Jeremy H. M. Wong, Huayun Zhang, and Nancy F. Chen††thanks: This work was
supported by the A$\star$STAR Computational Resource Centre through the use of
its high performance computing facilities.

# Multiple output samples for each input in a single-output Gaussian process

###### Abstract

The standard Gaussian Process (GP) only considers a single output sample per
input in the training set. Datasets for subjective tasks, such as spoken
language assessment, may be annotated with output labels from multiple human
raters per input. This paper proposes to generalise the GP to allow for these
multiple output samples in the training set, and thus make use of available
output uncertainty information. This differs from a multi-output GP, as all
output samples are from the same task here. The output density function is
formulated to be the joint likelihood of observing all output samples, and
latent variables are not repeated to reduce computation cost. The test set
predictions are inferred similarly to a standard GP, with a difference being
in the optimised hyper-parameters. This is evaluated on speechocean762,
showing that it allows the GP to compute a test set output distribution that
is more similar to the collection of reference outputs from the multiple human
raters.

Index Terms: Gaussian process, subjective, uncertainty, inter-rater agreement,
spoken language assessment

##  1 Introduction

The Gaussian Process (GP) [1] expresses a distributional uncertainty [2] that
naturally increases for inputs further away from the training data. It may
also be interpreted as marginalising over functions [1], and thus reduces the
influence of model uncertainty [3, 4]. However, a GP does not consider whether
outputs are similar for proximate inputs, and thus may lack in modelling data
uncertainty [3, 5]. Better uncertainty estimation may be especially desirable
for subjective tasks, such as Spoken Language Assessment (SLA), where multiple
human annotators may provide differing output labels for the same input [6]. A
collection of human annotations for the same input may be interpreted as a
reference of data uncertainty that an automatic model should also aim to
compute. In such settings, the uncertainties expressed by the model and the
human annotators can be explicitly compared.

The standard GP assumes that each input in the training set is paired with
only a single output, treated as the ground truth. This paper proposes to
extend the GP formulation, to use available information of data uncertainty
from multiple samples of outputs for each input. A naive approach is to repeat
the inputs for each output, but this greatly increases the computational cost.
A proposed omission of repeated latent variables avoids this expense. This
paper uses a GP with a Gaussian output, where analytical densities can be
computed. A related work in [7] instead extends a bounded GP with a non-
Gaussian output to use multiple output samples in the training set, where the
density functions can no longer be analytically expressed.

The aim of SLA is to assign a score related to oral proficiency, to an audio
input. Examples of proficiency aspects that are often assessed include
pronunciation accuracy, intonation, fluency, prosody, task completion, and
topic relevance. As previously mentioned, SLA is a subjective task, where
differing scores are often assigned by multiple expert raters for the same
input. The diversity of scores can be reduced by increasing the coverage of
rules in a rubric. However, the choice of rules may not agree with the needs
of diverse users. This paper instead embraces subjectivity, by using the
diversity of reference scores to better predict output uncertainty. This
uncertainty information may allow the system to better calibrate its feedback
to the user, by not penalising a student for a low score that raters would
have disagreed upon, and instead seek clarification or human intervention.
This also reduces the need for a strict rubric, thereby allowing better
generalisation to diverse user needs.

##  2 Related work

As opposed to a GP, Neural Networks (NN) may yield overly confident
predictions of uncertainty, even when the input is from a mismatched domain
[8]. The multi-output GP [9, 10] is formulated in a multi-task framework [11],
where each output is a separate task. This paper instead considers a single-
output GP with one output task, where multiple output samples per input are
present for that task. This paper uses a distribution of reference outputs to
train and evaluate a GP. When training a NN, the reference output can be in
the form of a distribution, as opposed to a scalar or single class. Full-sum
training in speech recognition [12] and handwriting recognition [13] trains a
NN toward the distributional reference formed by the soft forced alignment.
NNs can also be trained toward the distribution represented by multiple
reference scores [14]. The distributional output from one NN can also be used
as a reference to train another NN toward [15, 16].

##  3 Gaussian process regression

When given a training set of $N$ input feature vectors of dimension $D$,
$\mathbf{X}\in\mathbb{R}^{N\times D}$, a GP places a jointly Gaussian prior
over latent variables, $\bm{f}\in\mathbb{R}^{N}$, as

$p\left(\bm{f}\middle|\mathbf{X}\right)=\mathcal{N}\left(\bm{f};\bm{0},\mathbf{K}\left(\mathbf{X},\mathbf{X}\right)\right).$
(1)

The covariance of the latent variable is defined as a pair-wise similarity
between the inputs, computed by the kernel, $\mathbf{K}$. Here, the squared
exponential kernel is used, with elements defined as

$k_{ij}\left(\mathbf{X},\mathbf{X}\right)=s^{2}\exp\left[-\frac{\left(\bm{x}_{i}-\bm{x}_{j}\right)^{\top}\left(\bm{x}_{i}-\bm{x}_{j}\right)}{2l^{2}}\right],$
(2)

where $i$ and $j$ are the data point indexes, $l$ is a length hyper-parameter,
and $s$ is a scale hyper-parameter. The GP makes the assumption that the
outputs, $\bm{y}\in\mathbb{R}^{N}$, are conditionally independent of the
inputs when given the latent variables. The outputs are Gaussian distributed
about a mean of the latent variable, with an observation noise hyper-
parameter, $\sigma$,

$p\left(\bm{y}\middle|\bm{f}\right)=\mathcal{N}\left(\bm{y};\bm{f},\sigma^{2}\mathbf{I}\right),$
(3)

where $\mathbf{I}$ is the identity matrix.

Training a GP involves estimating the hyper-parameters $s$, $l$, and $\sigma$.
One approach is to maximise the marginal log-likelihood of the training data,

$\mathcal{F}=\log p\left(\bm{y}^{\text{ref}}\middle|\mathbf{X}\right),$ (4)

where $\bm{y}^{\text{ref}}$ are the reference outputs. The marginal likelihood
is

$\displaystyle p\left(\bm{y}\middle|\mathbf{X}\right)$ $\displaystyle=\int
p\left(\bm{y}\middle|\bm{f}\right)p\left(\bm{f}\middle|\mathbf{X}\right)d\bm{f}$
$\displaystyle=\mathcal{N}\left(\bm{y};\bm{0},K\left(\mathbf{X},\mathbf{X}\right)+\sigma^{2}\mathbf{I}\right).$
(5)

Gradient-based optimisation of the hyper-parameters requires inverting
$K(\mathbf{X},\mathbf{X})+\sigma^{2}\mathbf{I}$ [1], with a number of
operations scaling as $\mathcal{O}(N^{3})$. Algorithms such as [17] reduce the
polynomial power, but still require more than $\mathcal{O}(N^{2})$ operations.

For evaluation, outputs, $\widehat{\bm{y}}$, are predicted from test inputs,
$\widehat{\mathbf{X}}$, by first computing a density over predicted latent
variables,

$p\left(\widehat{\bm{f}}\middle|\widehat{\mathbf{X}},\bm{y},\mathbf{X}\right)=\frac{p\left(\bm{y},\widehat{\bm{f}}\middle|\mathbf{X},\widehat{\mathbf{X}}\right)}{p\left(\bm{y}\middle|\mathbf{X}\right)}=\mathcal{N}\left(\widehat{\bm{f}};\widehat{\bm{\mu}},\widehat{\mathbf{V}}\right),$
(6)

where

$\displaystyle\widehat{\bm{\mu}}$
$\displaystyle=\mathbf{K}^{\hat{x}x}\left[\mathbf{K}^{xx}+\sigma^{2}\mathbf{I}\right]^{-1}\bm{y}$
(7) $\displaystyle\widehat{\mathbf{V}}$
$\displaystyle=\mathbf{K}^{\hat{x}\hat{x}}-\mathbf{K}^{\hat{x}x}\left[\mathbf{K}^{xx}+\sigma^{2}\mathbf{I}\right]^{-1}\mathbf{K}^{x\hat{x}},$
(8)

which, similarly to training, also requires the computation of
$[\mathbf{K}^{xx}+\sigma^{2}\mathbf{I}]^{-1}$. For brevity, the notations of
$\mathbf{K}^{xx}=\mathbf{K}(\mathbf{X},\mathbf{X})$,
$\mathbf{K}^{x\hat{x}}=\mathbf{K}(\mathbf{X},\widehat{\mathbf{X}})={\mathbf{K}^{\hat{x}x}}^{\top}$,
and
$\mathbf{K}^{\hat{x}\hat{x}}=\mathbf{K}(\widehat{\mathbf{X}},\widehat{\mathbf{X}})$
are used. The predictive density function for the test output is then

$p\left(\widehat{\bm{y}}\middle|\widehat{\mathbf{X}},\bm{y},\mathbf{X}\right)=\mathcal{N}\left(\widehat{\bm{y}};\widehat{\bm{\mu}},\widehat{\mathbf{V}}+\sigma^{2}\mathbf{I}\right).$
(9)

Finally, the predicted scalar output for each test data point, $i$, can be
inferred from (9), using a decision rule of, for example, choosing the mean,
$\widehat{y}_{i}^{\star}=\widehat{\mu}_{i}$. The mean, mode, and median are
equivalent because of the Gaussian symmetry and uni-modality.

##  4 Multiple output samples

In the standard GP, each input is paired with a single output sample. In SLA,
using a GP with only a scalar output label as the reference [18] may forego
information about the inter-rater uncertainty. This paper proposes to extend
the GP to use the multiple output samples in the training set. Let each
training input be associated with $R$ output reference samples,
$\bm{y}_{i}^{\text{ref}}=[y_{i1}^{\text{ref}},\cdots,y_{iR}^{\text{ref}}]$,
where $y_{ir}^{\text{ref}}$ is the $r$th reference output sample for the $i$th
data point. It is assumed for simplicity that the number of output samples is
the same for all inputs, but an analogous formulation can be derived to allow
for varying numbers of raters. These outputs are stacked together across all
data points to yield an output matrix
$\mathbf{Y}^{\text{ref}}\in\mathbb{R}^{N\times R}$.

A naive approach to accommodate multiple output samples is to flatten
$\mathbf{Y}^{\text{ref}}$ into a vector and repeat the inputs for each output
sample. This will yield a $\mathbb{R}^{NR\times NR}$ kernel111A $\sim$ is used
to emphasise repeated elements. of
$\widetilde{\mathbf{K}}=\mathbf{K}(\mathbf{X},\mathbf{X})\otimes\mathbf{1}^{(R)}$,
where $\mathbf{1}^{(R)}$ is a $\mathbb{R}^{R\times R}$ matrix of 1s and
$\otimes$ represents a Kronecker (tensor) product. There is no need to
explicitly invert this non-full-rank kernel. Instead, only the inverse of
$\widetilde{\mathbf{K}}+\sigma^{2}\mathbf{I}$ needs to be computed during
training and inference, which is full-rank. This inversion requires a number
of operations that scales as $\mathcal{O}(N^{3}R^{3})$, as opposed to
$\mathcal{O}(N^{3})$ when using only a single output sample. Computational
constraints may then impose the need for sparse approximation [19], even for
modest training set sizes.

When using this repeated kernel, the latent variables are distributed as
$p(\widetilde{\bm{f}}|\mathbf{X})=\mathcal{N}(\widetilde{\bm{f}}|\bm{0},\widetilde{\mathbf{K}})$.
The repeated structure of $\widetilde{\mathbf{K}}$ causes all latent variables
that are associated with the same training data point to be perfectly
correlated, and thus should have the same value. Therefore, it seems
computationally redundant to separately model the repeated latent variables.
This paper proposes that there is no need to separately compute each of the
repetitions, and to instead only express a single instance of each latent
variable per training data point using (1). The joint density of the multiple
output samples is then conditioned on these non-repeated latent variables,
$p(\mathbf{Y}|\bm{f})$. The diagonal covariance in the output density function
of a standard GP in (3) implies that the outputs from different training data
points are conditionally independent of each other when given the latent
variables. This conditional independence, applied to the case of repeated
inputs in the naive approach, yields in this proposed approach a joint output
density that can be factorised as

$\displaystyle p\left(\mathbf{Y}\middle|\bm{f}\right)\\!=$
$\displaystyle\prod_{r=1}^{R}\mathcal{N}\left(\bm{y}_{r};\bm{f},\sigma^{2}\mathbf{I}\right)$
(10) $\displaystyle=$
$\displaystyle\,\mathcal{N}\left(\overline{\bm{\mu}}\\!;\bm{f},\frac{\sigma^{2}}{R}\mathbf{I}\right)\mathcal{N}\left(\overline{\bm{\eta}}\\!;\bm{0},\frac{\sigma^{2}}{R}\mathbf{I}\right)\frac{\left(2\pi\sigma^{2}\right)^{\left(2-R\right)\frac{N}{2}}}{R^{N}}\\!,$

where

$\overline{\bm{\mu}}=\frac{1}{R}\sum_{r=1}^{R}\bm{y}_{r}\quad\text{and}\quad\overline{\eta}_{i}=\sqrt{\frac{1}{R}\sum_{r=1}^{R}\left(y_{ir}-\overline{\mu}_{i}\right)^{2}},$
(11)

and $\bm{y}_{r}\in\mathbb{R}^{N}$ is a vector of the $r$th output sample for
all inputs. The factorisation and diagonal covariance in (10) assume that the
multiple output samples for the same input and across different inputs are
independent of each other, when given the latent variable. As a result, the
output density does not depend on the order of the multiple output samples for
each input. This can be explicitly seen in the re-parameterisation of the
multiple outputs as the empirical mean $\overline{\bm{\mu}}$ and empirical
biased standard deviation $\overline{\bm{\eta}}$, both of which are
independent of the ordering of the output samples. Also, (10) may seem
unnormalised over $\overline{\bm{\mu}}$ and $\overline{\bm{\eta}}$. However,
it should be noted that $p(\mathbf{Y}|\bm{f})$ is expected to sum to one when
integrated over $\mathbf{Y}$, and not over $\overline{\bm{\mu}}$ and
$\overline{\bm{\eta}}$.

The marginal likelihood can be computed from the non-redundant prior of (1)
and the output density of (10), as

$\displaystyle p\left(\mathbf{Y}\middle|\mathbf{X}\right)$ $\displaystyle=\int
p\left(\mathbf{Y}\middle|\bm{f}\right)p\left(\bm{f}\middle|\mathbf{X}\right)d\bm{f}$
$\displaystyle=\mathcal{N}\left(\overline{\bm{\mu}};\bm{0},\mathbf{K}^{xx}+\frac{\sigma^{2}}{R}\mathbf{I}\right)g\left(\mathbf{Y}\right),$
(12)

where

$g\left(\mathbf{Y}\right)=\mathcal{N}\left(\overline{\bm{\eta}};\bm{0},\frac{\sigma^{2}}{R}\mathbf{I}\right)\left(2\pi\sigma^{2}\right)^{\left(2-R\right)\frac{N}{2}}R^{-N}.$
(13)

Analogously to the standard GP, and equivalently to the naive approach, the
hyper-parameters here can be optimised by maximising the joint marginal log-
likelihood of all output samples,

$\mathcal{F}_{\text{joint}}=\log
p\left(\mathbf{Y}^{\text{ref}}\middle|\mathbf{X}\right).$ (14)

Unlike a naive implementation, gradient-based optimisation of the proposed
method has a number of operations that scales as $\mathcal{O}(N^{3})$,
similarly to a standard GP with a single output sample.

When performing evaluation, the predicted output can be inferred from the
predictive density function. First, the joint density between the training set
outputs and the test latent variables needs to be computed. When the training
set comprises multiple output samples per input, this paper again proposes
that there is no need to separately model redundant latent variables, by
expressing this joint density as

$\displaystyle
p\left(\mathbf{Y},\widehat{\bm{f}}\middle|\mathbf{X},\widehat{\mathbf{X}}\right)$
$\displaystyle=\int
p\left(\bm{f},\widehat{\bm{f}}\middle|\mathbf{X},\widehat{\mathbf{X}}\right)p\left(\mathbf{Y}\middle|\bm{f}\right)d\bm{f}$
(15)
$\displaystyle=\mathcal{N}\left(\left[\begin{matrix}\overline{\bm{\mu}}\\\
\widehat{\bm{f}}\end{matrix}\right]\\!;\bm{0},\left[\begin{matrix}\mathbf{K}^{xx}\\!+\frac{\sigma^{2}}{R}\mathbf{I}&\mathbf{K}^{x\hat{x}}\\\
\mathbf{K}^{\hat{x}x}&\mathbf{K}^{\hat{x}\hat{x}}\end{matrix}\right]\right)\\!g\left(\mathbf{Y}\right)\\!.$

The predictive density over the test latent variables is

$p\left(\widehat{\bm{f}}\middle|\widehat{\mathbf{X}},\mathbf{Y},\mathbf{X}\right)=\frac{p\left(\mathbf{Y},\widehat{\bm{f}}\middle|\mathbf{X},\widehat{\mathbf{X}}\right)}{p\left(\mathbf{Y}\middle|\mathbf{X}\right)}=\mathcal{N}\left(\widehat{\bm{f}};\breve{\bm{\mu}},\breve{\mathbf{V}}\right),$
(16)

analogously to (6), where here

$\displaystyle\breve{\bm{\mu}}$
$\displaystyle=\mathbf{K}^{\hat{x}x}\left[\mathbf{K}^{xx}+\frac{\sigma^{2}}{R}\mathbf{I}\right]^{-1}\overline{\bm{\mu}}$
(17) $\displaystyle\breve{\mathbf{V}}$
$\displaystyle=\mathbf{K}^{\hat{x}\hat{x}}-\mathbf{K}^{\hat{x}x}\left[\mathbf{K}^{xx}+\frac{\sigma^{2}}{R}\mathbf{I}\right]^{-1}\mathbf{K}^{x\hat{x}}.$
(18)

The predictive density over the test outputs is then

$p\left(\widehat{\bm{y}}\middle|\widehat{\mathbf{X}},\mathbf{Y},\mathbf{X}\right)=\mathcal{N}\left(\widehat{\bm{y}};\breve{\bm{\mu}},\breve{\mathbf{V}}+\sigma^{2}\mathbf{I}\right).$
(19)

Finally, the output can be inferred for each test input data point by choosing
the mean of this predictive density, similarly to how a score is chosen from
(9). The matrix inverted in (17) and (18) is of size $\mathbb{R}^{N\times N}$,
as opposed to $\mathbb{R}^{NR\times NR}$ in the naive repetition approach. The
number of operations thus scales as $\mathcal{O}(N^{3})$, similarly to a
standard GP with a single output sample.

The covariance of the predictive densities in both (9) and (19) are
independent of the training set outputs, only depending on the training set
inputs and the hyper-parameters. As such, information about the proximity
between the multiple output samples for the same input in the training set
does not influence the resulting predictive covariance. Furthermore, whether
the training set output values are near or far from each other, for two inputs
that are near as measured by the kernel, does not influence the resulting
predictive covariance for test inputs near to these training inputs. The
predictive covariance is only affected by the distance between the test and
training set inputs. This implies that a GP may be more suited for computing
distributional uncertainty, and less so for the data uncertainty [5] that may
be more relevant to the uncertainty expressed between the raters. A future
research direction may be to allow the GP to take training set outputs into
account for the predictive covariance.

As can be seen from (17), the predictive mean under the proposed formulation
is dependent on the mean of the multiple output samples,
$\overline{\bm{\mu}}$, and independent of $\overline{\bm{\eta}}$. This is
similar to a standard GP formulation, when using a mean combination of the
multiple rater scores to get the reference. As mentioned previously, the
predictive covariance is independent of the training set outputs. Thus, during
inference, the remaining difference between the standard GP and the proposal
is in the hyper-parameters. The proposal optimises the hyper-parameters by
maximising the joint marginal log-likelihood of all training set outputs,
while the standard approach maximises the marginal log-likelihood of the
combined output reference.

##  5 Experiments

Experiments were performed on speechocean762 [6]. This comprises training and
test sets, each with 2500 sentences and 125 disjoint speakers. The native
Mandarin speakers read the sentences in English. Each sentence is annotated by
5 raters with a variety of score types at different linguistic levels, but
only the sentence-level pronunciation accuracy was used in this paper, which
has an integer score range between 0 and 10.

The scalar reference scores were computed as a mean of the multiple rater
scores. This differs from the median used in [6]. The mean may yield a
baseline that better matches (10). Following [6], the predicted and combined
reference scores were first rounded to integers, before computing the
Pearson's Correlation Coefficient (PCC) and Mean Squared Error (MSE). These
metrics only compare scalar references and hypotheses. A model with a
probabilistic output can also be assessed on how well it matches the
distribution of scores from multiple raters, and thus its ability to predict
the inter-rater uncertainty. The reference distribution over integer scores,
$c$, can be expressed as a mixture of Kronecker $\delta$-functions at each
rater's score,

$P^{\text{ref}}\left(c\middle|\widehat{\bm{x}}_{i}\right)=\frac{1}{R}\sum_{r=1}^{R}\delta\left(c,y^{\text{ref}}_{ir}\right).$
(20)

This Monte Carlo approximation to a Bayesian NN [4] marginalises over the
human equivalent of model uncertainty, yielding a purer reference of data
uncertainty. It also represents the fraction of raters who assigned each
score. The model's output predictive density can be discretised as

$P\left(c\middle|\widehat{\bm{x}}_{i},\mathbf{Y},\mathbf{X}\right)=\frac{\int_{c-0.5}^{c+0.5}p\left(\widehat{y}_{i}\middle|\widehat{\bm{x}}_{i},\mathbf{Y},\mathbf{X}\right)d\widehat{y}_{i}}{\sum\limits_{c^{\prime}}\int_{c^{\prime}-0.5}^{c^{\prime}+0.5}p\left(\widehat{y}_{i}^{\prime}\middle|\widehat{\bm{x}}_{i},\mathbf{Y},\mathbf{X}\right)d\widehat{y}_{i}^{\prime}},$
(21)

to express the probability of each integer score. The reference and
hypothesised distributions can then be compared using a discrete Kullback-
Leibler (KL) divergence,

$\text{KL}=\frac{1}{M}\sum_{i=1}^{M}\sum_{c}P^{\text{ref}}\left(c\middle|\widehat{\bm{x}}_{i}\right)\log\frac{P^{\text{ref}}\left(c\middle|\widehat{\bm{x}}_{i}\right)}{P\left(c\middle|\widehat{\bm{x}}_{i},\mathbf{Y},\mathbf{X}\right)}.$
(22)

The test set size is $M$. Unlike a continuous KL divergence, (22) is lower-
bounded by 0, easing interpretability, and follows the standard practice [6]
to round scores before evaluating.

The statistical significance, $\rho$, for MSE and KL divergence was computed
using a two-tailed paired $t$-test. The PCC is not easily expressed as a sum
over data points, making it difficult to apply the central limit theorem.
Instead, the $Z_{1}^{\star}$ approach was used [20]. Here, an approximately
normally distributed transformation [21] was first computed from the two PCCs
being compared. Then the significance was computed as the two-tailed
cumulative density of this transformed quantity.

The feature extraction for a NN SLA model followed [22]. First, hybrid speech
recognition models force-aligned the audio with the transcriptions. These were
trained using Kaldi [23] on 960 hours Librispeech [24] using cross-entropy,
following [6]. From the forced alignment, Goodness Of Pronunciation (GOP)
[25], Log Phone Posterior (LPP) [26], Log Posterior Ratio (LPR) [26], and
tempo [22] features were extracted for each phone. Pitch features [27] were
also extracted. A continuous skip-gram model [28], with a 32-node recurrent NN
[29] hidden layer, was trained on the training set non-silence phone
sequences, and used to extract phone embeddings [22]. The GOP, LPP, LPR,
tempo, pitch, and phone embedding features were concatenated to form one
feature vector per phone.

In baseline NN SLA models, the sequence of feature vectors, with length equal
to the number of phones in the sentence, were fed into a Bidirectional Long
Short-Term Memory (BLSTM) layer [30], with 32 nodes per direction. The output
was then pooled across all phones in the sequence with equal weights, and fed
through a linear layer. Two forms of NN output layers were used. In the first
variation, referred to as NNscalar, the linear layer had a single dimensional
output, which then passed through a sigmoid [22]. This sigmoid output was then
scaled to the bounds of the output score range, and the computed output was
treated as the predicted score. Such a regression model was trained by
minimising the MSE toward the combined reference score. In the second
variation, referred to as NNcategorical, the linear layer output dimension was
equal to the number of integer scores, and the output was fed through a
softmax [31]. This classification model was trained using the cross-entropy
criterion toward the combined reference score. Both models used dropout [32]
with an omission probability of 60%, before the BLSTM and linear layers. For
NN training, a 10% validation set was held out without speaker disjointment.

It may not be trivial to allow a GP to use sequential inputs. Work in [33]
designs kernels to operate on sequences, while [18] extracts hand-crafted
sentence-level features for SLA. This paper uses the activations after the NN
pooling layer as bottleneck features to be sentence-level inputs to the GP
[34]. The GP and NN feature extractor were not jointly fine-tuned, as doing so
may overfit easily [35]. Principle component analysis whitening, estimated on
the training set, was applied to these features to better abide by the
diagonal covariance assumption of the scalar length hyper-parameter in the
kernel in (2).

Table 1: Comparison of NN and GP SLA models SLA model | Bottleneck extractor | PCC$\uparrow$ | MSE$\downarrow$
---|---|---|---
NNscalar | - | 0.711 | 1.232
NNcategorical | - | 0.701 | 1.208
GPbase | NNscalar | 0.710 | 1.149
GPbase | NNcategorical | 0.694 | 1.209

The first experiment compares GP and NN SLA models, for each NN feature
extractor. The standard GP was used, with only a single reference output
sample per training data point, referred to as GPbase. The GP hyper-parameters
were optimised separately for each feature type, using gradient descent to
maximise (4). The mean score between the multiple raters was used as the
training set reference output. The results in table 1 show that the GP and NN
models exhibit comparable performances. A GP that uses the NNscalar bottleneck
features may yield better PCC and MSE performances than when using
NNcategorical features, but $\rho_{\text{PCC}}=0.033$ and
$\rho_{\text{MSE}}=0.078$ suggest that the difference may not be significant.
All subsequent experiments use bottleneck features from NNscalar as inputs to
the GP.

The next experiment assesses the proposal, of allowing the GP to use the
separate output scores from multiple raters in the training set, referred to
as GPjoint. This is compared against GPbase, and also against a naive multiple
output extension of repeating the training set input features, referred to as
GPrepeat. The GPjoint hyper-parameters were optimised by maximising (14), and
the predicted scores were inferred using (19).

Table 2: Using training set scores from multiple raters in a GP Model | PCC | MSE | KL$\downarrow$ | Inference time (s)$\downarrow$
---|---|---|---|---
GPbase | 0.710 | 1.149 | 3.10 | 136 $\pm$ 3
GPrepeat | 0.713 | 1.136 | 0.85 | 8476 $\pm$ 510
GPjoint | 138 $\pm$ 4

In table 2, the PCC and MSE results may suggest that the proposed GPjoint
approach yields performance gains over GPbase, but these may not be
significant, with $\rho_{\text{PCC}}=0.065$ and $\rho_{\text{MSE}}=0.036$. The
PCC and MSE compare the prediction against the mean combined reference,
assuming that this scalar reference is correct, and not considering how well
the model's output uncertainty matches the uncertainty between the raters. The
KL divergence is used to assess this uncertainty matching. As a baseline, the
KL divergence of the NNcategorical model is 1.26. The results show that using
the separate scores from multiple training set raters improves the KL
divergence of the GP, with $\rho_{\text{KL}}<0.001$. This suggests that the
proposed extension allows the GP's output distribution to better match that
represented by the collection of scores from the multiple raters. Such a more
accurate uncertainty prediction may better inform the type of feedback that
should be given to the user.

The computational savings of GPjoint over GPrepeat was assessed by repeating
inference 10 times on a 48 core Intel Xeon Platinum 8268 2.90GHz CPU using the
Numpy pseudo-inverse implementation. The mean and standard deviation of the
total time to infer the whole test set accumulated over all threads are shown
in the right-side column in table 2. Inference using a GP can be run
efficiently by pre-computing the matrix inverse before any test data points
are seen. It therefore seems more interpretable to view the total time across
the whole test set, rather than the time per data point or real-time factor,
assuming that the computation time is dominated by matrix inversion. The
results show that GPjoint is indeed faster to infer from than GPrepeat.
GPjoint and GPbase require comparable durations to infer from. Although not
shown here, this computational saving also benefits the matrix inversion
computation during training.

##  6 Conclusion

This work extends the GP to allow for multiple output samples per input in the
training set with computational efficiency. The hyper-parameters can be
optimised by maximising the joint marginal likelihood of all output samples.
Inference is performed in a fairly similar way to a standard GP, with the main
deviation being in the hyper-parameters that are optimised differently. This
allows the predictive density to better match the uncertainty expressed by the
multiple raters. Such information may better inform the type of feedback given
to the user.

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# Brauer group of moduli stack of stable parabolic
$\textnormal{PGL}(r)$-bundles over a curve

Indranil Biswas, Sujoy Chakraborty, Arijit Dey  School of Mathematics, Tata
Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005,
India<EMAIL_ADDRESS>Department of Mathematics, Indian Institute
of Technology Chennai, Chennai, India<EMAIL_ADDRESS>Department of
Mathematics, Indian Institute of Technology Chennai, Chennai, India
<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be an algebraically closed field of characteristic zero. We prove that
the Brauer group of moduli stack of stable parabolic
$\textnormal{PGL}(r,k)$-bundles with full flag quasi-parabolic structures at
an arbitrary parabolic divisor on a curve $X$ coincides with the Brauer group
of the smooth locus of the corresponding coarse moduli space of parabolic
$\textnormal{PGL}(r,k)$-bundles. We also compute the Brauer group of the
smooth locus of this coarse moduli for more general quasi-parabolic types and
weights satisfying certain conditions.

###### Key words and phrases:

Brauer group; Moduli space; Moduli stack; Parabolic bundle.

###### 2010 Mathematics Subject Classification:

14D20, 14D22, 14D23, 14F22, 14H60

##  1\. introduction

Let $Y$ be a quasi-projective variety defined over an algebraically closed
field $k$ of characteristic zero. The cohomological Brauer group of $Y$ is
defined to be the torsion part

$\operatorname{H^{2}(Y_{\text{\'{e}t}},\,\mathbb{G}_{m})_{\text{torsion}}}\,\subset\,H^{2}(Y_{\text{\'{e}t}},\,\mathbb{G}_{m}),$

and it is denoted by $\operatorname{\textnormal{Br}}^{\prime}(Y)$. When the
variety $Y$ is smooth, it is known that the above group
$\operatorname{H^{2}(Y_{\text{\'{e}t}},\,\mathbb{G}_{m})}$ is actually
torsion. The Brauer group of $Y$, which is denoted by
$\operatorname{\textnormal{Br}}(Y)$, is defined to be the Morita equivalence
classes of Azumaya algebras over $Y$. Giving an Azumaya algebra over $Y$ is
equivalent to giving a Brauer-Severi scheme (i.e., a projective bundle) over
$Y$ in the étale topology, which is also equivalent to giving a principal
$\textnormal{PGL}(r,k)$-bundle over $Y$. The Brauer class of a projective
bundle $\mathbb{P}\,\longrightarrow\,Y$ can be thought of as the obstruction
for $\mathbb{P}$ to be the projectivization of a vector bundle on $Y$. For
each integer $r\,\geq\,2$, the sequence of cohomologies associated to the
exact sequence

$\\{e\\}\,\longrightarrow\,\mathbb{\mu}_{r}\,\longrightarrow\,\text{SL}_{r}\,\longrightarrow\,\text{PGL}_{r}\,\longrightarrow\,\\{e\\}$

gives a homomorphism
$H^{1}(Y_{\text{\'{e}t}},\,\textnormal{PGL}_{r})\,\longrightarrow\,\text{Br}^{\prime}(Y)$,
which induces an injective group homomorphism
$i\,:\,\text{Br}(Y)\,\longrightarrow\,\operatorname{\textnormal{Br}}^{\prime}(Y)$.
By a theorem of Gabber it is known that this homomorphism $i$ is surjective
[dJ]. For an algebraic stack $Y$, by Brauer group of $Y$ we mean the
cohomological Brauer group of $Y$.

Let $X$ be an irreducible smooth projective curve of genus $g\,\geq\,2$ over
$k$. Fix an integer $r\geq 2$ and a line bundle $\xi$ of degree $d$ on $X$. If
$g=2,$ we assume that $r\,\geq\,3$. let $M(r,d)$ denote the moduli space of
stable vector bundles on $X$ of rank $r$ and degree $d$; it is a smooth quasi-
projective variety. Let $M_{\xi}(r,d)\,\subset\,M(r,d)$ denote the moduli
space of stable vector bundles on $X$ with rank $r$ and determinant $\xi$. Let
$\mathcal{M}(r,d)$ and $\mathcal{M}_{\xi}(r,d)$ denote the corresponding
moduli stack of stable vector bundles. The Brauer group
$\operatorname{\textnormal{Br}}(M_{\xi}(r,d))$ is cyclic of order
g.c.d.($r,d$) [BBGN]. A generator for
$\operatorname{\textnormal{Br}}(M_{\xi}(r,d))$ is obtained by restricting the
universal projective bundle over $X\times M_{\xi}(r,d)$ to $\\{x_{0}\\}\times
M_{\xi}(r,d)$ , where $x_{0}$ is a closed point of $X$.

Parabolic vector bundles on $X$, denoted by $E_{*}$, were introduced by Mehta
and Seshadri in [MS]. These are vector bundles $E$ on $X$ with a filtration of
fibers of $E$ at a collection of finitely many points $S$ of $X$ and certain
real numbers, called weights, attached to these filtrations. Let
$M^{\alpha}(r,d)$ denote the moduli space of rank $r$ stable parabolic vector
bundles $E_{*}$ on $X$ whose underlying bundle $E$ is of degree $d$ and the
parabolic structure at each parabolic point $p_{i}\,\in\,S$ is of the
following type:

$E_{p_{i}}\,=:\,F_{i,1}\,\supsetneq\,\cdots\,\,\supsetneq\,F_{i,a_{i}}\,\not=\,0$

with $\dim F_{i,j}/F_{i,j+1}\,:=\,r_{i,j}$; the parabolic weight of $F_{i,j}$
is $\alpha_{i,j}\,\in\,\mathbb{R}$ with

$0\,\leq\,\alpha_{i,1}\,<\,\cdots\,<\,\alpha_{i,a_{i}}\,<\,1\,.$

(see § 2 for more details). It is known that $M^{\alpha}(r,d)$ is a smooth
quasi–projective variety [MS]. Let ${M}^{\alpha}_{\xi}(r,d)$ denote the moduli
space of stable parabolic vector bundles of rank $r$ and above parabolic type
with determinant $\xi$, i.e., $\wedge^{r}E\,=\,\xi$. Let
$\mathcal{M}^{\alpha}(r,d)$ and $\mathcal{M}^{\alpha}_{\xi}(r,d)$ denote the
corresponding moduli stack of rank $r$ stable parabolic vector bundles with
above parabolic data.

The Brauer group $\operatorname{\textnormal{Br}}(M^{\alpha}_{\xi}(r,d))$ is
isomorphic to the cyclic group ${\mathbb{Z}}/m{\mathbb{Z}}$ [BD], where

$\displaystyle m\,=\,{\rm
g.c.d.}(d\,,r\,,r_{1,1}\,,\cdots\,,r_{1,a_{1}}\,,\cdots\,,r_{\ell,1}\,,\cdots\,,r_{\ell,a_{\ell}}).$
(1.1)

Like in the case of vector bundles, the cyclic group ${\rm
Br}(M^{\alpha}_{\xi}(r,d))$ is generated by the Brauer class of the
Brauer–Severi variety ${\mathbb{P}}_{x_{0}}$, where $\mathbb{P}$ is the
universal projective bundle on $X\times M^{\alpha}_{\xi}(r,d)$ and
$x_{0}\,\in\,X$.

Let $\mathcal{N}(r)$ be the moduli stack of stable $\textnormal{PGL}(r,k)$
bundles and $N(r)$ the corresponding coarse moduli space. We have the natural
morphism

$f\,:\,\mathcal{N}(r)\,\longrightarrow\,N(r).$

The connected components of both $\mathcal{N}(r)$ and $N(r)$ are indexed by
integers $i\,\in\,[0,\,r-1]$; the $i$-th component of $\mathcal{N}(r)$
(respectively, $N(r)$) is denoted by $\mathcal{N}(r)_{i}$ (respectively,
$N(r)_{i}$). This $\mathcal{N}(r)_{i}$ (respectively, $N(r)_{i}$) is the
quotient of $\mathcal{N}_{\xi}(r,d)$ (respectively, $N_{\xi}(r,d)$) by the
group of stack of $\mu_{r}$-bundles $\rm Bun_{\mu_{r}}$ ((respectively, by the
subgroup $\Gamma\,\subset\,\operatorname{\textnormal{Pic}}^{0}(X)$ of
$r$-torsion points), where $i\,\equiv\,d\,\ (\text{mod}\ r)$ (see [GO, Remark
2.8]). The above morphism $f$ sends $\mathcal{N}(r)_{i}$ onto $N(r)_{i}$. The
Brauer groups of $\mathcal{N}(r)_{i}$ and $N(r)_{i}$ coincide [BH10], and it
fits in the exact sequence

$0\,\longrightarrow\,\frac{H^{2}(\Gamma,k^{*})}{H}\,\longrightarrow\,\operatorname{\textnormal{Br}}(\mathcal{N}(r)_{i})\,\longrightarrow\,\mathbb{Z}/\delta\mathbb{Z}\,\longrightarrow\,0\,,$
(1.2)

where $H\,\subset\,H^{2}(\Gamma,\,k^{*})$ is a subgroup of order
$\delta\,=\,\text{g.c.d}(r,d)$ [BH10, Theorem 1.2].

As in the case of vector bundles, a parabolic stable
$\textnormal{PGL}(r,k)$-bundle is an equivalence class of parabolic stable
vector bundles $[E_{*}]$, where two stable parabolic vector bundles $E_{*}$
and $E^{\prime}_{*}$ are equivalent if there exists a line bundle $L$ such
that $E^{\prime}_{*}\,\cong\,E_{*}\otimes L$. The parabolic structure on
$E^{\prime}_{*}$ is obtained from the parabolic structure of $E_{*}$ by the
above isomorphism.

Let $\mathcal{PN}^{\alpha}(r)$ be the moduli stack of stable parabolic
$\textnormal{PGL}(r,k)$-bundles with full-flag parabolic structures at each
parabolic points of $S$, and let $PN^{\alpha}(r)$ denote the corresponding
coarse moduli space. We have the natural morphism

$\displaystyle\widetilde{g}\,:\,\mathcal{PN}^{\alpha}(r)\,\longrightarrow\,PN^{\alpha}(r).$
(1.3)

As before, the connected components of $\mathcal{PN}^{\alpha}(r)$ and
$PN^{\alpha}(r)$ are indexed by integers $i\,\in\,[0,\,r-1]$; the $i$-th
connected component of $\mathcal{PN}^{\alpha}(r)$ (respectively,
$PN^{\alpha}(r)$) is denoted by $\mathcal{PN}^{\alpha}(r)_{i}$ (respectively,
$PN^{\alpha}(r)_{i}$). The scheme $PN^{\alpha}(r)_{i}$ is the quotient of
$M^{\alpha}_{\xi}(r,d)$ by the above mentioned group $\Gamma$, while the stack
$\mathcal{PN}^{\alpha}(r)_{i}$ is the quotient of the stack
$\mathcal{M}^{\alpha}_{\xi}(r,d)$ by $\rm Bun_{\mu_{r}}$, where
$i\,\equiv\,d\,\ (\textnormal{mod}\ r)$. In both cases, the action is given by
parabolic tensor product with line bundles equipped with the trivial parabolic
structure. The morphism $\widetilde{g}$ sends $\mathcal{PN}^{\alpha}(r)_{i}$
onto $PN^{\alpha}(r)_{i}$.

Our aim here is to study the Brauer group of $\mathcal{PN}^{\alpha}(r)_{i}$
and the smooth locus of $PN^{\alpha}(r)_{i}$. We prove that the two Brauer
groups in question coincide, and is in fact isomorphic to the kernel in (1.2).

The main results are as follows:

######  Theorem 1.1.

  1. (1)

In the case of full flags (i.e. $r_{i,j}=1\,\forall\,(i,j)$), the Brauer
groups of $\mathcal{PN}^{\alpha}(r)_{i}$ and the smooth locus of
$PN^{\alpha}(r)_{i}$ coincide.

  2. (2)

Let $m$ as in (1.1) equals 1, and $PN^{\alpha,sm}(r)_{i}$ denote the smooth
locus of $PN^{\alpha}(r)_{i}$. We have
$\textnormal{Br}(PN^{\alpha,sm}(r)_{i})\,\cong\,\frac{H^{2}(\Gamma,k^{*})}{H},$
where $H\,\subset\,H^{2}(\Gamma,k^{*})$ is as in (1.2).

Theorem 1.1(1) is proved by producing an open subscheme $U$ of
$\textnormal{PN}^{\alpha}(r)$ whose complement is of codimension at least 3
such that the map $\widetilde{g}$ (1.3) is an isomorphism over $U$ (cf.
Corollary 4.3). For Theorem 1.1(2), we first assume the the parabolic weights
are sufficiently small, so that there exists a forgetful map from an open
subscheme $\mathcal{U}_{\alpha}\,\subset\,M^{\alpha}_{\xi}(r,d)$ to
$M_{\xi}(r,d)$ by simply forgetting the parabolic structure. This map is an
étale-locally trivial fibration. Next, we restrict our attention over a
suitably chosen open subscheme of $M^{\alpha}_{\xi}(r,d)$ on which $\Gamma$
acts freely. Quotienting by $\Gamma$ then gives rise to a finite étale
morphism, from which we obtain our result using the Hochschild-Serre spectral
sequence corresponding to this finite étale map (cf. Proposition 5.1).
Finally, for arbitrary generic weights, we use a result of Thaddeus [Th, § 6,
6.2] which says that the moduli spaces with different weights are actually
birational.
Finally, as a remark (Remark 6.2), we discuss the case when the weights are
not generic, and show that under the same conditions as Theorem 1.1, the
Brauer groups can be described for moduli of parabolic stable $PGL(r)$-bundles
for those non-generic weights which belong to a single wall.

##  2\. Preliminaries

Let $X$ be an irreducible smooth projective curve of genus $g\,\geq\,2$,
defined over an algebraically closed field $k$ of characteristic zero. A
vector bundle on $X$ will always mean an algebraic vector bundle. Fix an
integer $r\,\geq\,2;$ if $g\,=\,2,$ then set $r\,\geq\,3$. Fix a line bundle
$\xi$ on $X$ of degree $d$. By a point of $X$ we will always mean a closed
point. Fix a subset

$S\,=\,\\{p_{1},\,p_{2},\,\cdots,\,p_{s}\\}\,\subset\,X$

of $s$ distinct points; they will be referred to as parabolic points. Let $E$
be a vector bundle of rank $r$ on $X$. The fiber of $E$ over a point
$z\,\in\,X$ will be denoted by $E_{z}$.

######  Definition 2.1.

A parabolic data of rank r for points of $S$ consists of the following
collection: for each $p_{i}\,\in\,S$

  * •

a string of positive integers
$(m^{i}_{1},\,m^{i}_{2},\,\cdots,\,m^{i}_{l_{i}})$ such that
$\sum_{j=1}^{l_{i}}m^{i}_{j}\,=\,r$, and

  * •

an increasing sequence of real numbers
$0\,\leq\,\alpha^{i}_{1}\,<\,\alpha^{i}_{2}\,<\,\cdots\,<\,\alpha^{i}_{l_{i}}\,<\,1$.

If $m^{i}_{j}\,=\,1$ for all $1\,\leq\,j\,\leq\,l_{i}$ and
$1\,\leq\,i\,\leq\,s$, we say that it is a full-flag parabolic data.

A parabolic structure on $E$ over $S$ with parabolic data of rank $r$ as above
consists of the following: for each $p_{i}\,\in\,S$, a weighted filtration

$\displaystyle E_{p_{i}}\,=\,E^{i}_{1}\,\supsetneq\,E^{i}_{2}\,\supsetneq\,$
$\displaystyle\,\cdots\,\supsetneq\,E^{i}_{l_{i}}\,\supsetneq\,E^{i}_{l_{i}+1}\,=\,0$
$\displaystyle 0\,\leq\,\alpha^{i}_{1}\,<\,\alpha^{i}_{2}\,<\,$
$\displaystyle\,\cdots\,<\,\alpha^{i}_{l_{i}}\,<\,\alpha^{i}_{l_{i}+1}\,=\,1$

such that
$m^{i}_{j}\,=\,\dim(E^{i}_{j}/E^{i}_{j+1})\,\,\,\forall\,1\,\leq\,j\,\leq\,l_{i},\,1\,\leq\,i\,\leq\,s$.

The above collection
$\alpha\,:=\,\\{(\alpha^{i}_{1}\,<\,\alpha^{i}_{2}\,<\,\cdots\,<\,\alpha^{i}_{l_{i}})_{1\leq
i\leq s}\\}$ is called the weights, and the above integer $m^{i}_{j}$ is
called the multiplicity of the weight $\alpha^{i}_{j}$.

By a parabolic bundle we mean a collection of data $(E,\,m,\,\alpha)$, where
$E$ is a vector bundle on $X$, while $m$ and $\alpha$ are as described above.
For notational convenience, such a parabolic vector bundle will also be
referred to as $E_{*}$; the vector bundle $E$ is called the underlying bundle.
A full-flag parabolic data is also called a full-flag parabolic structure.

######  Definition 2.2.

Let $E_{*}$ be a parabolic bundle on $X$ of rank $r$. The parabolic degree of
$E_{*}$ is defined as

$\textnormal{pardeg}(E_{*})\,:=\,deg(E)+\sum_{i=1}^{s}\sum_{j=1}^{l_{i}}m^{i}_{j}\alpha^{i}_{j},$

and the parabolic slope of $E_{*}$ is defined as

$\textnormal{par}\mu(E_{*}):=\dfrac{\textnormal{par}deg(E_{*})}{r}.$

The parabolic bundle $E_{*}$ is called parabolic semistable (respectively,
parabolic stable) if for every proper sub-bundle $F\,\subset\,E$ we have

$\textnormal{par}\mu(F_{*})\,\leq\,\textnormal{par}\mu(E_{*})\ \
\,(\text{respectively, }\
\textnormal{par}\mu(F_{*})\,<\,\textnormal{par}\mu(E_{*}),$

where $F_{*}$ denotes the parabolic bundle defined by $F$ equipped with the
induced parabolic structure from $E_{*}$ (see [MS] for the details).

See [Se], [MS] for homomorphisms of parabolic bundles. The class of parabolic
semistable bundles with fixed parabolic slope forms an abelian category. We
refer to [Se, p. 68] for further details. In particular, it makes sense to
take direct sum of parabolic semistable bundles with same parabolic slope.

######  Definition 2.3.

A system of weights $\alpha$ is called generic if every semistable parabolic
bundle $E_{*}$, of given rank and degree, with weights $\alpha$ is parabolic
stable. We refer to [BY] for more details.

###  2.1. Parabolic push-forward and pull-back

Let $X$ and $Y$ be two irreducible smooth projective curves, and let
$\gamma\,:\,Y\,\longrightarrow\,X$ be a finite étale Galois morphism. If $F$
is a vector bundle on $Y$ of rank $n$, then $\gamma_{*}F$ is a vector bundle
on $X$ of rank $mn$, where $m$ is the degree of the map $\gamma$. Given a
parabolic structure on $F$, there is a natural way to construct a parabolic
structure on $\gamma_{*}F$. We refer to [BM, § 3] for details.

Let us mention a special case of parabolic push-forward which will be used
here. Let $p\in X$ be a point. Let $m$ be the degree of $\gamma$. Since
$\gamma$ is unramified, the inverse image $\gamma^{-1}(p)$ consists of $m$
distinct points of $Y$. Suppose we are given a full-flag parabolic data of
rank $n$ with $\gamma^{-1}(p)$ as the set of parabolic points, so that the
parabolic data looks like

$\displaystyle q\,\in\,\gamma^{-1}(p),\,\,F_{q}\,=\,F^{q}_{1}\,$
$\displaystyle\supsetneq\,F^{q}_{2}\,\supsetneq\,\cdots\,\supsetneq\,F^{q}_{n}\,\supsetneq\,F^{q}_{n+1}\,=\,0,$
$\displaystyle\alpha^{q}_{1}\,$
$\displaystyle<\,\alpha^{q}_{2}\,<\,\cdots\cdots<\,\alpha^{q}_{n}\,<\,\alpha^{q}_{n+1}\,=\,1.$

Moreover, we assume that in the collection
$\\{\alpha^{q}_{j}\,\mid\,\,q\,\in\,\gamma^{-1}(p),\,\,1\,\leq\,j\,\leq\,n\\}$
all numbers are distinct. Let $F_{*}$ be a parabolic bundle on $Y$ with this
parabolic data. We shall construct a full-flag parabolic structure for
$E\,=\,\gamma_{*}F$ at the point $p$ from this data. Note that

$E_{p}\,=\,\bigoplus_{q\,\in\,\gamma^{-1}(p)}F_{q}.$

Let $\beta^{p}_{1}\,<\,\beta^{p}_{2}\,<\,\cdots\,<\,\beta^{p}_{mn}$ be the
increasing sequence of length $mn$ obtained by ordering the numbers
$\\{\alpha^{q}_{j}\,\mid\,\,q\,\in\,\gamma^{-1}(p),\,1\,\leq\,j\,\leq\,n\\}$.
For each integer $1\,\leq\,k\,\leq\,mn$, define a filtration of $E_{p}$ by the
subspaces

$E^{p}_{k}\,:=\,\bigoplus_{q\in\gamma^{-1}(p)}F^{q}_{\omega(q,k)},$

where $\omega(q,k)$, for each point $q$, is the smallest integer
$1\,\leq\,j(q)\,\leq\,n$ satisfying the condition
$\beta^{p}_{k}\,\leq\,\alpha^{q}_{j(q)}.$ It is straight-forward to see that
$\dim(E^{p}_{k}/E^{p}_{k+1})\,=\,1\,\ \forall\ 1\,\leq\,k\,\leq\,mn-1$ and
$\dim E^{p}_{mn}\,=\,1$. Consequently,

$\displaystyle E_{p}\,=\,E^{p}_{1}$
$\displaystyle\,\supsetneq\,E^{p}_{2}\,\supsetneq\,\cdots\,\supsetneq\,E^{p}_{mn}$
$\displaystyle\beta^{p}_{1}$
$\displaystyle\,<\,\beta^{p}_{2}\,<\,\cdots\cdots\,<\,\beta^{p}_{mn}$

is a full-flag parabolic structure of rank $mn$ at $p$.

Finally, for multiple parabolic points on $X$, we perform exactly the same
construction for each parabolic point to define the parabolic push-forward.

Let $\gamma:Y\rightarrow X$ be as above. If $E_{*}$ is a parabolic bundle on
$X$, then $\gamma^{*}E$ has an induced parabolic structure as follows: let
$S\subset X$ be the set of parabolic points for $E$; we define
$\gamma^{-1}(S)$ as the set of parabolic points for $\gamma^{*}E$, and for any
$p\in S$ and $q\in\gamma^{-1}(p)$, give the fiber $(\gamma^{*}E)_{q}=E_{p}$
the same parabolic structure as $E_{p}$.

###  2.2. Moduli of parabolic ${\rm PGL}(n,k)$-bundles

######  Definition 2.4.

A parabolic $\textnormal{PGL}(n,k)$-bundle is an equivalence class of
parabolic vector bundles, where two parabolic bundles $E_{*}$ and
$E^{\prime}_{*}$ are considered equivalent if there exists a line bundle $L$
such that $E^{\prime}_{*}\,\simeq\,E_{*}\otimes L$ as parabolic bundles.

Let $\mathcal{M}^{\alpha}_{\xi}(r,d)$ denote the moduli stack of stable
parabolic bundles on $X$ of rank $r$ and fixed determinant $\xi$ of degree
$d$; the corresponding coarse moduli space will be denoted by
$M^{\alpha}_{\xi}(r,d)$. The natural morphism

$\rho\,:\,\,\mathcal{M}^{\alpha}_{\xi}(r,d)\,\longrightarrow\,M^{\alpha}_{\xi}(r,d)$

is a $\mathbb{G}_{m}$-gerbe; this follows from the fact that the automorphisms
of a stable parabolic bundle are nonzero scalar multiplications.

Let

$\Gamma\,\subset\,{\rm Pic}^{0}(X)$

be the group of isomorphism classes of line bundles on $X$ of order $r$; in
other words, $\Gamma$ is the $r$–torsion points of the Jacobian of $X$. This
group $\Gamma$ acts naturally on $PN^{\alpha}_{\xi}(r,d)$ by parabolic tensor
product. More precisely, the action of any $L\,\in\,\Gamma$ sends any
$E_{*}\,\in\,PN^{\alpha}_{\xi}(r,d)$ to the parabolic tensor product

$L\cdot(E_{*})\,:=\,E_{*}\otimes L,$ (2.1)

where $L$ has the trivial parabolic structure.

Let ${\rm Bun}_{\mu_{r}}$ denote the group stack of $\mu_{r}$-bundles on $X$.
The natural map

$h\,:\,{\rm Bun}_{\mu_{r}}\,\longrightarrow\,\Gamma$

is a $\mu_{r}$-gerbe. The tensor product operation defines an action of the
stack $\rm Bun_{\mu_{r}}$ on $\mathcal{M}^{\alpha}_{\xi}(r,d)$. This action is
clearly compatible with the $\Gamma$-action on $M^{\alpha}_{\xi}(r,d)$,
meaning the following diagram commutes:

$\textstyle{\rm
Bun_{\mu_{r}}\times\mathcal{M}^{\alpha}_{\xi}(r,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h\times\rho}$$\textstyle{\mathcal{M}^{\alpha}_{\xi}(r,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\Gamma\times
M^{\alpha}_{\xi}(r,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\alpha}_{\xi}(r,d)}$

Let $i\,\equiv\,d\ (\textnormal{mod r})$ with $i\in\,[0,\,d-1]$. The moduli
stack of parabolic stable $\textnormal{PGL}(r,k)$-bundles on $X$ of
topological type $i$, denoted by $\mathcal{PN}^{\alpha}(r)_{i}$, is defined to
be the quotient of $\mathcal{M}^{\alpha}_{\xi}(r,d)$ by the action of $\rm
Bun_{\mu_{r}}$.
The moduli space of parabolic stable $\textnormal{PGL}(r,k)$-bundles on $X$ of
topological type $i$ is defined similarly as the quotient of
$M^{\alpha}_{\xi}(r,d)$ by $\Gamma$:

$PN^{\alpha}(r)_{i}\,:=\,M^{\alpha}_{\xi}(r,d)/\Gamma\,.$

We have the following commutative diagram

$\textstyle{\mathcal{M}^{\alpha}_{\xi}(r,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{M^{\alpha}_{\xi}(r,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{PN}^{\alpha}(r)_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\rho}}$$\textstyle{PN^{\alpha}(r)_{i}.}$

See [GO, § 2] for more details.

##  3\. The fixed point loci

We fix a positive integer $r$, a subset of parabolic points of $X$ and a line
bundle $\xi$ of degree $d$ on $X$. Assume that $i\,\equiv\,d\,\ (\text{mod\,
r}),\,i\,\in\,[0,\,d-1]$ in what follows. We adopt the following notations:

$\displaystyle\text{M}_{\xi}^{ss}:=$ $\displaystyle\,\,\text{coarse moduli
space of semistable vector bundles of rank}\,\,r\,\,\text{and determinant
iso-}$ $\displaystyle\text{morphic to}\,\,\xi.$
$\displaystyle\operatorname{M_{\xi}}:=$ $\displaystyle\,\,\text{moduli space
of stable vector bundles of rank}\,\,r\,\,\text{and determinant isomorphic
to}\,\,\xi.$ $\displaystyle\text{N}(r)_{i}:=$ $\displaystyle\,\,\text{moduli
space of stable}\,\,\text{PGL}(r,k)\text{--bundles of rank}\,\,r\,\,\text{and
topological type}\,\,i,$ $\displaystyle\operatorname{M_{\alpha}}:=$
$\displaystyle\,\,\text{moduli space of stable parabolic vector bundles of
rank}\,\,r,\,\,\text{fixed determinant}\,\,\xi$ $\displaystyle\,\,\text{with
full flag and weights}\,\,\alpha.$ $\displaystyle\text{PN}_{\alpha,i}:=$
$\displaystyle\,\,\text{moduli space of stable
parabolic}\,\text{PGL}(r,k)\text{--bundles of rank}\,\,r,\,\,\text{topological
type}$ $\displaystyle\,i\,\,\text{with full flag and weights}\,\,\alpha.$
$\displaystyle\mathcal{PN}_{\alpha,i}:=$ $\displaystyle\,\,\text{moduli stack
of stable parabolic}\,\text{PGL}(r,k)\text{--bundles of
rank}\,\,r,\,\,\text{topological type}$ $\displaystyle\,i\,\,\text{with full
flag and weights}\,\,\alpha.$ $\displaystyle\Gamma:=$
$\displaystyle\,\,\\{L\,\in\,\text{Pic}^{0}(X)\,\,\mid\,\,L^{r}\,\simeq\,\mathcal{O}_{X}\\}\simeq(\mathbb{Z}/r\mathbb{Z})^{2g}.$

The aim in this section is to estimate the codimension of the fixed point
locus of $\operatorname{M_{\alpha}}$ under any $r$-torsion line bundle $L$.
This would ensure the existence of a closed subscheme of
$\operatorname{M_{\alpha}}$ (namely, the union of the fixed point locus for
all $L\in\Gamma\setminus\\{\mathcal{O}_{X}\\}$) of codimension at least three
such that $\Gamma$ acts freely on its complement. We remark that $\alpha$ need
not be a generic weight. We first prove a result in linear algebra which will
be used.

######  Lemma 3.1.

Let $\psi$ be a diagonalizable automorphism of a $k$-vector space $V$ of
dimension $r$ equipped with a filtration of subspaces

$V\,=\,V_{r}\,\supsetneq\,V_{r-1}\,\supsetneq\,V_{r-2}\,\supsetneq\,\cdots\,\supsetneq\,V_{1}\,\supsetneq\,0$

such that $\psi(V_{i})\,=\,V_{i}$ for all $1\,\leq\,i\,\leq\,r$. Then there
exists a basis of of $V$ consisting of eigenvectors
$\\{v_{1},\,v_{2},\,\cdots,\,v_{r}\\}$ of $\psi$ such that $V_{j}\,=\,\langle
v_{1},\,\cdots,\,v_{j}\rangle$ for all $1\,\leq\,j\,\leq\,r$.

###### Proof.

Since $\psi$ is diagonalizable, for any subspace $W\,\subset\,V$ such that
$\psi(W)\,=\,W$, the restriction of $\psi$ to $W$ is diagonalizable and,
moreover, since diagonalizable maps are semisimple, there is a subspace
$W^{\prime}\,\subset\,V$ satisfying the conditions
$\psi(W^{\prime})\,=\,W^{\prime}$ and $V\,=\,W\bigoplus W^{\prime}$. Choose
any basis vector $v_{1}$ for $V_{1}$. Suppose $\\{v_{1},v_{2},\cdots,v_{j}\\}$
has been chosen satisfying the hypothesis up to $j\leq r-1$. Thus there exists
a vector $v_{j+1}\in V_{j+1}$ such that $V_{j+1}=\langle
v_{j+1}\rangle\bigoplus V_{j}$. Then $\\{v_{1},\cdots,v_{j+1}\\}$ satisfy the
hypothesis till $(j+1)$. Repeating this process, the lemma follows. ∎

For $L\,\in\,\Gamma\setminus\mathcal{O}_{X}$, let
$\operatorname{M^{L}_{\xi}}\,\subset\,\operatorname{M_{\xi}}$ (respectively,
$\operatorname{M^{L}_{\alpha}}\,\subset\,\operatorname{M_{\alpha}}$) be the
locus of fixed points for the action of $L$ on $\operatorname{M_{\xi}}$
(respectively, $\operatorname{M_{\alpha}}$); see (2.1). If
$m\,=\,\text{ord}(L)$, choosing a nonzero section
$s_{0}\,\in\,H^{0}(X,\,L^{\otimes m})$, define the spectral curve

$Y_{L}\,:=\,\\{v\,\in\,L\,\,\mid\,\,v^{\otimes m}\,\in\,s_{0}(X)\\}\,.$

The natural projection $\gamma\,:\,Y_{L}\,\longrightarrow\,X$ is an étale
Galois covering with Galois group ${\mathbb{Z}}/m{\mathbb{Z}}$. The
isomorphism class of this covering $\gamma$ does not depend on the choice of
the section $s_{0}$. Let $N_{L}\,\subset\,M_{Y_{L}}(r/m,d)$ denote the moduli
space of stable vector bundles $F$ over $Y_{L}$ of rank $r/m$ and degree $d$
such that $\det(\gamma_{*}F)\,\simeq\,\xi$. We recall the following:

######  Lemma 3.2 ([BH10, Lemma 2.1]).

There is a nonempty Zariski open subset $U\,\subset\,N_{L}$ such that
$\gamma_{*}F\,\in\,\operatorname{M^{L}_{\xi}}$ for all $F\,\in\,U$. Moreover,
the morphism $U\,\longrightarrow\,\operatorname{M^{L}_{\xi}}$ defined by
$F\,\longmapsto\,\gamma_{*}F$ is surjective. To be more precise, $U$ is the
subset of $N_{L}$ consisting of those bundles $F$ such that $\gamma_{*}F$ is
also stable.

###  3.1. Estimate of codimension of fixed points

Given a set $\alpha\,=\,\\{\alpha_{1},\,\alpha_{2},\,\cdots,\,\alpha_{r}\\}$
of $r$ distinct elements, let $P(\alpha)$ denote the set of all possible
partitions of its elements into $m$ subsets, each containing $n\,=\,r/m$
elements. Clearly, we have $|P(\alpha)|\,=\,{r\choose n}{r-n\choose
n}{r-2n\choose n}\cdots{n\choose n}.$

Consider the spectral curve $\gamma\,:\,Y_{L}\,\longrightarrow\,X$. Given a
full-flag parabolic data of rank $r$ at the parabolic points $S$ of $X$, we
would like to describe a full-flag parabolic data of rank $n$ on
$\gamma^{-1}(S)$ for each element of $P(\alpha)$. For simplicity, let us start
with the case of a single parabolic point $S\,=\,\\{p\\}$. So, we are given a
full-flag parabolic data of rank $r$ at $p$. Let $\alpha$ denote its set of
weights. Let

$\textnormal{Gal}(\gamma)\,=\,\\{1,\,\mu,\,\mu^{2},\,\cdots,\,\mu^{m-1}\\}\,\subset\,k^{*}.$

The action of $\textnormal{Gal}(\gamma)$ on $\gamma^{-1}(p)$ is via
multiplication. Using this, we can define a full-flag parabolic data at the
points of $\gamma^{-1}(p)$ as follows: let us fix an ordering on the points of
$\gamma^{-1}(p)$, say
$\gamma^{-1}(p)\,=\,\\{q_{1},\,q_{2},\,\cdots,\,q_{m}\\}$, such that $\mu^{i}$
acts on $\gamma^{-1}(p)$ as the cyclic permutation sending $q_{j}$ to
$q_{j+i}$ , where the subscript $(j+i)$ is to be understood mod $m$. For
$t\,\in\,P(\alpha)$, suppose

$\alpha\,=\,\coprod_{j=1}^{m}\Lambda_{j}$

be the partition of the set of weights $\alpha$ according to $t$. Clearly each
$\Lambda_{j}$ can be arranged into an increasing sequence of length $n$. We
designate $\Lambda_{j}$ as the set of weights at $q_{j}$ for each
$1\,\leq\,j\,\leq\,m$. This gives a full-flag parabolic data of rank $n$ at
the points of $\gamma^{-1}(p)$. Finally, for multiple parabolic points, say
$S\,=\,\\{p_{1},\,\cdots,\,p_{s}\\}$ and
$\gamma^{-1}(p_{i})\,=\,\\{q_{1i},\,q_{2i},\,\cdots,q_{mi}\\}$, then we
perform the same procedure as above for each $p_{i}$ to get the parabolic
structure upstairs. In particular, the number of possible parabolic data
upstairs would be $|P(\alpha)|\,=\,({r\choose n}{r-n\choose n}{r-2n\choose
n}\cdots{n\choose n})^{s}.$

For each $t\,\in\,P(\alpha)$, let $M^{t}_{Y_{L}}(n,d)$ denote the moduli space
of stable parabolic bundles over $Y_{L}$ of rank $n$, degree $d$ and having
full-flag parabolic structures at the points of $\gamma^{-1}(p)$ according to
$t$ as described above. Let $\mathcal{N}^{t}_{L}\,\subset\,M_{Y_{L}}^{t}(n,d)$
denote the subvariety consisting of stable parabolic bundles $F_{*}$ such that
$\det(\gamma_{*}F)\,\simeq\,\xi$. Define $\mathcal{N}_{L}\,:=\,\underset{t\in
P(\alpha)}{\coprod}\mathcal{N}^{t}_{L}.$

######  Lemma 3.3.

There is a surjective morphism
$f\,:\,\mathcal{N}_{L}\,\longrightarrow\,\operatorname{M^{L}_{\alpha}}$ given
by parabolic push-forward for the étale map $\gamma$.

###### Proof.

First, for simplicity let us assume that we have only one parabolic point
$S\,=\,\\{p\\}$. The map $f$ in the statement of the send any $F_{*}$ to the
parabolic push-forward of $F_{*}$ constructed in Section 2.1.

We claim that $f(F_{*})$ is parabolic semistable. To prove this, note that if
$E_{*}\,:=\,f(F_{*})$, then

$\gamma^{*}(E_{*})\,\cong\,\bigoplus_{\sigma\in\textnormal{Gal}(\gamma)}\sigma^{*}(F_{*}),$

where $\sigma^{*}(F_{*})$ has the obvious parabolic structure coming from
$F_{*}$. Clearly $\text{par}\mu(\sigma^{*}(F_{*}))\,=\,\text{par}\mu(F_{*})$
for all $\sigma$. Therefore $\gamma^{*}(E_{*})$ is a direct sum of parabolic
stable bundles of same parabolic slope, which implies that $\gamma^{*}(E_{*})$
is parabolic semistable. Thus $E_{*}$ must be parabolic semistable as well,
since any sub-bundle $E^{\prime}_{*}\,\subset\,E_{*}$ with strictly larger
parabolic slope would give rise to a subbundle
$\gamma^{*}(E^{\prime}_{*})\,\subset\,\gamma^{*}(E_{*})$ of strictly larger
parabolic slope, contradicting parabolic semistability of $\gamma^{*}(E_{*})$.
Hence $f(F_{*})$ is parabolic semistable.

It can be shown that $E_{*}$ is actually parabolic stable. To prove this, take
any non-trivial sub-bundle $E^{\prime}\,\subset\,E$ such that
$\textnormal{par}\mu(E^{\prime}_{*})\,=\,\textnormal{par}\mu(E_{*})$. Then

$\gamma^{*}(E^{\prime}_{*})\,\subset\,\gamma^{*}(E_{*})\,=\,\underset{\sigma\in\textnormal{Gal}(\gamma)}{\bigoplus}\sigma^{*}(F_{*})$

is a sub-bundle with same parabolic slope. Now, it is clear that
$\gamma^{*}(E^{\prime}_{*})$ is also parabolic semistable. Thus it contains a
parabolic stable sub-bundle $F^{\prime}_{*}$. We note that the sub-bundles

$\\{\sigma^{*}(F_{*})\,\,\mid\,\,\sigma\in\textnormal{Gal}(\gamma)\\}$

are mutually non-isomorphic. Indeed, otherwise there would exist a parabolic
isomorphism $\widetilde{f}$ from $F_{*}$ to $\sigma^{*}F_{*}$ for some
$\sigma\,\in\,{\rm Gal}(\gamma)\setminus\\{e\\}$; at a parabolic point $y$ it
should preserve the filtrations of $F_{y}$ and $F_{\sigma(y)}$. But since the
weights of $F_{*}$ at the parabolic points $y$ and $\sigma(y)$ are
collectively distinct, and a parabolic isomorphism preserves weights, this
leads to a contradiction. Therefore, it follows that all projections
$F^{\prime}_{*}\,\longrightarrow\,\sigma^{*}(F_{*})$ except one must be zero,
so that $\gamma^{*}(E^{\prime}_{*})$ contains one of the $\sigma^{*}(F_{*})$.
But $\gamma^{*}(E^{\prime}_{*})$ is $\textnormal{Gal}(\gamma)$-equivariant,
which would force that $\gamma^{*}(E^{\prime}_{*})\,=\,\gamma^{*}(E_{*})$;
this clearly implies that $E^{\prime}_{*}\,=\,E_{*}$. Consequently, $E_{*}$ is
parabolic stable.

Next, we argue that
$\textnormal{Im}(f)\,\subseteq\,\operatorname{M^{L}_{\alpha}}$. Let
$E_{*}=f(F_{*})$. There exists a tautological trivialization of $\gamma^{*}L$
over $Y_{L}$, which induces an isomorphism

$\displaystyle\theta:\mathcal{O}_{Y_{L}}\rightarrow\gamma^{*}L\,.$ (3.1)

For each $1\leq i\leq m$, the induced map
$\theta_{q_{i}}:k\rightarrow(\gamma^{*}L)_{q_{i}}=L_{p}$ is given by sending
$\lambda\mapsto\lambda q_{i}.$ This induces an isomorphism

$\displaystyle Id_{F}\otimes\theta:F\xrightarrow{\simeq}F\otimes\gamma^{*}L.$

This, in turn, produces a canonical isomorphism $\psi:E\simeq E\otimes L$ on
the underlying bundle arising from $\gamma_{*}(Id_{F}\otimes\theta)$ followed
by projection formula:

$\psi\,:\,E=\gamma_{*}F\xrightarrow[\simeq]{\gamma_{*}(Id_{F}\otimes\theta)}\gamma_{*}(F\otimes\gamma^{*}L)\cong
E\otimes L.$

Now $E_{p}\,=\,\bigoplus_{i=1}^{m}F_{q_{i}}$ , and the map
$\psi_{p}:E_{p}\rightarrow E_{p}\otimes L_{p}$ on the fiber takes $F_{q_{i}}$
to $F_{q_{i}}\otimes L_{p}$, which clearly implies that $\psi_{p}$ preserves
the filtration induced on $E_{p}$. Thus
$\text{Im}(f)\subseteq\operatorname{M^{L}_{\alpha}}.$

To show the surjectivity of $f$, let
$E_{*}\,\in\,\operatorname{M^{L}_{\alpha}}$, so there exists a parabolic
isomorphism

$\varphi_{*}\,:\,E_{*}\,\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\,E_{*}\otimes
L.$

Suppose $(\alpha^{p}_{1}\,<\,\alpha^{p}_{2}\,<\,\cdots\,<\,\alpha^{p}_{r})$
are the weights at $p$. We would like to show that there exists
$F_{*}\,\in\,\mathcal{N}_{L}$ such that $f(F_{*})\,\simeq\,E_{*}$. Since
$E_{*}$ is parabolic stable, it is simple, and hence any parabolic
endomorphism of $E_{*}$ is a constant scalar multiplication. As a consequence,
any two parabolic isomorphisms from $E_{*}$ to $E_{*}\otimes L$ will differ by
a constant scalar multiplication. Thus, we can re-scale $\varphi_{*}$ by
multiplying with a nonzero scalar, so that the $m$–fold composition

$\underset{m-times}{\varphi_{*}\,\circ\cdots\circ\,\varphi_{*}}\,:\,E_{*}\,\longrightarrow\,E_{*}\otimes
L^{m}$

coincides with $\textnormal{Id}_{E_{*}}\otimes\tau$, where $\tau$ is the
chosen nowhere vanishing section of $L^{m}$. This produces the isomorphism

$\underset{m-times}{\varphi\,\circ\cdots\circ\,\varphi}\,=\,\textnormal{Id}_{E}\otimes\tau\,:\,E\,\longrightarrow\,E\otimes
L^{m}$

on the underlying bundles. Then, the argument given in the proof of [BH10,
Lemma 2.1] will produce a vector bundle $F$ on $Y_{L}$ with
$\gamma_{*}F\,\cong\,E$. Let us briefly recall the argument [BH10] for
convenience. Consider the pull-back $\gamma^{*}\varphi$, and compose it with
the tautological trivialization of $\gamma^{*}L$ to get a morphism

$\phi\,:\,\gamma^{*}E\,\longrightarrow\,\gamma^{*}E.$

Since $Y_{L}$ is irreducible, the characteristic polynomial of $\phi_{y}$
remains unchanged as $y\,\in\,Y_{L}$ moves. This allows us to decompose
$\gamma^{*}E$ into generalized eigenspace sub-bundles. If $F$ is an eigenspace
sub-bundle of $E$, then we have $\gamma_{*}F\,\cong\,E$. Moreover, the
decomposition $\gamma^{*}(E)=\bigoplus_{i=1}^{m}(\mu^{i})^{*}F$ is precisely
the decomposition of $\gamma^{*}(E)$ into generalized eigenspace sub-bundles.

Our next task is to produce a parabolic structure on $F$ so that the parabolic
push-forward $\gamma_{*}(F_{*})$ (in the sense of Remark 2.1) coincides with
$E_{*}$. To see this, recall the description of $\theta$ in (3.1), and notice
that for any choice of $q\in\gamma^{-1}(p)$, the map $\phi_{q}$ is precisely
the composition

$\displaystyle(\gamma^{*}E)_{q}=E_{p}\xrightarrow{\varphi_{p}}E_{p}\otimes
L_{p}=(\gamma^{*}E)_{q}\otimes(\gamma^{*}L)_{q}\xrightarrow{Id\otimes(\theta_{q})^{-1}}(\gamma^{*}E)_{q}\,,\,$
(3.2)

where $\theta_{q}:k\rightarrow(\gamma^{*}L)_{q}=L_{p}$ is given by sending
$\lambda\mapsto\lambda q$. Thus, if

$\displaystyle
E_{p}\,=\,E^{p}_{1}\,\supsetneq\,E^{p}_{2}\,\supsetneq\,\cdots\,\supsetneq\,E^{p}_{r}\,\supsetneq\,0$

be the given parabolic filtration of the fiber $E_{p}$, then as $\varphi_{*}$
is a parabolic isomorphism,

$\displaystyle\forall\,j\in[0,r],\,\,\,\varphi_{p}(E^{p}_{j})\,=\,E^{p}_{j}\otimes
L_{p}\,\implies\,\phi_{q}(E^{p}_{j})\,=\,E^{p}_{j}\,\,\,\,[\text{from}\,(\ref{phi_q})].$
(3.3)

let $\phi^{s}_{q}$ be the semisimple part of $\phi_{q}$ under its Jordan-
Chevalley decomposition. It is well-known that $\phi^{s}_{q}$ can be expressed
as a polynomial in $\phi_{q}$ without constant coefficient. Thus

$\phi^{s}_{q}(E^{p}_{j})=E^{p}_{j}\,\,\,\forall\,j\in[0,r-1].$

Moreover, the generalized eigenspaces of $\phi_{q}$ (namely $F_{q_{i}}$’s) are
the eigenspaces for $\phi^{s}_{q}$. Thus $\phi^{s}_{q}:E_{p}\rightarrow E_{p}$
and the filtration $E_{p}=E^{p}_{1}\supsetneq\cdots\supsetneq
E^{p}_{r}\supsetneq 0$ allow us to apply Lemma 3.1, which gives us a basis of
$E_{p}$ , say $\\{v_{1},\cdots,v_{r}\\}$, such that each $E^{k}_{p}=\langle
v_{k},\cdots,v_{r}\rangle$ and each $v_{j}$ is contained in a unique
$F_{q_{l}}$. From this data, we can produce a full-flag parabolic structure on
the fibers $F_{q_{1}},\cdots,F_{q_{m}}$ of $F$ as follows:

Choose a basis $B\,=\,\\{v_{1},\,v_{2},\,\cdots,\,v_{r}\\}$ of $E_{p}$ as in
Lemma 3.1, and for each $q_{i}$, consider the set $B_{i}\,:=\,B\cap
F_{q_{i}}$. By symmetry, each $B_{i}$ consists of $n$ elements and spans
$F_{q_{i}}$. Suppose that
$B_{i}\,=\,\\{v_{i_{1}},\,v_{i_{2}},\,\cdots,\,v_{i_{n}}\\},$ where
$i_{1}\,<\,i_{2}\,<\,\cdots\,<\,i_{n}.$ Then consider the following weighted
full-flag filtration at $F_{q_{i}}$:

$\displaystyle F_{q_{i}}\,=\,\langle
v_{i_{1}},\,v_{i_{2}},\,\cdots,\,v_{i_{n}}\rangle$
$\displaystyle\supsetneq\langle
v_{i_{2}},\cdots,v_{i_{n}}\rangle\supsetneq\cdots\supsetneq\langle
v_{i_{n}}\rangle\supsetneq 0,$ $\displaystyle\alpha^{p}_{i_{1}}$
$\displaystyle<\alpha^{p}_{i_{2}}<\cdots\cdots\cdots\cdots<\alpha^{p}_{i_{n}}.$

By repeating this for all $1\,\leq\,i\,\leq\,m$, a parabolic bundle $F_{*}$ on
$Y_{L}$ is constructed. Note that $F_{*}$ must be parabolic stable, because if
$F^{\prime}\,\subset\,F$ is any sub-bundle such that
$\text{Par}\mu(F^{\prime}_{*})\,\geq\,\text{Par}\mu(F_{*})$, then the
equalities

$\text{par}deg(\gamma_{*}(F^{\prime}_{*}))\,=\,\text{par}deg(F^{\prime}_{*})\
\ \text{ and }\ \
\text{rank}(\gamma_{*}(F^{\prime}))\,=\,m\cdot\text{rank}(F^{\prime})$

would imply that $\gamma_{*}(F^{\prime})\,\subset\,E$ violates the condition
of parabolic stability for $E_{*}$. Thus $F_{*}\,\in\,\mathcal{N}^{t}_{L}$ for
some $t\in P(\alpha).$ Moreover, the parabolic push-forward of $F_{*}$ under
$\gamma$ as in Remark 2.1 coincides with $E_{*}$.

Finally, if the number of parabolic points on $X$ is more than 1, then an
exactly similar argument with the obvious modifications will give the result.
∎

We note that if $|D|\,=\,s$, then each $\mathcal{N}^{t}_{L}$ is of dimension

$\displaystyle\dim(\mathcal{N}^{t}_{L})$
$\displaystyle=\frac{r^{2}}{m^{2}}(\text{genus}(Y_{L})-1)+1-g+sm\cdot\dfrac{\frac{r}{m}(\frac{r}{m}-1)}{2}$
$\displaystyle=(g-1)(\frac{r^{2}}{m}-1)+\dfrac{sr(\frac{r}{m}-1)}{2}\,\,\,\,\,[\because\text{genus}(Y_{L})=m(g-1)+1].$

Since
$\dim(\operatorname{M_{\alpha}})\,=\,(g-1)(r^{2}-1)+s\cdot\dfrac{r(r-1)}{2}$,
from Lemma 3.3 it follows that

$\dim(\operatorname{M_{\alpha}})-\dim(\operatorname{M^{L}_{\alpha}})\,\geq\,\dim(\operatorname{M_{\alpha}})-\dim(\mathcal{N}_{L})\,=\,r^{2}(\dfrac{m-1}{m})(g-1+\frac{s}{2})\,\geq\,3.$

######  Corollary 3.4.

The codimension of the closed subscheme

$Z_{\alpha}\,=\,\underset{L\in\Gamma\setminus\\{\mathcal{O}_{X}\\}}{\bigcup}\operatorname{M^{L}_{\alpha}}\,\subset\,\operatorname{M_{\alpha}}$

is at least three.

##  4\. The Brauer group of the stack $\mathcal{PN}_{\alpha,i}$

We continue with the notation of Section 2. For a parabolic bundle $E_{*}$,
let $\mathbb{P}(E)_{*}$ denote the corresponding parabolic
$\text{PGL}(r,k)$-bundle. If $E_{*}$ is stable parabolic, then
$\mathbb{P}(E)_{*}$ is a stable parabolic $\text{PGL}(r,k)$-bundle.

######  Lemma 4.1.

Let $U_{\alpha}\,:=\,\operatorname{M_{\alpha}}\setminus Z_{\alpha}$ (see
Corollary 3.4). Take any $E_{*}\,\in\,U_{\alpha}$, and let
$E^{\prime}_{*}\,:=\,E_{*}\otimes L$ for some $L\,\in\,\Gamma$. Then there
exists a unique isomorphism of parabolic ${\rm PGL}(r,k)$-bundles between
$\mathbb{P}(E)_{*}$ and $\mathbb{P}(E^{\prime})_{*}$.

###### Proof.

Since the parabolic structure of $E^{\prime}_{*}$ is induced from $E_{*}$, the
canonical isomorphism between the underlying $\text{PGL}(r,k)$-bundles
$\mathbb{P}(E)$ and $\mathbb{P}(E^{\prime})$ actually gives an isomorphism of
parabolic $\text{PGL}(r,k)$-bundles. If there are two isomorphisms between
$\mathbb{P}(E)_{*}$ and $\mathbb{P}(E^{\prime})_{*}$, then we get an
automorphism of $\mathbb{P}(E)_{*}$.

Let $f_{*}$ be a nontrivial automorphism of $\mathbb{P}(E)_{*}$. It evidently
induces an automorphism $f$ of the underlying projective bundle
$\mathbb{P}(E)$. Then there exists a line bundle $L_{0}$ and an isomorphism of
vector bundles

$\widetilde{f}\,:\,E\,\longrightarrow\,E\otimes L_{0}$

which induces $f$. Taking determinant it follows that $L_{0}\,\in\,\Gamma$.
Moreover, it is clear that $\widetilde{f}$ respects the parabolic structures
on $E$ and $E\otimes L_{0}$, so we get an isomorphism of parabolic bundles

$\widetilde{f}_{*}\,:\,E_{*}\,\longrightarrow\,E_{*}\otimes L_{0}.$

But $E_{*}\,\in\,U_{\alpha}$, thus $L_{0}$ must be trivial. Hence
$\widetilde{f}_{*}$ is an automorphism of $E_{*}$. Since $E_{*}$ is parabolic
stable, it must be a scalar multiple of the identity map. But then the induced
map $f_{*}$ must be the identity morphism. ∎

The following proposition should be well-known to the experts, but we are
providing a proof as we could not find it in the literature, and also for
completeness.

######  Proposition 4.2.

Let $\mathcal{X}$ be a locally noetherian Deligne-Mumford stack, and let
$\mathcal{Z}\hookrightarrow\mathcal{X}$ be a closed substack of codimension at
least 2. Let
$H^{2}_{\text{\'{e}t}}(\mathcal{X}\setminus\mathcal{Z},\mathbb{G}_{m})\simeq
H^{2}_{\text{\'{e}t}}(\mathcal{X},\mathbb{G}_{m}).$

###### Proof.

Consider an étale cover $V\rightarrow\mathcal{X}$. The pull-back of
$\mathcal{Z}$ produces a closed subscheme $Z^{\prime}$ of $V$. We still have
$\operatorname{\textnormal{codim}}_{V}(Z^{\prime})\geq 2$ [GS, Lemma 5.1].

Let $V^{\times i}=V\times V\times\cdots\times V$ ($i+1$ times). Now, consider
the descent spectral sequence

$E_{1}^{i,j}=H^{j}_{\text{\'{e}t}}(V^{\times i},\mathbb{G}_{m})\implies
H^{i+j}_{\text{\'{e}t}}(\mathcal{X},\mathbb{G}_{m}),\,\,\,(i)$

and similarly

$E_{1}^{i,j}=H^{j}_{\text{\'{e}t}}((V\setminus Z^{\prime})^{\times
i}\,\,,\mathbb{G}_{m})\implies
H^{i+j}_{\text{\'{e}t}}(\mathcal{X}\setminus\mathcal{Z}\,\,\,,\mathbb{G}_{m}).\,\,\,(ii)$

By purity statements for étale cohomology for smooth schemes [Ce, Theorem
1.1], we know that the open embedding $V\setminus Z^{\prime}\hookrightarrow V$
induces isomorphisms on the group of units, the Picard group and the
cohomological Brauer group. Thus the $E_{1}^{i,j}$-terms in $(i)$ and $(ii)$
are isomorphic for all $j\leq 2$ and all $i$. This in turn induces
isomorphisms on the convergence terms for all $i+j\leq 2$, and thus we have
$H^{2}_{\text{\'{e}t}}(\mathcal{X}\setminus\mathcal{Z}\,\,,\mathbb{G}_{m})\simeq
H^{2}_{\text{\'{e}t}}(\mathcal{X}\,\,,\mathbb{G}_{m}).$ ∎

Next, consider

$\textstyle{\mathcal{PN}_{\alpha,i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\operatorname{M_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\alpha}}$$\textstyle{\operatorname{\textnormal{PN}_{\alpha,i}}}$

where
$\phi_{\alpha}\,:\,\operatorname{M_{\alpha}}\,\longrightarrow\,\operatorname{\textnormal{PN}_{\alpha,i}}$
denotes the quotient by $\Gamma$, and
$p:\mathcal{PN}_{\alpha,i}\rightarrow\operatorname{\textnormal{PN}_{\alpha,i}}$
is the coarse moduli space map. $\mathcal{PN}_{\alpha,i}$ is a smooth Deligne-
Mumford stack, as the automorphism groups of its points are finite.
Note that $\mathcal{PN}_{\alpha,i}$ is the quotient stack
$[\operatorname{M_{\alpha}}/\Gamma]$, while
$\operatorname{\textnormal{PN}_{\alpha,i}}=\operatorname{M_{\alpha}}/\Gamma$
is the GIT quotient, and thus the map $p$ is natural map

$p:[\operatorname{M_{\alpha}}/\Gamma]\rightarrow\operatorname{M_{\alpha}}/\Gamma.$

Thus, by [St, Tag 075T, Lemma 100.33.7], $p$ is smooth. Since $\phi_{\alpha}$
is a finite morphism, clearly
$\operatorname{\textnormal{codim}}_{\operatorname{\textnormal{PN}_{\alpha,i}}}(\phi_{\alpha}(Z_{\alpha}))\geq
2$ by Corollary 3.4. Since smooth morphisms of Artin stacks are codimension-
preserving [GS, Lemma 5.1], we conclude that the closed substack

$\mathcal{Z}_{\alpha}:=p^{-1}(\phi_{\alpha}(Z_{\alpha}))\hookrightarrow\mathcal{PN}_{\alpha,i}$

also has codimension 2.

######  Corollary 4.3.

Let $PN^{sm}_{\alpha,i}$ denote the smooth locus of
$\operatorname{\textnormal{PN}_{\alpha,i}}$. We have
$\operatorname{\textnormal{Br}}(\mathcal{PN}_{\alpha,i})\simeq\operatorname{\textnormal{Br}}(PN^{sm}_{\alpha,i}).$

###### Proof.

Use Proposition 4.2 for
$\mathcal{Z}_{\alpha}\hookrightarrow\mathcal{PN}_{\alpha,i}$, together with
Lemma 4.1, which says that the map $p$ restricts to an isomorphism over
$\phi_{\alpha}(U_{\alpha})=\operatorname{\textnormal{PN}_{\alpha,i}}\setminus\phi_{\alpha}(Z_{\alpha})$.
∎

##  5\. Description of the Brauer groups for sufficiently small weights

In this section, we allow ourselves to work with parabolic structures which
can be consisting of partial flags as well. Assume that $\alpha$ be a
sufficiently small generic weight corresponding to a parabolic structure such
that

$g.c.d.(r,d,m^{1}_{1},\cdots,m^{1}_{l_{1}},\cdots,m^{i}_{j},\cdots,m^{s}_{1},\cdots,m^{s}_{l_{s}})=1.$

(See Definition 2.1). By [BY, Proposition 3.2], such a generic weight exists.
Since $\alpha$ is chosen sufficiently small, there is a morphism

$\pi\,:\,\operatorname{M_{\alpha}}\,\longrightarrow\,\textnormal{M}^{ss}_{\xi}$
(5.1)

that sends a parabolic bundle to the underlying vector bundle by simply
forgetting the parabolic structure. $\Gamma$ acts on both
$\operatorname{M_{\alpha}}$ and $M^{ss}_{\xi}$, and $\pi$ in (5.1) is
equivariant under this action. Restrict $\pi$ over the stable locus
$\operatorname{M_{\xi}}\subset M^{ss}_{\xi}$. Each fiber of
$\pi:\pi^{-1}(\operatorname{M_{\xi}})\rightarrow\operatorname{M_{\xi}}$ is a
product of flag varieties. Note that $\pi$ is flat (this is sometimes called
as ’miracle flatness theorem’ [St, Tag 00R4, Lemma 10.128.1]). In fact, $\pi$
is an étale locally trivial fibration [PP, Theorem 1.3]. The closed subset

$\mathcal{Z}\,:=\,\underset{L\in\Gamma\setminus\\{\mathcal{O}_{X}\\}}{\bigcup}M^{L}_{\xi}$

has codimension at least three [BH10, Corollary 2.2]. Define

$\mathcal{U}\,:=\,\operatorname{M_{\xi}}\setminus\mathcal{Z}.$

Consider $\pi^{-1}(\mathcal{U})\,\subset\,\pi^{-1}(\operatorname{M_{\xi}})$.
Clearly the action of $\Gamma$ on $\operatorname{M_{\xi}}$ restricts to a free
action on $\mathcal{U}$, and hence the action of $\Gamma$ on
$\pi^{-1}(\mathcal{U})$ is also free; indeed, if $E_{*}$ is an $L$-fixed
point, then the underlying bundle $E$ is also fxed by $L$. Thus, if
$PN^{sm}_{\alpha,i}$ (resp. $N(r)^{sm}_{i}$) be the smooth locus of
$\operatorname{\textnormal{PN}_{\alpha,i}}$ (resp. $N(r)_{i}$), then we have

$\mathcal{U}/\Gamma\subset
N(r)^{sm}_{i}\,\,\text{and}\,\,\pi^{-1}(\mathcal{U})/\Gamma\subset
PN^{sm}_{\alpha,i}.$

A straightforward codimension estimate shows that

$\displaystyle\operatorname{\textnormal{codim}}_{\operatorname{M_{\alpha}}}(\operatorname{M_{\alpha}}\setminus\pi^{-1}(\mathcal{U}))\,=\,\operatorname{\textnormal{codim}}_{\operatorname{M_{\xi}}}(\operatorname{M_{\xi}}\setminus\mathcal{U})\,\geq\,3.$
(5.2)

In view of these codimension estimates, we have

$\displaystyle\operatorname{\textnormal{Br}}(\mathcal{U}/\Gamma)=\operatorname{\textnormal{Br}}(N(r)^{sm}_{i})\,\,\text{and}\,\,\operatorname{\textnormal{Br}}(\pi^{-1}(\mathcal{U})/\Gamma)=\operatorname{\textnormal{Br}}(PN_{\alpha,i}^{sm}).$
(5.3)

This enables us to work with finite étale morphisms in what follows. Consider
the diagram

(5.8)

where $\phi$ and $\phi^{\prime}$ are both finite étale (due to the removal of
all the fixed points), and $\pi,\,\pi^{\prime}$ are both fibrations in the
étale topology. The typical fibers of $\pi$ and $\pi^{\prime}$ are the same,
with both being isomorphic to the product of flag varieties. In fact the
diagram in (5.8) is Cartesian. Denote the common typical fiber of the two
fiber bundles by $F$.

######  Proposition 5.1.

Let $\alpha$ be a sufficiently small weight, and $PN^{sm}_{\alpha,i}$ (resp.
$N(r)^{sm}_{i}$) be the smooth locus of
$\operatorname{\textnormal{PN}_{\alpha,i}}$ (resp. $N(r)_{i}$), where $i\equiv
d(mod\,r)$. Let $\mathcal{U}\subset\operatorname{M_{\xi}}$ be as above. Then
$\operatorname{\textnormal{Br}}(\,\text{PN}^{sm}_{\alpha,i}\,)$ can be
identified with the kernel of the natural surjective map

$\operatorname{\textnormal{Br}}(\,\textnormal{N}(r)^{sm}_{i}\,)=\operatorname{\textnormal{Br}}(\mathcal{U}/\Gamma)\,\longrightarrow\,\operatorname{\textnormal{Br}}(\mathcal{U})=\operatorname{\textnormal{Br}}(\operatorname{M_{\xi}})\,\cong\,\mathbb{Z}/\delta\mathbb{Z},$

where $\delta\,=\,{\rm g.c.d.}(r,\,d).$

###### Proof.

The Leray spectral sequence for the map
$\pi:\pi^{-1}(\mathcal{U})\rightarrow\mathcal{U}$ gives us the exact sequence

$\displaystyle 0\rightarrow
H^{1}(\mathcal{U},\pi_{*}\mathbb{G}_{m})\xrightarrow{\pi^{*}}\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U}))\rightarrow
H^{0}(\mathcal{U},R^{1}\pi_{*}\mathbb{G}_{m})$ (5.9)

We have $\pi_{*}\mathcal{O}_{\pi^{-1}(\mathcal{U})}=\mathcal{O}_{\mathcal{U}}$
since $\pi$ is flat and global sections of the fibers are trivial; this
implies $\pi_{*}\mathbb{G}_{m}=\mathbb{G}_{m}.$ Also, since $\mathcal{U}$ is
simply connected, $R^{1}\pi_{*}\mathbb{G}_{m}$ is the constant sheaf with
stalk $\operatorname{\textnormal{Pic}}(F)$, where $F$ is a typical fiber of
$\pi$ (for details, see [BD, Lemma 3.1].)
Similarly, for
$\pi^{\prime}:\pi^{-1}(\mathcal{U})/\Gamma\rightarrow\mathcal{U}/\Gamma$ we
get:

$\displaystyle 0\rightarrow
H^{1}(\mathcal{U}/\Gamma,\pi^{\prime}_{*}\mathbb{G}_{m})\xrightarrow{\pi^{\prime*}}\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U})/\Gamma)\rightarrow
H^{0}(\mathcal{U}/\Gamma,R^{1}\pi^{\prime}_{*}\mathbb{G}_{m})$ (5.10)

Again for similar reasons as above,
$\pi^{\prime}_{*}\mathbb{G}_{m}=\mathbb{G}_{m}$. We claim that
$R^{1}\pi^{\prime}_{*}\mathbb{G}_{m}$ is the constant sheaf with stalk
$\operatorname{\textnormal{Pic}}(F)$ . This is because, as remarked above,
$R^{1}\pi^{\prime}_{*}\mathbb{G}_{m}$ becomes the constant sheaf
$\underline{\operatorname{\textnormal{Pic}}(F)}_{\mathcal{U}}$ after pull-back
to the étale cover $\mathcal{U}\xrightarrow{\phi}\mathcal{U}/\Gamma$, and the
action of $\Gamma$ on $\operatorname{\textnormal{Pic}}(F)$ is trivial.

Thus the exact sequences of (5.9) and (5.10) become the following,
respectively:

$\displaystyle 0\rightarrow
Pic(\mathcal{U})\xrightarrow{\pi^{*}}\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U}))\rightarrow\operatorname{\textnormal{Pic}}(F)$
(5.11) $\displaystyle
0\rightarrow\operatorname{\textnormal{Pic}}(\mathcal{U}/\Gamma)\xrightarrow{\pi^{\prime*}}\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U})/\Gamma)\rightarrow\operatorname{\textnormal{Pic}}(F)$
(5.12)

Consequently, we get the following diagram:

(5.17)

where $j,\,j^{\prime}$ denote the isomorphism of $F$ with some fiber, and
$g^{\prime}$ is the induced map.

######  Claim 5.1.

The homomorphism $g^{\prime}$ in (5.17) is the identity map.

###### Proof of claim.

First, note that for a point $y\,\in\,\mathcal{U}/\Gamma$, identifying $F$
with $\pi^{\prime-1}(y)$,

$\displaystyle\phi^{\prime-1}(F)=\phi^{\prime-1}(\pi^{\prime-1}(y))=\pi^{-1}(\phi^{-1}(y))$
$\displaystyle\simeq\pi^{-1}(\\{y\\}\times\Gamma)$
$\displaystyle=\pi^{-1}(y)\times\Gamma$ $\displaystyle\simeq F\times\Gamma,$

and hence
$\operatorname{\textnormal{Pic}}(\phi^{\prime-1}(F))\,=\,\bigoplus_{i=1}^{|\Gamma|}\operatorname{\textnormal{Pic}}(F)$.
Therefore, $g^{\prime}$ can be expressed as the composition

$\operatorname{\textnormal{Pic}}(F)\,=\,\operatorname{\textnormal{Pic}}(\pi^{\prime-1}(y))\,\xrightarrow{\,\phi^{\prime*}\,}\,\bigoplus_{i=1}^{|\Gamma|}\operatorname{\textnormal{Pic}}(F)\,\longrightarrow\,\operatorname{\textnormal{Pic}}(F),$

where the last arrow is the projection to any one of the components. It is
easy to see that this is the identity map, which proves the claim. ∎

Note that $\Gamma$ acts trivially on both
$\operatorname{\textnormal{Pic}}(\mathcal{U})$ and
$\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U}))$ and both are torsion-
free, which implies that

$H^{1}(\Gamma,\operatorname{\textnormal{Pic}}(\mathcal{U}))=Hom(\Gamma,\operatorname{\textnormal{Pic}}(\mathcal{U}))=0\,\,\text{and}\,\,H^{1}(\Gamma,\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U})))=Hom(\Gamma,\operatorname{\textnormal{Pic}}(\pi^{-1}(\mathcal{U})))=0,$

since $\Gamma$ is finite. The five-term exact sequence corresponding to the
Hochschild-Serre spectral sequence associated to $\phi$ and $\phi^{\prime}$
[Mi, III Theorem 2.20] then produces following two exact sequences:

(5.20) (5.23)

whose rows and columns are exact. Note that due to the codimension estimates
in (5.2),

$\operatorname{\textnormal{Br}}(\mathcal{U})\cong\operatorname{\textnormal{Br}}(\operatorname{M_{\xi}})\cong\mathbb{Z}/\delta\mathbb{Z},\,\text{and}\,\operatorname{\textnormal{Br}}(\pi^{-1}(\mathcal{U}))\cong\operatorname{\textnormal{Br}}(\operatorname{M_{\alpha}})\,=\,0\,\,\text{\cite[cite]{[\@@bibref{}{BD11}{}{},
Theorem 1.1]}}.$

Let $M\,:=\,\text{coker}(f)$ and $M^{\prime}\,:=\,\text{coker}(f^{\prime})$.
Using the snake lemma in the diagram

we conclude that

$\displaystyle\operatorname{\textnormal{Pic}}(F)\,\simeq\,\dfrac{M^{\prime}}{M}.$
(5.24)

Set $H\,:=\,\text{coker}(\phi^{*})$ and
$H^{\prime}\,:=\,\text{coker}(\phi^{\prime*})$ (see the diagram in (5.8)). The
group $H$ is cyclic of order $\delta$ [BH10, Theorem 1.2]. Using the snake
lemma in the diagram

and from (5.17), it follows that

$\displaystyle
0\,\longrightarrow\,\dfrac{M^{\prime}}{M}\,\longrightarrow\,\operatorname{\textnormal{Pic}}(F)\,\longrightarrow\,\dfrac{H^{\prime}}{H}\,\longrightarrow\,0.$
(5.25)

Composing the first map with (5.24) gives

$\operatorname{\textnormal{Pic}}(F)\simeq\dfrac{M^{\prime}}{M}\rightarrow\operatorname{\textnormal{Pic}}(F)$

which is the identity map in Claim 5.1, and hence

$\dfrac{H^{\prime}}{H}=0\implies H\simeq H^{\prime}.$

From the exact sequences in (5.20) and (5.23) we get the following two exact
sequences:

  1. (i)

$0\,\longrightarrow\,H\,\longrightarrow\,H^{2}(\Gamma,\,k^{*})\,\longrightarrow\,\operatorname{\textnormal{Br}}(\mathcal{U}/\Gamma)\,\longrightarrow\,\operatorname{\textnormal{Br}}(\mathcal{U})\simeq\mathbb{Z}/\delta\mathbb{Z}\,\longrightarrow\,0$,
and

  2. (ii)

$0\,\longrightarrow\,H^{\prime}\,\longrightarrow\,H^{2}(\Gamma,\,k^{*})\,\longrightarrow\,\operatorname{\textnormal{Br}}(\pi^{-1}(\mathcal{U})/\Gamma)\,\longrightarrow\,0.$

Since both $\operatorname{\textnormal{PN}_{\alpha,i}}$ and $N(r)_{i}$ are
normal projective varieties and the open subsets $\mathcal{U}/\Gamma$ and
$\pi^{-1}(\mathcal{U})/\Gamma$ both have complement of codimension 2, it
follows that $\pi^{*}$ induces isomorphism

$k^{*}=H^{0}(\mathcal{U}/\Gamma,\mathbb{G}_{m})\rightarrow
H^{0}(\pi^{-1}(\mathcal{U})/\Gamma,\mathbb{G}_{m})=k^{*},$

and thus it induces isomorphism $H^{2}(\Gamma,k^{*})\rightarrow
H^{2}(\Gamma,k^{*})$, which takes $H$ to $H^{\prime}$. Thus, from the exact
sequences (i) and (ii) we finally conclude that

$\displaystyle\operatorname{\textnormal{Br}}(\mathcal{U}/\Gamma)\,\simeq\,\operatorname{\textnormal{Br}}(\pi^{-1}(\mathcal{U})/\Gamma).$

Applying (5.3) proves our claim. ∎

##  6\. Brauer groups for arbitrary generic weights

We once again work with full-flag parabolic structures. For any such generic
weight $\alpha$, let us denote as usual the smooth locus of
$\operatorname{\textnormal{PN}_{\alpha,i}}$ by $PN^{sm}_{\alpha,i}$. Let
$\alpha$ and $\beta$ be two generic weights separated by a single wall $W$
(cf. [BY, § 2]). In [Th, § 5, page 11] it is shown that there exists a closed
subscheme $Y_{\alpha}\,\hookrightarrow\,\operatorname{M_{\alpha}}$ whose image
is precisely the locus of those parabolic bundles that are $\alpha$-stable but
not $\beta$-stable, and thus $\operatorname{M_{\alpha}}\setminus Y_{\alpha}$
is the open subset of $\operatorname{M_{\alpha}}$ consisting precisely of
those parabolic bundles which are both $\alpha$-stable as well as
$\beta$-stable. Similarly, there exists a closed subscheme
$Y_{\beta}\,\hookrightarrow\,\operatorname{M_{\beta}}$ with exactly similar
properties as $Y_{\alpha}$ when $\alpha$ and $\beta$ are interchanged. Thus
there exists a natural isomorphism

$\displaystyle\operatorname{M_{\alpha}}\setminus
Y_{\alpha}\,\simeq\,\operatorname{M_{\beta}}\setminus Y_{\beta}.$ (6.1)

Moreover, both $Y_{\alpha}$ and $Y_{\beta}$ have codimension at least 2 in
$\operatorname{M_{\alpha}}$ and $\operatorname{M_{\beta}}$ respectively.

From their descriptions it is clear that both
$U_{\alpha}\,=\,\operatorname{M_{\alpha}}\setminus Y_{\alpha}$ and
$U_{\beta}\,=\,\operatorname{M_{\beta}}\setminus Y_{\beta}$ are
$\Gamma$-invariant, and their identification in (6.1) is $\Gamma$-equivariant.
Thus, if we consider the closed subschemes $Z_{\alpha}$ and $Z_{\beta}$ of
$\operatorname{M_{\alpha}}$ and $\operatorname{M_{\beta}}$ respectively as in
Corollary 3.4, then $V_{\alpha}:=U_{\alpha}\setminus Z_{\alpha}$ and
$V_{\beta}:=U_{\beta}\setminus Z_{\beta}$ are $\Gamma$-invariant open subsets
of $\operatorname{M_{\alpha}}$ and $\operatorname{M_{\beta}}$ respectively;
but now $\Gamma$ acts freely on these sets. It follows that the isomorphism in
(6.1) descends to an isomorphism

$\displaystyle V_{\alpha}/\Gamma\,\simeq\,V_{\beta}/\Gamma,$ (6.2)

and moreover, under the quotients by the $\Gamma$-action
$\phi_{\alpha}:\operatorname{M_{\alpha}}\,\longrightarrow\,\operatorname{\textnormal{PN}_{\alpha,i}}$
and
$\phi_{\beta}:\operatorname{M_{\beta}}\,\longrightarrow\,\textnormal{PN}_{\beta,i}$
, we have $V_{\alpha}/\Gamma\subset PN^{sm}_{\alpha,i}$ and
$V_{\beta}/\Gamma\subset PN^{sm}_{\beta,i}$. Of course, since $\phi_{\alpha}$
is a finite morphism, the complement of $V_{\alpha}/\Gamma$ in
$PN^{sm}_{\alpha,i}$ will still be of codimension at least 2; the same is true
if we replace $\alpha$ by $\beta$. Hence we conclude that when $\alpha$ and
$\beta$ are in adjacent chambers separated by a single wall,

$\displaystyle\operatorname{\textnormal{Br}}(PN_{\alpha,i}^{sm})\,\cong\,\operatorname{\textnormal{Br}}(PN^{sm}_{\beta,i}).$
(6.3)

Now, since there are only finitely many walls, we can arrange the collection
of chambers in a sequence, say $C_{1},\,C_{2},\,\cdots,\,C_{N}$, such that
$C_{1}$ contains a sufficiently small weight, and for each $1\,\leq\,j\,<\,N$,
the chambers $C_{j}$ and $C_{j+1}$ are separated by a single wall. Choose
generic weights $\alpha_{j}\,\in\,C_{j}$ for each $j$, such that $\alpha_{1}$
is a sufficiently small weight. By (6.3), we have

$\operatorname{\textnormal{Br}}(\textnormal{PN}^{sm}_{\alpha_{j},i})\,\cong\,\operatorname{\textnormal{Br}}(\textnormal{PN}^{sm}_{\alpha_{j+1},i})\
\ \,\,\forall\ \ 1\,\leq\,j\,\leq\,N.$

Thus we obtain the following:

######  Corollary 6.1.

For any two generic weights $\alpha$ and $\beta$ corresponding to full-flag
type,

$\operatorname{\textnormal{Br}}(\,PN^{sm}_{\alpha,i}\,)\,\cong\,\operatorname{\textnormal{Br}}(\,PN^{sm}_{\beta,i}\,).$

Thus, Poposition 5.1 remains valid for arbitrary generic weights of full-flag
type.

######  Remark 6.2.

In fact, under the condition
$g.c.d.(r,d,m^{1}_{1},\cdots,m^{1}_{l_{1}},\cdots,m^{i}_{j},\cdots,m^{s}_{1},\cdots,m^{s}_{l_{s}})=1$,
we can also describe the Brauer groups for the non-generic weights which lie
on a single wall (that is, those weights which do not lie on the intersection
of two or more hyperplanes). Let $\gamma$ be such a weight lying on a single
wall $W$. Under the hypothesis, we know a generic weight exists [BY,
Proposition 3.2]. It follows that in a small enough neighborhood of $\gamma$,
we can choose two generic weights $\alpha$ and $\beta$ which are separated by
$W$ only, and $\gamma$ is the point of intersection between $W$ and the line
joining $\alpha$ and $\beta$. By [BH95, Lemma 3.6], any $\alpha$-parabolic
stable bundle is $\gamma$-parabolic semistable, which induces a
$\Gamma$-equivariant morphism $\varphi_{\alpha}:M_{\alpha}\rightarrow
M_{\gamma}$ by simply replacing the weights. Note that $M_{\gamma}$ is no
longer smooth. Let $U_{\gamma}\subset M_{\gamma}$ be the open subset
consisting of $\gamma$-parabolic stable bundles. From [BH95, Theorem 3.1] it
follows that $\varphi_{\alpha}$ is an isomorphism between the open subsets
$U_{\alpha}:=\varphi_{\alpha}^{-1}(U_{\gamma})$ and $U_{\gamma}$ , and
moreover, both of them have complements of codimension at least 2. Since
$U_{\gamma}$ is $\Gamma$-invariant, this isomorphism is $\Gamma$-equivariant.
The rest of the argument is similar to above. Namely, if we consider the
closed subschemes $Z_{\alpha}$ and $Z_{\gamma}$ of $M_{\alpha}$ and
$M_{\gamma}$ respectively as in Corollary 3.4, then
$V_{\alpha}:=U_{\alpha}\setminus Z_{\alpha}$ and
$V_{\gamma}:=U_{\gamma}\setminus Z_{\gamma}$ are also $\Gamma$-equivariantly
isomorphic under $\varphi_{\alpha}$, and thus descends to an isomorphism

$V_{\alpha}/\Gamma\simeq V_{\gamma}/\Gamma.$

Since $\Gamma$ acts freely on both $V_{\alpha}$ and $V_{\gamma}$, we have
$V_{\alpha}/\Gamma\subset PN^{sm}_{\alpha,i}$ and its complement will still
have codimension 2. Of course, the same is true for $V_{\gamma}/\Gamma\subset
PN^{sm}_{\gamma,i}$ as well. From this we conclude that

$\operatorname{\textnormal{Br}}(PN^{sm}_{\alpha,i})\simeq\operatorname{\textnormal{Br}}(PN^{sm}_{\gamma,i}).$

Note that $\operatorname{\textnormal{Br}}(PN^{sm}_{\alpha,i})$ is already
known from Proposition 5.1. Of course, the same remains true if $\alpha$ is
replaced by $\beta$.

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