}f_{*}c_{1}(\mathcal{L})^{n+1}-(n+1)f_{*}((-K_{\mathcal{X}/B}-\mathcal{D})\cdot
c_{1}(\mathcal{L})^{n}).$ (2)

######  Remark 2.1.

In [90], there is another definition of the log CM line bundle for the case of
one divisor with weight $1-\beta$. And in the same work (Theorem 2.7 first
assertion) there is a calculation of the first Chern class of the defined log
CM line bundle. We point out that the definition we gave in this section it is
exactly the same for the case of one divisor. Indeed, for the first assertion
of Theorem 2.7 we just require to let

$\mu(\mathcal{L}_{b}):=-\frac{-K_{\mathcal{X}_{b}}\cdot\mathcal{L}^{n-1}}{(\mathcal{L})^{n}},\
\
\mu(\mathcal{L}_{b},\mathcal{D}_{b}):=\frac{\mathcal{D}_{b}\cdot\mathcal{L}_{b}^{n-1}}{(\mathcal{L}^{n})}$

and of course $\mathcal{D}=(1-\beta)D.$

The definition of the log CM line bundle given in the same work [90]
definition 2.2 is also a functorial definition, namely

$\Lambda_{CM,\mathcal{D}}:=\lambda_{n+1}^{n(n+1)+\frac{2a_{1}-(1-\beta)\tilde{a}_{0}}{a_{0}}}\otimes\lambda_{n}^{-2(n+1)}\otimes\tilde{\lambda}_{n}^{(1-\beta)(n+1)}$

The element $\tilde{\lambda}_{n}$ refers to the leading order term of the
Hilbert-Mumford expansion for $f_{!|\mathcal{D}}\mathcal{L}$, and
$\tilde{a}_{0}$ to the leading order term coefficient of the corresponding
Hilbert polynomial, that is by the Hirzebruch-Riemann-Roch expansion

$\tilde{a}_{0}=\frac{1}{n!}\mathcal{L}_{b}^{n}\cdot\mathcal{D}_{b}.$

Unravelling the other terms via the Hirzebruch-Riemann-Roch expansion, we
obtain definition 2.1

When computing the intersection number of the log CM line bundle on a
$\mathbb{Q}-$Gorenstein flat family on the moduli of pairs, it is important to
choose a ample line bundle on such family. In the state of the art of these
possible choices, mainly we can make two of such. These choices are presented
in the following

######  Corollary 2.1.

Let $f:(\mathcal{X},\mathcal{D})\rightarrow B$ be a $\mathbb{Q}-$Gorenstein
flat family over a normal proper variety $B.$ We have

  1. 1.

[89] Given $\mathcal{L}=-K_{\mathcal{X}/B}-\mathcal{D}$, then

$c_{1}(\lambda_{CM,\mathcal{D}})=-f_{*}c_{1}(-K_{\mathcal{X}/B}-\mathcal{D})^{n+1}.$

  2. 2.

[90] Given $\mathcal{L}=-K_{\mathcal{X}/B},$ and
$\mathcal{D}_{\mathcal{X}_{b}}\in|-K_{\mathcal{X}_{b}}|,\ \forall b\in B$,
then

$c_{1}(\lambda_{CM,\mathcal{D}})=f_{*}\left(c_{1}(-K_{\mathcal{X}/B})^{n}\cdot(-c_{1}(-K_{\mathcal{X}/B})+\sum_{i=1}^{k}d_{i}((n+1)\mathcal{D}_{i}-nc_{1}(-K_{\mathcal{X}/B}))\right).$

Depending on the choice of $\mathcal{L}$ we will get different results in
terms of intersection number on a generic curve into the moduli of pairs we
want to study. In the below subsection, as an example, we will point out this
difference when calculating the intersection number of the log CM line bundle
with a generic $\mathbb{P}^{1}$-_curve_ in the moduli space of log Fano
hyperplane arrangements. Finally, we wish to make a remark on the log CM line
bundle in the case of families of general type varieties. As a direct
consequence of Proposition 2.1, we get the following

######  Corollary 2.2.

Let $f\colon(\mathcal{X},\mathcal{D})\rightarrow B$ be a
$\mathbb{Q}$-Gorenstein family of log general type varieties of relative
dimension n, and with relatively ample line bundle
$\mathcal{L}=K_{\mathcal{X}/B}+\mathcal{D}.$ Then,

$c_{1}(\lambda_{\mathrm{CM}})=f_{*}c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n+1}$
(3)

Proof By applying directly Proposition 2.1 we get

$n\frac{(-K_{\mathcal{X}_{b}}-\mathcal{D}_{b})\cdot(K_{\mathcal{X}_{b}}+\mathcal{D})^{n-1}}{(K_{\mathcal{X}_{b}+\mathcal{D}_{b})^{n}})}=-n.$

Thus,

$\displaystyle c_{1}(\lambda_{\mathrm{CM},\mathcal{D}})$
$\displaystyle=-nf_{*}c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n+1}-(n+1)f_{*}(-(K_{\mathcal{X}/B}+\mathcal{D})c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n})$
$\displaystyle=-nf_{*}c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n+1}+(n+1)f_{*}c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n+1}$
$\displaystyle=f_{*}c_{1}(K_{\mathcal{X}/B}+\mathcal{D})^{n+1}.$

$\Box$

###  2.2 The intersection number of the log CM line bundle

In this section we compute the intersection number of the log CM line bundle
with a family of log Fano hyperplane arrangements with base $\mathbb{P}^{1}.$
Consider the product $\mathcal{X}:=\mathbb{P}^{n}\times\mathbb{P}^{1},$ We
define a divisor $\mathcal{D}\subset\mathcal{X},$ as

$\mathcal{D}:=\sum_{i=1}^{m-1}d_{i}\mathrm{pr}_{2}^{*}h+d_{m}(\mathrm{pr}_{1}^{*}l+\mathrm{pr}_{2}^{*}h)$

Where the $\mathrm{pr}_{i}^{\prime}s,\ i=1,2$ denotes the projections onto the
first and second factor of $\mathcal{X}.$ We can think about $\mathcal{D}$ as
$m-1$ fixed hyperplanes of $\mathbb{P}^{n}$ together with a line $l$ with
weight $d_{m}$free to move along the _diagonal_ of $\mathcal{X}.$ We assume
that $d_{i}\in(0,1)\cap\mathbb{Q},\ \forall i\in\\{1,2,..,m\\},$
$\sum_{i=1}^{m}d_{i}<n+1,$ and the chosen line and hyperplanes are in general
position.

######  Proposition 2.2.

With the above data we have

$c_{1}(\lambda_{CM,\mathcal{D}})=(n+1)d_{j}\left(n+1-\sum_{i=1}^{m}d_{i}\right)^{n},\
\forall j\in\\{1,2,...,m\\}.$

Proof We see that the projection
$\mathrm{pr}_{2}:\mathcal{X}\rightarrow\mathbb{P}^{1}$ gives a proper and flat
family of relative dimension $n.$ Choose
$\mathcal{L}=-K_{\mathcal{X}/B}-\mathcal{D}$and use assertion $1.$ of 2.1, we
find that

$\mathcal{L}=\left(n+1-\sum_{i=1}^{m}d_{i}\right)\mathrm{pr}_{1}^{*}h-d_{m}\mathrm{pr}_{2}^{*}l$

Using the binomial expansion, the only term that survives is

$c_{1}(\lambda_{CM,\mathcal{D}})=-\mathrm{pr}_{2*}c_{1}(\mathcal{L})^{n+1}=(n+1)d_{m}\left(n+1-\sum_{i=1}^{m}d_{i}\right)^{n}\mathrm{pr}_{2*}\left(\mathrm{pr}_{1}^{*}h\cdot\mathrm{pr}_{2}^{*}l\right).$

By using the projection formula and the arbitrariness of the choice of the
weight on the line, the claim follows. $\Box$

######  Remark 2.2.

If we would have chosen $\mathcal{L}=-K_{\mathcal{X}/B},$ we can see that the
hypothesis $\mathcal{D}_{|\mathcal{X}_{b}}\in|-K_{\mathcal{X}_{b}}|$ would not
be satisfied. Indeed, it is true if and only if $\sum_{i=1}^{m}d_{i}=(n+1),$
i.e. a Calabi-Yau log hyperplane arrangement, but this is in contradiction
with the log Fano assumption. However, $-K_{\mathcal{X}/B}$ it is still a
relatively ample line bundle on
$\mathrm{pr}_{2}:\mathcal{X}\rightarrow\mathbb{P}^{1}.$ We can compute the CM
line bundle with this choice, and relaxing the hypothesis for which
$\mathcal{D}_{|\mathcal{X}_{b}}\in|-K_{\mathcal{X}_{b}}|$ as the next result
will show.

######  Proposition 2.3.

With the above data and $\mathcal{L}=-K_{\mathcal{X}/B},$ we have

$c_{1}(\lambda_{CM,\mathcal{D}})=(n+1)^{2}d_{j}.$

Proof The first summand is zero, since $\mathcal{X}/\mathbb{P}^{1}$ is a
trivial fibration. From the second summand we get

$c_{1}(\lambda_{CM,\mathcal{D}})=\\\
=-(n+1)\mathrm{pr}_{2*}\left[\left(n+1-\sum_{i=1}^{m}d_{i}\right)\mathrm{pr}_{2}^{*}h-d_{m}\mathrm{pr}_{1}^{*}l\right]\cdot(n+1)\mathrm{pr}_{2}^{*}h=\\\
=(n+1)^{2}d_{m}.$

Because of the arbitrariness of choices the claim follows $\forall
j\in\\{1,2,...,m\\}.$ $\Box$

From the choice of the two ample line bundles on the given flat and proper
family we see that they mainly differ by a factor which involves the volume of
the fiber, that is

$\mathrm{Vol}(\mathrm{pr}_{2,t}):=\left(n+1-\sum_{i=1}^{m}d_{i}\right)^{n}.$

This difference might become important when calculating the volume of the
moduli space of weighted hyperplane arrangements. To see that, we shall
calculate the log CM line on to the mentioned moduli space. Recall that the
moduli space of weighted log Fano hyperplane arrangement is the GIT quotient

$M_{d}:=(\mathbb{P}^{n})^{m}//_{\mathcal{L}_{d}}\mathrm{SL}_{n+1}$

where $\mathcal{L}_{d}=\mathcal{O}(d_{1},...,d_{m}),$ is the linearization and
$d_{i}\in(0,1)\cap\mathbb{Q},\forall i\in\\{1,2,...,m\\}.$ It is known that,

$\mathrm{Pic}(M_{d})=\mathbb{Z}^{m}$

see for example [8, Chapter 11, Lemma 11.1]. Therefore
$c_{1}(\lambda_{CM,\mathcal{D}})\in\mathrm{Pic}(M_{d})\Rightarrow
c_{1}(\lambda_{CM,\mathcal{D}})=\mathcal{O}(r_{1},...,r_{m}),$ for some
$(r_{1},...,r_{m})\in\mathbb{Z}^{m}.$ In order to compute the
$r_{j}^{\prime}$s we can use the results of 2.2 and 2.3. Namely, in both cases
we compute the following intersection number

$c_{1}(\lambda_{CM,\mathcal{D}})\cdot(\mathcal{X}\rightarrow\mathbb{P}^{1})=k,$

which means, $\forall j\in\\{1,2,...,m\\}$

$\int_{(\mathbb{P}^{1})_{j}}\mathcal{O}(r_{1},...,r_{m})=r_{j}=k.$

Therefore, from 2.2 we get $r_{j}=(n+1)d_{j}\mathrm{Vol}(\mathrm{pr}_{2,t}),$
and from 2.3 we get $r_{j}=(n+1)^{2}d_{j}.$ Hence, we just proved the
following result.

######  Proposition 2.4.

The log CM line bundle on the moduli space of log Fano hyperplane arrangement
is given by

  * •

When $\mathcal{L}=-K_{\mathcal{X}/B}-\mathcal{D},$ we have that

$c_{1}(\lambda_{CM,\mathcal{D}})=\mathrm{Vol}(\mathrm{pr}_{2,t})\mathcal{O}((n+1)d_{1},...,(n+1)d_{m})$

  * •

When $\mathcal{L}=-K_{\mathcal{X}/B},$ we have

$c_{1}(\lambda_{CM,\mathcal{D}})=(n+1)\mathcal{O}((n+1)d_{1},...,(n+1)d_{m}).$

######  Remark 2.3.

In both cases of 2.4 we can see that the log CM line bundle is a multiple of
the _weighted prequantum_ line bundle on $(\mathbb{P}^{n})^{m}$hence of its
Kähler form, we recall that

$\left[\frac{\omega}{2\pi}\right]=c_{1}(-K_{(\mathbb{P}^{n})^{m}})=c_{1}(\mathcal{O}((n+1)d_{1},..,(n+1)d_{m})))$

Where the Kähler form $\omega$ is given by the weighted sum of the Fubini
Study metrics on each $\mathbb{P}^{n}$, namely
$\omega=\sum_{i=1}^{m}d_{i}\omega_{FS,i}.$

##  3 A first study: the case of four points on the complex projective line.

As a first study, we consider the case of four points $\\{p_{i}\\}_{i=1}^{4}$
in $\mathbb{P}^{1}$ with rational weights
$d_{i}\in(0,1)\cap\mathbb{Q},i\in\\{1,2,3,4\\}.$ Given the pair
$(\mathbb{P}^{1},\sum_{i=1}^{4}d_{i}p_{i}),$ a choice of the rational weights
correspond to three distinguished _geometries_

$\begin{cases}\sum_{i=1}^{4}d_{i}<2&\ \text{log Fano,}\\\
\sum_{i=1}^{4}d_{i}=2&\ \text{log Calabi-Yau,}\\\ \sum_{i=1}^{4}d_{i}>2&\
\text{log General type.}\end{cases}$

We want to study how the _Volume_ of the moduli space of four points in the
projective line varies in the above geometries. We begin with the Calabi-Yau
geometry. The moduli space of four points in $\mathbb{P}^{1}$ can be described
as the following GIT quotient

$M_{d}=(\mathbb{P}^{1})^{4}//_{\mathcal{L}_{d}}\mathrm{SL}_{2}\mathbb{C},$ (4)

with linearization $\mathcal{L}_{d}=\mathcal{O}(d_{1},d_{2},d_{3},d_{4}).$
Furthermore, $M_{d}$ is isomorphic to the moduli space of marked curves of
genus $0,$ denoted by $\mathcal{M}_{0,4}$. For clarity, we recall the
following result

######  Theorem 3.1.

([33], Theorem 8.1) Let $\mathcal{M}_{0,n}$ be the moduli space of $n$ ordered
points on the Riemann sphere. Then the complex hyperbolic volume of
$\mathcal{M}_{0,n}$ is given by

$\mathrm{Vol}(\mathcal{M}_{0,n})=C_{n-3}\sum_{\mathcal{P}}(-1)^{|\mathcal{P}|+1}(|\mathcal{P}|-3)!\prod_{B\in\mathcal{P}}\mathrm{max}\left(0,1-\sum_{i\in
B}\mu_{i}\right)^{|B|-1},$ (5)

where $\sum_{i=1}^{n}\mu_{i}=2,$ $0<\mu_{i}<1,\ \forall i\in\\{1,2,...,n\\},$
and $C_{n}=\frac{(-4\pi)^{n}}{(n+1)!}.$ Here $\mathcal{P}$ ranges over all
partitions of the indices $(1,...,n)$ into blocks $B.$

From the above theorem, we can seee that the only partitions making sense are
those whose size is greater or equal than three. Therefore, in this considered
case, are those of size $3,$ and $4.$ Recall that the size of a partition
$\mathcal{P}$ equals the number of blocks $B$ of the same partition.
Therefore, if the partition has size $4$ then there is only one partition with
four blocks each of which has size one. If the size of the partition is 3,
then we have six different partitions of two blocks of size one and one block
of size two. From 3.1, we have $C_{1}=-2\pi,$ set
$\alpha_{B}=\mathrm{max}\left(0,1-\sum_{i\in B}d_{i}\right).$ If a block has
size one then $\alpha^{|B|-1}=1.$ The contribution to the sum from the
partition of size 4 is just $-2\pi,$ and the contribution to the sum from the
partitions of size 3 is the sum of all $d_{i}`$s whose indexing set is a block
of size two. Namely $-2\pi\sum_{B,|B|=2}\alpha_{B}.$ Then, as a final result
we get

$\mathrm{Vol}(\mathcal{M}_{0,4})=-2\pi\sum_{I\subset\subset\\{1,2,3,4\\}:|I|=2}(1-\sum_{i\in
I}d_{i})$ (6)

######  Remark 3.1.

The same result given in Equation 6 can be achieved from a result of Thurston
[32] which shows that the general moduli space of marked curves of genus $0,$
with the additional condition that the weights sum up to $2$, is a complex
hyperbolic cone manifold. The latter, in general, have a natural
_stratification_ , and the strata of a cone manifold are connected and
Riemannian. The _solid angle_ of a cone manifold is constant on each strata.
We know that for a closed complex hyperbolic manifold $M$ of dimension $n,$
its volume can be expressed by its Euler characteristic

$\mathrm{Vol}(M)=\frac{(-4\pi)^{n}}{(n+1)!}\chi(M)$ (7)

The dimension of $\mathcal{M}_{0,4}$ is $1,$ and it is homeomorphic to
$\mathbb{P}^{1}$. Considering the analogy with the $2$-sphere with four marked
points with weights $d_{1},d_{2},d_{3},d_{4},$ respectively, we can associate
to these latter a cell complex, where its skeleton consist of four distinct
points and six triangles. These latter correspond to the four points
partitions of size two and three. With this in mind, and by applying 7 we get
Equation 6.

In the case of four points in $\mathbb{P}^{1}$ the result provided by Madini
in [38] becomes

$\mathrm{Vol}(M_{d})=-\pi\sum_{k=0}^{3}(-1)^{k}\sum_{I\subset\mathcal{I}_{+},|I|=k}\left(D_{I}^{+}-D_{I}^{-}\right).$
(8)

Now assume that $\sum_{i=1}^{4}d_{i}=2.$ Unravelling the definition of
$D_{I}^{+}$ and $D_{I}^{-}$, we have

$\displaystyle\sum_{i\not\in I}d_{i}-\sum_{i\in I}d_{i}$
$\displaystyle=2-\sum_{i\in I}d_{i}-\sum_{i\in I}d_{i}$
$\displaystyle=2(1-\sum_{i\in I}d_{i}).$

By substituting in Equation 8, we find

$\mathrm{Vol}(M_{d})=-2\pi\sum_{k=1}^{4}(-1)^{k}\sum_{I\subset\mathcal{F},|I|=k}\mathrm{max}\left(0,1-\sum_{i\in
I}d_{i}\right)$ (9)

######  Proposition 3.1.

The volume function of the moduli space of four points in $\mathbb{P}^{1}$ is
continuous when passing from the Fano geometry to the Calabi-Yau geometry.

Proof All we need to show is that Equation 9 equals Equation 6. When looking
at Equation 9 the subsets $I$ of size one will contribute $-2$ to the sum.
Indeed, by assumption $d_{i}\in(0,1)\cap\mathbb{Q},\forall i\in\\{1,2,3,4\\},$
therefore $\alpha_{I}=1-d_{i},\forall i\in\\{1,2,3,4\\}.$ By taking the sum of
the four subset of size one, and observing that the sum of all the $d_{i}$ is
by assumption two, we have

$-\pi\sum_{i=1}^{4}(1-d_{i})=-2\pi(4-2)=-2\pi.$

The subsets of size three have zero contribute to the sum. Indeed, without
loss of generality, suppose $I=\\{1,2,3\\},$ then
$1-d_{1}-d_{2}-d_{3}=1-2+d_{4}=d_{4}-1.$ Therefore,

$\alpha_{\\{1,2,3\\}}=\mathrm{max}\left(0,d_{4}-1\right)=0.$

Hence, it remains to sum the contributions coming from the subsets of size
$2.$ But, putting all together we have

$\mathrm{Vol}(M_{d})=-2\pi\sum_{I\subset\subset\\{1,2,3,4\\}:|I|=2}(1-\sum_{i\in
I}d_{i}).$

That proves the claim. $\Box$

This proves, in the four point case, that the _volume function_ is continuous
when passing from the Fano geometry to the Calabi-Yau geometry. When
$\sum_{i=1}^{4}d_{i}>2$ we may loose this behaviour, since we do not have a
GIT quotient, therefore the previous techniques for calculating the volume can
not be applied. The work of Alexeev [25] describes the moduli space of
hyperplane arrangements by generalising the work of Hasset [24] so that the
one dimensional case coincides with the _Hasset moduli space_. The moduli
space of weighted hyperplane arrangements of dimension $1,$ is the moduli
space of points in $\mathbb{P}^{1}.$ Since the dimension of the considered
moduli space is $1,$ for any choice of weights, then we can calculate the
volume in the general type geometry, by computing the degree of the log CM
line bundle. To do so, we notice that $M_{d}$ has a _universal family_
obtained by considering the product $\mathbb{P}^{1}\times\mathbb{P}^{1}$,
fixing three points, $0,1,\infty$, and allowing the last one to move in the
diagonal $\Delta=\\{(s,t)\in\mathbb{P}^{1}\times\mathbb{P}^{1}\ |\ s=t\ \\}.$
We set

$\mathcal{X}=\mathbb{P}^{1}\times\mathbb{P}^{1},\ \mathrm{and}\
\mathcal{D}=d_{1}\cdot\mathbb{P}^{1}\times\\{0\\}+d_{2}\cdot\mathbb{P}^{1}\times\\{1\\}+d_{3}\cdot\mathbb{P}^{1}\times\\{\infty\\}+d_{4}\cdot\Delta.$

Since we fixed three points, then as divisors we can assume that they have the
same hyperplane class, that will be called $h_{2}$. Hence, we can rewrite
$\mathcal{D}$ as follows

$\mathcal{D}=\sum_{i=1}^{4}d_{i}\mathrm{pr}_{2}^{*}h_{2}+d_{4}\mathrm{pr}_{1}^{*}h_{1}.$

Because of the construction of the Hasset’s moduli space we must distinguish
several situations. Heuristically, we have the following phenomena: when the
point on the diagonal with weight $d_{4}$ meets one, two or three points, then
a _wall_ is crossed and therefore the universal family becomes singular at
that point. This phenomena is encoded in the notion of _stability_ given in
[24] that we recall in the following

######  Definition 3.1.

Let $\pi:(C,s_{1},...,s_{n})\rightarrow B$ be a proper and flat morphism of
nodal curves of arithmetic genus $g,$ and $s_{1},...,s_{n}$ are the sections
of $\pi$ corresponding to the marked points on $C.$ A collection of data
$(g,\mathcal{A})=(g,a_{1},...,a_{n})$ consist of an integer $g\geq 0$ and
weights $(a_{1},...,a_{n})\in\mathbb{Q}^{n},$ such that $0<a_{j}\leq 1,\forall
j\in\\{1,2,...,n\\}$ and $2g-2+a_{1}+a_{2}+\cdots+a_{n}>0.$ We say that $\pi$
is _stable_ if the following conditions are satisfied

  1. 1.

The sections $s_{1},...,s_{n}$ are in the smooth locus of $\pi,$ and for every
subset $\\{s_{i_{1}},...s_{i_{r}}\\}$ with nonempty intersection we have
$a_{i_{1}}+...+a_{i_{r}}\leq 1.$

  2. 2.

$K_{\pi}+\sum_{i=1}^{n}a_{i}s_{i}$ is relatively ample.

Clearly, in our case $g=0,$ and $M_{0,4}\simeq\mathbb{P}^{1}.$ The above
definition, and the heuristic are rather _intuitive_ as the following picture
shows

$\mathrm{pr}_{2}$$d_{4}\cdot\Delta_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$$d_{1}\cdot\mathbb{P}^{1}\times\\{0\\}$$d_{2}\cdot\mathbb{P}^{1}\times\\{1\\}$$d_{3}\cdot\mathbb{P}^{1}\times\\{\infty\\}$$\\{0\\}$$\\{1\\}$$\\{t\\}$$\\{\infty\\}$
In this picture, the dashed vertical lines represents the fibers of the map
$\mathrm{pr}_{2}.$ To the red dots correspond singular points in the fibers,
namely those for which the diagonal meets one or more divisors. The blue point
is a smooth point for the fiber of $\\{t\\}$

When looking at the above picture, we shall assume that the weight $d_{4}$ is
the smallest among the weights. So, while $d_{4}$ is free to move, and meet
one (or more) points in the diagonal, then the rest of the points remain
fixed.

  * •

The following inequalities holds, when the diagonal does not meet any of the
fixed points

$\begin{cases}d_{i}+d_{j}>1&\ \text{$i,j\in\\{1,2,3\\}$}\\\ d_{k}+d_{4}<1&\
\text{for all $k\in\\{1,2,3,4\\}$}\end{cases}$ (10)

  * •

If $d_{4}$ grows, then it meets one (or more) points along the diagonal, and a
desingularization is needed

$\begin{cases}d_{i}+d_{4}>1&\ \text{$i\in\\{1,2,3\\}$}\\\ d_{j}+d_{4}\leq 1&\
\text{$i\neq j$.}\end{cases}$ (11)

As the above picture suggest we can take the projection onto the second factor
of $\mathcal{X}$, namely
$\mathrm{pr}_{2}:(\mathcal{X},\mathcal{D})\rightarrow\mathbb{P}^{1},$ to get a
$\mathbb{Q}$-Gorenstein family of log general type varieties. The stability
conditions of Hasset suggest that we should choose, as a relatively ample line
bundle for $\mathrm{pr}_{2},$ the log canonical polarization
$K_{\mathcal{X}/\mathbb{P}^{1}}+\mathcal{D}.$

We begin by studying the 10. We chose
$K_{\mathcal{X}/\mathbb{P}^{1}}+\mathcal{D}$ as a relative polarization for
the family
$\mathrm{pr}_{2}\colon(\mathcal{X},\mathcal{D})\rightarrow\mathbb{P}^{1}.$ By
a direct application of Corollary 2.2 we have that

$c_{1}(\lambda_{\mathrm{CM},\mathcal{D}})=2d_{4}\left(\sum_{i=1}^{4}d_{i}-2\right).$
(12)

In 11, it is the situation when the diagonal meets one point. Suppose, without
loss of generality, that the diagonal meets the zeroth fiber. The universal
family becomes singular at that point, therefore we take the blowup at
$\\{0\\}$ of the total space of the universal family. Set
$\tilde{\mathcal{X}}=\mathrm{Bl}_{0}(\mathcal{X}).$ The divisor $\mathcal{D},$
modifies as follows

$\displaystyle\mathcal{D}$
$\displaystyle=d_{1}(\mathrm{pr}_{2}^{*}h_{2}-E)+(d_{2}+d_{3})\mathrm{pr}_{2}^{*}h_{2}+d_{4}(\mathrm{pr}_{2}^{*}h_{2}+\mathrm{pr}_{1}^{*}h_{1}-E)$
$\displaystyle=\sum_{i=1}^{4}d_{i}\mathrm{pr}_{2}^{*}h_{2}+d_{4}\mathrm{pr}_{1}^{*}h_{1}-(d_{1}+d_{4})E.$

Then,

$K_{\tilde{\mathcal{X}}/\mathbb{P}^{1}}+\mathcal{D}=\left(\sum_{i=1}^{4}d_{i}-2\right)\mathrm{pr}_{2}^{*}h_{2}+d_{4}\mathrm{pr}_{1}^{*}h_{1}-(d_{1}+d_{4}+1)E.$

By applying directly Corollary 2.2 we find Equation 12. Indeed the cohomology
ring of the blow up suggests that the intersection products
$\mathrm{pr}_{i}^{*}h_{i}\cdot E=0,i=1,2$. Moreover, the pushforward of the
constant term $(d_{1}+d_{4}+1)$ coming from $E^{2}=-1$ term, is zero,
therefore the only term that survives is only
$\mathrm{pr}_{2*}2d_{4}\left(\sum_{i=1}^{4}d_{i}-2\right)\mathrm{pr}_{2}^{*}h_{2}\cdot\mathrm{pr}_{1}^{*}h_{1}$,
that yields to 12. The same holds if more than one desingularization is
needed. The formula of the volume function for the log general type case is
therefore constant with respect to the log canonical polarization. We notice
also that when Equation 12 is evaluated in the Calabi-Yau zone then the volume
function is zero. With respect to the anticanonical log polarization given in
the Fano case we have a change of sign. This shows that the volume function
has a discontinuity point in the Calabi-Yau zone. However, according to [11,
Theorem 5.5.2], and [25, Theorem 1.5], if we choose weights whose sum is
slightly greater than $2$ the moduli space do not change, and it is described
as the GIT quotient 4. Then, at least for 10, we can chose as relative
polarization $-K_{\mathcal{X}/\mathbb{P}^{1}}$. Therefore, a fast computation
proves that

$c_{1}(\lambda_{\mathrm{CM},\mathcal{D}})=4d_{4}.$

We can easily observe that up to a normalization factor, $\pi$, it coincides
with the result obtained by applying 6, and 9 with the following order of
weights

$d_{i}<\sum_{j\neq i}d_{j},\ \forall i\in\\{1,2,3,4\\},$

$\displaystyle d_{4}+d_{2}<d_{3}+d_{1}$ $\displaystyle
d_{4}+d_{3}<d_{2}+d_{1}$ $\displaystyle d_{4}+d_{4}<d_{2}+d_{1}.$

We resume all these results in the following

######  Proposition 3.2.

Let $M_{d}$ be the moduli space of weighted hyperplane arrangements of
dimension $1.$ Suppose that the weights
$d_{i}\in(0,1)\cap\mathbb{Q},i\in\\{1,2,3,4\\}$ satisfy the following
conditions

  * •

$d_{j}<\sum_{i\neq j}d_{i},\forall j\in\\{1,2,3,4\\};$

  * •

$\displaystyle d_{4}+d_{2}<d_{3}+d_{1}$ $\displaystyle
d_{4}+d_{3}<d_{2}+d_{1}$ $\displaystyle d_{4}+d_{4}<d_{2}+d_{1}.$

Then, for small weights the volume of $M_{d}$ changes continuously along the
Fano, Calabi-Yau and general type geometry and it is given by $4\pi d_{4}.$

######  Remark 3.2.

In the condition of Proposition 3.2, when the sum of the weights largely
exceed two, then we must chose another polarization, i.e. the log canonical
polarization, as the Hasset compactification suggests. The calculations shows
that the volume of $M_{d}$ in this case is given by
$2d_{4}\left(\sum_{i=1}^{4}d_{i}-2\right).$ Note that when approaching the
Calabi-Yau geometry from the _far away_ general type geometry, then the volume
of $M_{d}$ goes to zero.

##  4 The general case

In order to prove Theorem 1.1, we recall the following definition

######  Definition 4.1.

[94] Let $f:(\mathcal{X},\mathcal{D})\longrightarrow B$ be a family of log
K-polystable Fano varieties of pure dimension $n.$ Assume that
$\mathcal{D}:=\sum_{k}(1-\beta_{k})[s_{k}=0]$ is simple normal crossing. Then,
we define the log Weil-Petersson metric as the following fiber integral

$\omega_{\mathrm{WP}}^{0}:=-\int_{\mathcal{X}^{0}/B^{0}}\omega_{\mathcal{X}^{0}}^{n+1}.$
(13)

In [94, Theorem 2.3] it is proven that Equation 13 can be extended to the
whole family $f:(\mathcal{X},\mathcal{D})\longrightarrow B,$ and it extends as
a curvature for a metric on the descended log CM line bundle
$\lambda_{\mathrm{CM},\mathcal{D}}.$

Proof (Of Theorem 1.1). Consider $n$ fixed points $\\{p_{i}\\}_{i=1}^{n}$ in
general position in $\mathbb{P}^{1}.$ Each of these points has weight
$d_{i}\in(0,1)\cap\mathbb{Q},$ where $d_{i}=1-\beta_{i}$ is related to the
conic angle $\beta_{i}.$ Fix a conic KE metric in $\mathbb{P}^{1},$
$\omega_{d}\in c_{1}(\mathcal{O}(1)),$ where $d=(d_{1},...,d_{n}).$ Note that
when $d=(1,...,1)$ then $\omega_{d}$ coincides with the Fubini-Study metric on
$\mathbb{P}^{1}.$ When $\sum_{i=1}^{n}d_{i}=2$ then $\omega_{d}$ is a Calabi-
Yau metric, that is

$\omega_{d}=i\Omega\wedge\overline{\Omega}$ (14)

Where, by choosing an affine coordinate $z$ for the points
$\\{p_{i}\\}_{i=1}^{n}$, such that $z(p_{i})=c_{i}\in\mathbb{C},\forall
i\in\\{1,2,...,n\\},$ the holomorphic $1$-form $\Omega$ is given by

$\Omega=\frac{dz}{\prod_{i=1}^{n}(z-c_{i})^{d_{i}}}.$

In [79, section 4.1] it is proven that 14 is indeed the metric used by
McMullen in [33], that is the Weil-Peterson metric. Namely,

$\omega_{WP}=-i\partial\overline{\partial}\log\int_{\mathbb{P}^{1}}i\Omega\wedge\overline{\Omega}.$

Now, let $f:(\mathcal{X},\mathcal{D})\longrightarrow\mathbb{P}^{1}$ be a
familily of $K$-polystable log Fano hyperplane arrangements of dimension one,
like in section 2.2. That is,
$\mathcal{X}=\mathbb{P}^{1}\times\mathbb{P}^{1},$
$\mathcal{D}=\sum_{i=1}^{n-1}d_{i}\mathrm{pr}_{2}^{*}h+d_{n}(\mathrm{pr}_{1}^{*}l+\mathrm{pr}_{2}^{*}h).$
Then $f$ is the projection map onto the second factor of $\mathcal{X}.$
Furthermore, we assume for this family that the sum of all the weights is two.
We choose as a relative ample line bundle for $f$
$\mathcal{L}=-K_{\mathcal{X}/\mathbb{P}^{1}}.$ Fiberwise we have

$-i\partial\overline{\partial}\log\int_{\mathcal{X}_{t}}i\Omega_{t}\wedge\overline{\Omega_{t}}=-i\partial\overline{\partial}\log\int_{\mathcal{X}_{t}}\omega_{d,t}^{n+1}.$

Then, because of Definition 4.1 the above is exactly the fiberwise definition
of the log Weil Petersson metric, and hence the curvature for a metric on
$\lambda_{\mathrm{CM},\mathcal{D}}.$ That means,

$\mathrm{Vol}(\mathcal{M}_{0,n})=\mathrm{Vol}(\lambda_{\mathrm{CM},\mathcal{D}},\omega_{WP}).$
(15)

Because of the second assertion of Proposition 2.4, we see that the log CM
line bundle it is a multiple of the anti-canonical bundle of $\mathbb{P}^{1}.$
Therefore, the volume can be achieved with the Jeffrey-Kirwan residue theorem,
that leads to equation in [33] (Theorem 8.1) as wanted. $\Box$

######  Remark 4.1.

A combinatorial proof like in the case of four points it is rather
complicated. As a check we provide a Python script that shows numerically the
equality of the two formulas, that can be found in APPENDIX A.

##  5 Final remarks

We wish to conclude this section with an observation. Consider the log Fano
pair $(\mathbb{P}^{1},\sum_{i=1}^{4}d_{i}p_{i})$. Chose some coordinate
$z\in\mathbb{P}^{1},$ such that for a fixed $i\in\\{1,2,3,4\\}$ we have
$z(p_{i})=0$. Consider the vector field $v=z\partial_{z},$ and notice that it
generates a one parameter subgroup
$\lambda:\mathbb{C}^{*}\hookrightarrow\mathrm{GL}_{1}(\mathbb{C})$, that acts
on $\mathbb{P}^{1}$ in the following fashion $(\lambda(t),z):=t\cdot z.$ This
actions easily translates on the divisor,
$\mathcal{D}_{t}:=t\cdot(\sum_{i=1}^{4}d_{i}p_{i})$, and on the anticanonical
polarization of $\mathbb{P}^{1},$ namely $\mathcal{O}_{\mathbb{P}^{1}}(2)$.
This data define a test configuration
$\\{(\mathbb{P}^{1}_{t},\mathcal{D}_{t},-K_{\mathbb{P}^{1}_{t}})\\}_{t\in\mathbb{C}^{*}}=(\mathcal{X},\mathcal{D},K_{\mathcal{X}})$
whose central fiber $\mathcal{X}_{0}$ is just a point and the divisor, at the
central fiber, behave as

$\mathcal{D}_{0}=d_{i}\\{\infty\\}+\sum_{i\neq j}d_{j}\\{0\\}.$ (16)

Then, this latter defines an _integral test configuration_. As we mentioned in
previous discussions and in [78] there exist a bijection between integral test
configurations and _dreamy prime divisors_. We observe that since we are
dealing with toric varieties, then every prime divisor is also dreamy ([78]).
In particular in our central fiber 16 we have the sum of two dreamy prime
divisor

$W_{1}=\\{0\\},\ \mathrm{and}\ W_{2}=\\{\infty\\}.$

We have that

$\mathrm{ord}_{W_{1}}(\mathcal{D}_{0})=d_{i},\ \mathrm{and}\
\mathrm{ord}_{W_{2}}(\mathcal{D}_{0})=\sum_{i\neq j}d_{j}$

The divisor $W_{1}$ leads to
$\mathrm{DF}_{\mathcal{D}}(\mathcal{X},\mathcal{L})=\sum_{i\neq
j}d_{j}-d_{i}.$ We will now show that the volume of $M_{0,4}$ can be obtained
by summing all the Donaldson-Futaki invariant of the corresponding integral
test configuration. In 8 every single term $D_{F}^{+}-D_{F}^{-}$ , by the
Jeffrey-Kirwan _residue theorem_ [19] is associated to

$\frac{i_{B}^{*}c_{1}(\mathcal{L}_{d})(X)}{e_{B}(X)}=(-1)^{4-k}\mu(B)X^{-3}$
(17)

where $B\in\mathcal{F}=\\{(z^{1},...,z^{4})\in(\mathbb{P}^{1})^{4}|\
z^{j}\in\\{0,\infty\\}\\}$ is a fixed point for the maximal torus action. In
17 we used the fact that since $\mathcal{L}_{d}$ is a prequantum line bundle
and the maximal torus is the unit circle then by [12], Lemma 9.31, the image
of the moment map $\mu(B)$ is given by the weighted sum of the height function
on each $\mathbb{P}^{1},$ namely

$\mu(B)=\sum_{i=1}^{4}d_{i}\mu^{i}(z^{i}),$

where $\mu^{i}(z^{i})$ is the height function, i.e. the moment map for the
maximal torus action on each $\mathbb{P}^{1}.$ Clearly, the fixed points for
this action are the standard basis element $e_{1},e_{2}$ of $\mathbb{C}^{2}$
that corresponds to $0,\infty$ respectively. We have

$\mu^{i}(e_{i})=\begin{cases}1&\ \text{if $i=1$}\\\ -1&\ \text{if
$i=2.$}\end{cases}$

It follows in 8 that the index $F$ is the one that tells how many $e_{2}$ we
have in the fixed point $B.$

If $|F|\geq 2$ we can always think of a integral test configuration for which
two points came together into one point, where the weight of the latter is the
sum of two or more weights. With this in mind, it is easy to convince
ourselves that the size of the index set $F$ tells how many points should come
together into one point, and hence how many weights should be summed at that
point. Therefore, the general term

$\frac{i_{B}^{*}c_{1}(\mathcal{L})}{e_{B}(X)}=-\frac{(-1)^{4-k}}{x^{3}}\mathrm{DF}_{\mathcal{D}}(\mathcal{X},\mathcal{L})(B),$

as wanted. This observation, can be immediately generalised for a
configurations of $n-$points in $\mathbb{P}^{1}.$ The key point is that in the
work of [80] it is shown that in the case of log Fano hyperplane arrangements
the notion of K-stability coincides with the notion of GIT-stability. Moreover
by changing the weights, the corresponding moduli spaces are _well behaved_ so
that the Jeffrey-Kirwan localization formula of [19] can be applied for
calculating the volume. The GIT quotient describing the hyperplane
arrangements of dimension one is the following

$M_{d}=(\mathbb{P}^{1})^{n}//_{\mathcal{L}_{d}}\mathrm{SL}_{2}\mathbb{C}.$

The dimension of $M_{d}$ is $n-3.$ We apply directly the Jeffrey-Kirwan
theorem for the case of $\mathrm{SU}(2),$ on the top class namely

$k(c_{1}(\mathcal{L}_{d})^{n-3})e^{\omega_{0}}[M_{d}]=\frac{n_{0}}{2}\mathrm{Res}_{X=0}\left(4X^{2}\sum_{F\in\mathcal{F}_{+}}\int_{F}\frac{i_{F}^{*}c_{1}(\mathcal{L}_{d})^{n-3}(X)}{e_{F}(X)}dX\right)$

since $\mathcal{L}_{d}$ is a prequantum line bundle and the maximal torus is
the unit circle then by Lemma 9.31 of [12]
$i_{f}^{*}c_{1}(\mathcal{L}_{d})^{n-3}(X)=(-1)^{n-3}\mu(F)^{n-3}(X).$ Hence,

$\frac{i_{F}^{*}c_{1}(\mathcal{L}_{d})^{n-3}}{e_{F}(X)}=-(-1)^{n-k}(\mu(F)(X))^{n-3}X^{-n-3}.$

Because of the above observation,
$\mu(F)=\mathrm{DF}_{\mathcal{D}}(\mathcal{X}_{F},\mathcal{L}_{F})=D_{F}^{+}-D_{F}^{-}$.
We have

$4X^{2}\frac{i_{F}^{*}c_{1}(\mathcal{L}_{d})^{n-3}}{e_{F}(X)}=-(-1)^{n-k}\mathrm{DF}_{\mathcal{D}}(\mathcal{X}_{F},\mathcal{L}_{F})^{n-3}X^{-1}.$
(18)

The residue of $X^{-1}$ in 18 is one. By taking the sum, we proved the
following result

######  Theorem 5.1.

The volume of the moduli space of log Fano hyperplane arrangements of
dimension one is the sum of the Donaldson-Futaki invariants associated to test
configurations $(\mathcal{X}_{F},\mathcal{L}_{F}).$ Where
$F\in\mathcal{F}_{+}$ is like in [38], and $(\mathcal{X}_{F},\mathcal{L}_{F})$
is the test configuration for which $|F|$ points come together into one point.

$\mathrm{Vol}(M_{d})=-\frac{(2\pi)^{n-3}}{2(n-3)!}\sum_{i=0}^{n-3}(-1)^{k}\sum_{F\in\mathcal{F}_{+},\
|F|=k}\mathrm{DF}_{\mathcal{D}}(\mathcal{X}_{F},\mathcal{L}_{F})^{n-3}.$

######  Remark 5.1.

It is natural to ask if more generally one can express the volumes as _sums_
of CM degrees of special unstable families, using the $\Theta$-stratification
[28] to generic _Kirwan’s_ conditions [28, Theorem 5.6].

## APPENDIX A

The Python script for calculating the equality of the two mentioned volume
formulas is given below.

Listing 1: Test of Equivalence

⬇

import numpy as np

import math

from fractions import Fraction

import sympy

import itertools

from itertools import combinations

import random

def subsets_k(collection, k):

yield from partition_k(collection, k, k)

def partition_k(collection, min, k):

if len(collection) == 1:

yield [ collection ]

return

first = collection[0]

for smaller in partition_k(collection[1:], min - 1, k):

if len(smaller) > k: continue

# insert ‘first‘ in each of the subpartition’s subsets

if len(smaller) >= min:

for n, subset in enumerate(smaller):

yield smaller[:n] + [[ first ] + subset] + smaller[n+1:]

# put ‘first‘ in its own subset

if len(smaller) < k: yield [ [ first ] ] + smaller

def MandiniVolume(V):

N = len(V)

num = Fraction(np.power(2, 2*N-7))

Prefactor = Fraction(num, math.factorial(N-3))*np.power(sympy.pi, N-3)

Result = 0

for I in range(0,N):

R = list(combinations(V,I))

factor = 0

for j in range(0, np.array(R, dtype = ’object’).shape[0]):

Q = 1

S = np.power(np.max([0, 1- np.sum(R[j])]), N-3)

Q *=S

factor += (-1)**(I+1)*Q

Result += Fraction(factor)

return (Result*Prefactor)

def McMullenVolume(V):

N = len(V)

Prefactor = Fraction((-4)**(N-3),math.factorial(N-2)) * sympy.pi**(N-3)

Result = 0

for P in range(3,N+1):

R = list(subsets_k(V, P))

factor = 0

for j in range(0,np.array(R, dtype=”object”).shape[0]):

Q = 1

for i in range(0,P):

B = len(R[j][i])

S = np.max([0,1-np.sum(R[j][i])])**(B-1)

Q *=S

factor += (-1)**(P+1) *math.factorial(P-3)*Q

Result += factor

return (Result*Prefactor)

class AnomalyTest:

def __init__(self,length, total_tests):

if not isinstance(length, int) and length >= 3:

raise ValueError(’lenght␣must␣be␣an␣integer␣greater␣or␣equal␣than␣3’)

if not isinstance(total_tests, int) and total_tests >0:

raise ValueError(f’total_test␣is␣a␣positive␣ingeger,␣found␣{total_test}’)

self.length = length

self.total_tests = total_tests

def RandomVector(self):

V = []

for i in range(0,self.length):

n = random.randint(1,30)

V.append(n)

V = list(np.asarray(V)*Fraction(2)/np.sum(V))

return V

def run(self):

test_iterations, anomalies = np.zeros((1,), dtype=int), np.zeros((1,),
dtype=int)

while test_iterations < self.total_tests:

V = self.RandomVector()

check= any(entry > 1 for entry in V)

if check:

continue

test_iterations+=1

Vol_1 = MandiniVolume(V)

Vol_2 = McMullenVolume(V)

if Vol_1 == 0 and Vol_2 == 0:

Ratio = 1.0

else:

Ratio = float(Vol_1 / Vol_2)

if Ratio != 1:

anomalies += 1

print (V)

print(’Vol_1,␣Vol_2:’, Vol_1, Vol_2)

print(’The␣ratio␣of␣the␣two␣values␣of␣Vol(M_d):’, r)

print(’Total␣anomalies’, anomalies)

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