11institutetext: Novartis Pharmaceutical Company, East Hanover, NJ, USA
11email: {yao.chen, aimee.readie, gregory.ligozio,
<EMAIL_ADDRESS>22institutetext: Big Data Institute, University
of Oxford, Oxford, UK
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>

# VertXNet: Automatic Segmentation and Identification of Lumbar and Cervical
Vertebrae from Spinal X-ray Images

Yao Chen ∗ Yuanhan Mo ∗ Aimee Readie Gregory Ligozio
Thibaud Coroller † Bartłomiej W. Papież † 112211111122

###### Abstract

Manual annotation of vertebrae on spinal X-ray imaging is costly and time-
consuming due to bone shape complexity and image quality variations. In this
study, we address this challenge by proposing an ensemble method called
VertXNet, to automatically segment and label vertebrae in X-ray spinal images.
VertXNet combines two state-of-the-art segmentation models, namely U-Net and
Mask R-CNN to improve vertebrae segmentation. A main feature of VertXNet is to
also infer vertebrae labels thanks to its Mask R-CNN component (trained to
detect ’reference’ vertebrae) on a given spinal X-ray image. VertXNet was
evaluated on an in-house dataset of lateral cervical and lumbar X-ray imaging
for ankylosing spondylitis (AS) patients. Our results show that VertXNet can
accurately label spinal X-rays (mean Dice of 0.9). It can be used to
circumvent the lack of annotated vertebrae without requiring human expert
review. This step is crucial to investigate clinical associations by solving
the lack of segmentation, common bottleneck for most computational imaging
projects.

###### Keywords:

deep learning spinal x-rays vertebrae detection segmentation

$\ast$$\ast$footnotetext: Contribute equally$\dagger$$\dagger$footnotetext:
Contribute equally

##  1 Introduction

Reliable vertebrae segmentation and identification from spinal X-ray images is
a prerequisite to quantitatively perform functional analysis of the spine.
However, this step is challenging due to the need for large datasets to
efficiently train machine learning models whilst relying on human annotation
of the images. Using clinical experts to annotate images is time-consuming and
expensive, but also prone to errors (i.e. inter- and intra-reader
variability). Therefore, a solution that could automatically segment and label
images would greatly decrease the cost, time and inter-observer errors caused
by human experts. In this paper, we propose a novel method,VertXNet, for
vertebrae segmentation and identification from spinal X-ray images that was
validated utilizing anonymized internal trial data (MEASURE1) with ankylosing
spondylitis (AS) patients [1].

Figure 1: VertXNet overview: 1) An ensemble method (U-Net and Mask R-CNN)
produces a robust segmentation for each vertebra, 2) Mask R-CNN locates a
reference vertebra (either C2 or S1) and infers remaining vertebrae.

##  2 Method

VertXNet employs two-step pipeline, segmentation and identification of spinal
vertebrae (see Fig. 1). X-ray images are input to a segmentation module to
first generate separable mask of each vertebra. Two state-of-the-art methods,
U-Net [2] and Mask R-CNN [3], have been investigated; however, neither of them
works perfectly individually. U-Net generates connected masks when two
neighbouring vertebrae are too close. Mask R-CNN misses vertebrae if they are
incomplete or have unclear boundaries. A rule-based ensemble method is then
proposed to combine outputs of both methods to produce complete vertebrae
segmentation.

Due to the similarity in shape, it is extremely challenging to correctly label
the vertebrae under multi-class task settings (see Table 1(b)). The spatial
relationship among vertebrae, such as the knowledge that L1 should be followed
by L2, can be used. Thus finding the reference vertebrae becomes essential to
properly label all remaining vertebrae in an X-ray image. Luckily, both lumbar
and cervical images contain two types of vertebrae that can be easily
distinguished from others, and are identified as the reference vertebrae. The
reference vertebrae include: 1) ‘C2’, cone-shaped and the first detectable
vertebra at the top of the cervical X-ray image, and 2) ‘S1’, a triangular-
shaped vertebra, which is commonly the last visible vertebra at the bottom of
a lumbar X-ray. To detect the reference vertebrae, the Mask R-CNN is trained
to distinguish the reference vertebra from other vertebrae. If C2 is detected
on an image, we simply ”zip” down the spine to infer the cervical (C) and
thoracic (T) labels (from C3 to C7 and T1). On the other hand, if ”S1” is
detected we ”zip” up the spinal vertebrae to infer the lumbar (L) and thoracic
(T) labels (from L5 to L1, and T12 to T11).

##  3 Experiments and Results

Data. The method was developed based on the anonymized dataset of a
secukinumab clinical trial for Ankylosing Spondylitis. A sample of images from
the MEASURE 1 CAIN457F2305 study [1] were utilized, in which sagittal cervical
and lumbar X-ray images were acquired at different visit (Baseline, Week 104,
and Week 208). All vertebra in the 512 X-ray images (293 cervical and 219
lumbar) were annotated by an expert radiologist.

Experiments. The 512 annotated X-rays were randomly split and stratified by
spinal acquisition type. 80% of the X-rays images were used for training the
pipeline and the remaining 20% for testing. All models were trained on the
training dataset and the Dice coefficients were calculated on the testing set
to compare all models. Our method’s mean Dice coefficient was 0.90, compared
with Mask R-CNN’s 0.73 and U-Net’s 0.72. Detailed Dice coefficients on each
vertebral type have been provided in Table 1(b). Drop in performances across
models were observed for T1, T11 and T12 due to data imbalance caused by the
lack of appearance of these vertebrae in X-rays. Greater performance from our
model was observed for T1, T11 and T12. In the experiment, the ensemble mask
not only solved the potential risk of overlapping masks or missing vertebral
masks by one model, but also improved the Dice coefficients of the outputs.

Table 1: Model Comparison

Model | C2 | C3 | C4 | C5 | C6 | C7 | T1
---|---|---|---|---|---|---|---
Mask RCNN | 0.852 | 0.882 | 0.889 | 0.842 | 0.767 | 0.736 | 0.325
U-Net | 0.823 | 0.846 | 0.848 | 0.822 | 0.809 | 0.717 | 0.067
VertXNet | 0.905 | 0.912 | 0.912 | 0.906 | 0.901 | 0.881 | 0.518
(a) Dice coefficients of Cervical vertebrae on test set

Model | T11 | T12 | L1 | L2 | L3 | L4 | L5 | S1
---|---|---|---|---|---|---|---|---
Mask RCNN | 0.379 | 0.577 | 0.706 | 0.730 | 0.615 | 0.707 | 0.732 | 0.744
U-Net | 0.338 | 0.519 | 0.643 | 0.726 | 0.785 | 0.772 | 0.726 | 0.603
VertXNet | 0.761 | 0.843 | 0.864 | 0.870 | 0.871 | 0.865 | 0.854 | 0.823
(b) Dice coefficients of Lumbar vertebrae on test set

Conclusion. We successfully demonstrated that our pipeline can automatically
segment and label vertebrae from spinal X-ray images and has outperformed two
benchmark models for the segmentation task according to the Dice coefficients.
The proposed method is the fundamental step to perform the further
quantitative analyses.

## References

  * [1] Baeten, D., et al.: Secukinumab, an interleukin-17a inhibitor, in ankylosing spondylitis. New England journal of medicine 373(26), 2534–2548 (2015)
  * [2] He, K., et al.: Mask R-CNN. In: Proceedings of the IEEE international conference on computer vision. pp. 2961–2969 (2017)
  * [3] Ronneberger, O., et al.: U-net: Convolutional networks for biomedical image segmentation. In: International Conference on Medical image computing and computer-assisted intervention. pp. 234–241. Springer (2015)


#  Eigenperiods and the moduli of points in the line

Haohua Deng  Department of mathematics, Duke University, Durham, NC, 27708
<EMAIL_ADDRESS>and  Patricio Gallardo  Department of mathematics,
University of California at Riverside, Riverside, CA, 92521<EMAIL_ADDRESS>
###### Abstract.

We study the period map of configurations of $n$ points on the projective line
constructed via a cyclic cover branching along these points. By considering
the decomposition of its Hodge structure into eigenspaces, we establish the
codimension of the locus where the eigenperiod map is still pure. Furthermore,
we show that the period map extends to the divisors of a specific moduli space
of weighted stable rational curves, and that this extension satisfies a local
Torelli map along its fibers.

##  1\. Introduction

Understanding the connection between geometric and Hodge-theoretic
compactifications of moduli spaces of varieties is a central quest within
algebraic geometry. In particular, attention is given to the role of these
compactifications in expanding a given period map and to Torelli-type results
for singular varieties. A classical example of such results is due to Deligne
and Mostow in 1986 [DM86]. They showed that certain geometric
compactifications of the moduli space for up to 12 partially labeled points in
$\mathbb{P}^{1}$, constructed via Geometric Invariant Theory, are isomorphic
to a Hodge-theoretic compactification of the image of their period domain -
the Baily-Borel compactification. This result sparked significant interest,
see [looijenga2007uniformization], and has been further extended to
isomorphisms to other geometric and Hodge-theoretic compactifications of such
moduli spaces and their period domains [gallardo2021geometric].

Yet, there is not understanding of the possible Hodge theoretical
compactifications of moduli space $M_{0,n}$ of points in the projectively
line. It remains unknown whether the Deligne-Mostow results can be generalized
to an arbitrary number of points. To understand the difficulties, let us
recall the construction of the period map for $n$ points in $\mathbb{P}^{1}$.
It begins with a cyclic cover $C$ of degree $d$ branching along these points.
The Galois group of this finite cover lifts to both the cohomology of and the
Hodge structure on $C$. So, the action of the Galois group induces an
eigenspace decomposition on $H^{1}(C)$ indexed by an integer $k$. One can
restrict the Hodge structure of $C$ to the $k$-th eigenspace, to construct a
potential eigenperiod map characterized by the data $(d,k)$. Two important
problems arise: for an arbitrary number of points the potential eigenperiod
map is no dominant and the monodromy representation may not even be discrete.
Nowadays, it is understood that most period maps will not be dominant into the
period domain, and work of McMullen describes conditions on $(d,k)$ ensuring
that the associated monodromy representation is discrete, see Theorem 2.10 and
Table 1. Therefore, it is possible to define a family of eigenperiods that
contain and generalize the Deligne-Mostow cases. In our first result, we show
that these maps extend to an open of codimension two.

######  Theorem 1.1.

Let $n\geq 5$ and $k/d$ be as specified in Table 1, and suppose that either
$d|n$ or $\gcd(d,n-1)=1$ holds. There exist an eigenperiod map, see
Proposition 2.13,

$\Phi_{n,d,k}:M_{0,n}\to\Gamma_{n,d,k}\backslash D_{n,d,k}$

such that

  * (i)

$\Phi_{n,d,k}$ extends to a partial compactification
$M_{0,n}^{\mathrm{pr}}\subset\overline{M}_{0,n}$, that is

$\Phi_{n,d,k}:M_{0,n}^{\mathrm{pr}}\to\Gamma_{n,d,k}\backslash D_{n,d,k}$

  * (ii)

there exists a finite-index normal neat subgroup
$\Gamma^{\mathrm{nm}}_{n,d,k}\leq\Gamma_{n,d,k}$, and a normalization of a
finite cover $\overline{M}^{\mathrm{nm}}_{0,n}\to\overline{M}_{0,n}$ such that
the lift of $\Phi_{n,d,k}$ admits a Kato-Usui-Toroidal type extension:

in codimension one. Here, $\Gamma_{N}$ denotes the stabilizer of $N$ in
$\Gamma^{\mathrm{nm}}_{n,d,k}$.

The extension in Theorem 1.1 satisfies the property that, for any point $p\in
M_{0,n}^{\mathrm{pr}}$ and any one-dimensional neighborhood $\Delta$ around
$p$ with $\Delta^{*}\subset M_{0,n}$, the local monodromy around $p$ is
finite. We say that $M_{0,n}^{\mathrm{pr}}$ is the pure locus because the
limiting mixed Hodge structures (LMHS) obtained from the local period maps
over $\Delta$ around $p\in M_{0,n}^{\mathrm{pr}}$ are pure as a direct
consequence of finite local monodromy.

To prove Theorem 1.1.(i), we analyze in Section 3 the eigenspectra of the
degenerations parametrized by the partial compactification
$M_{0,n}^{\mathrm{pr}}$. These perspective extends Steenbrink’s theory on the
spectrum of hypersurface singularities, see [KL20], [castor2022remarks]. The
proof is then completed by stuyding the behavior of the LMHS arising from
degenerations of eigenperiod maps, as detailed in Section 3.2.

Next, we also calculate the codimension of complement of the “pure” loci from
Theorem 1.1 in relevant GIT compactification.

######  Proposition 1.2.

Let $n,k$ and $d$ be as in Theorem 1.1, and let
$\overline{M}^{\mathrm{git}}_{0,\mathbf{w}}$ be the GIT compactification
relative to the weights $\mathbf{w}=(\frac{2}{n},...,\frac{2}{n})$ if $d|n$,
and
$\mathbf{w}=(\frac{1}{n-1}+\epsilon,...,\frac{1}{n-1}+\epsilon,1-(n-1)\epsilon)$
if $\gcd(d,n-1)=1$.

Given the reduction morphism
$\pi:\overline{M}_{0,n}\longrightarrow\overline{M}^{\mathrm{git}}_{0,\mathbf{w}}$,
then it holds that

$\displaystyle\text{cod}\left(\overline{M}^{\mathrm{git}}_{0,\mathbf{w}}\setminus\pi(M^{\mathrm{pr}}_{0,n})\right)=H(n,d,k).$

If $d|n$, then

$\displaystyle
H(n,d,k)=\left\\{\begin{array}[]{rcl}\frac{d}{\mathrm{gcd}(k,d)}-1&\mbox{{if}}&\textrm{gcd}(k,d)>\frac{2d}{n}\\\
n-3&\mbox{{if}}&\textrm{gcd}(k,d)=\frac{2d}{n}\\\
\infty&\mbox{{if}}&\textrm{gcd}(k,d)<\frac{2d}{n}\\\ \end{array}\right.$

On the other hand, $\gcd(d,n)=1$, then

$\displaystyle
H(n,d,k)=\left\\{\begin{array}[]{rcl}\frac{d}{\mathrm{gcd}(k,d)}-1&\mbox{{if}}&\textrm{gcd}(k,d)>\frac{d}{n-1}\\\
n-2&\mbox{{if}}&\textrm{gcd}(k,d)=\frac{d}{n-1}\\\
\infty&\mbox{{if}}&\textrm{gcd}(k,d)<\frac{d}{n-1}\\\ \end{array}\right.$

Note that if the codimension is infinite, it implies that the spaces are the
same.

Next, we describe another extension theory on the eigenperiod maps
$\phi_{n,d,k}$. For even $n$ a partial extension of the period map using Kato-
Usui’s theory [KU08]. Recall that there is a smooth geometric
compactification, commonly known as Hassett’s moduli spaces of weighted stable
curves [hassett2003moduli], with a unique divisor $\mathcal{D}$ in
$\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}/S_{n}$ that generically
parametrizes the following weighted stable curves: The union of two
$\mathbb{P}^{1}$ components, each with $r$ distinct marked points glued at
another point. In an open locus $\mathcal{D}^{\circ}\subset\mathcal{D}$ we
obtain a reduction map

$\displaystyle\mathcal{R}:$ $\displaystyle\mathcal{D}^{\circ}\longrightarrow
M_{0,r}/S_{r}\times M_{0,r}/S_{r}$

by disregarding the gluing points on each $\mathbb{P}^{1}$, see Equation 21.

On the other hand, there are Hodge-theoretical compactifications of the period
domain known as Kato and Usui’s compactifications, which align with the
classical toroidal compactification introduced by [AMRT10] in the classical
case. For our period domains, the construction of Kato-Usui’s space depends on
the selection of an admissible fan $\Sigma$, and it may require a finite cover
extension induced by selecting a neat subgroup
$\Gamma^{\mathrm{nm}}_{n,d,k}\subset\Gamma_{n,d,k}$ to obtain such a
compactification which we denote as $(\Gamma^{\mathrm{nm}}_{n,d,k}\backslash
D_{n,d,k})_{\Sigma}.$

By Theorem 1.1, we have a rational map

$\textstyle{\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}^{nm}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Gamma^{\mathrm{nm}}_{n,d,k}\backslash
D_{n,d,k})_{\Sigma}}$

where $\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}^{nm}$ is a finite cover
of $\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}/S_{n}$. Our next result
extends such rational map to the interior of the divisors $\mathcal{D}$, and
shows that such an extension satisfies a local Torelly-type theorem. We denote
the union of $M_{0,n}$ and all codimension one strata within
$\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}$ as
$\overline{M}^{\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}\subset\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}$.
The selection of a neat subgroup
$\Gamma^{\mathrm{nm}}_{n,d,k}\subset\Gamma_{n,d,k}$ induces a lift
$\mathcal{D}^{\mathrm{nm},\circ}$ and $\mathcal{R}^{\mathrm{nm}}$ for both the
divisor $\mathcal{D}^{\circ}$ and the map $\mathcal{R}$. In our case, $\Sigma$
is a polyhedral fan in $\mathrm{Lie}(U(r,s))$ with rays generated by the
monodromy logarithm $N$ around each divisor $\mathcal{D}$.

######  Theorem 1.3.

Let $(n,d,k)$ be as in Theorem 1.1(i) and with $n$ even. Then, there exists a
finite-index normal neat subgroup
$\Gamma^{\mathrm{nm}}_{n,d,k}\leq\Gamma_{n,d,k}$, and a normalization of a
finite cover
$\overline{M}^{\mathrm{nm},\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}\longrightarrow\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}^{\circ}/S_{n}$
such that:

  1. (1)

The eigenperiod map

$\phi_{n,d,k}:M_{0,n}/S_{n}\longrightarrow\Gamma_{n,d,k}\backslash D_{n,d,k}$

has a unique lift which admits a Kato-Usui type extension:

where $\Gamma_{N}$ denotes the stabilizer of $N$ in
$\Gamma^{\mathrm{nm}}_{n,d,k}$.

  2. (2)

For $k=1$, the restriction of $\phi^{\circ}_{n,d,k}$ to any fiber
$\left(\mathcal{R}^{\mathrm{nm}}\right)^{-1}(x)$ is an immersion. i.e the map
is locally injective. (the associated map between tangent space, which is
induced by the differential, at every smooth point is injective.

Our result includes all the Deligne-Mostow cases [gallardo2021geometric] and
some cases with $n>12$ that are not covered by Deligne-Mostow. The first
statement is a consequence of the techniques employed in
[gallardo2021geometric, Sec. 3.3]. The second part of the theorem follows from
a study of the associated Abel-Jacobi map.

### Acknowledgements

We would like to thank Matt Kerr and Gregory Pearlstein for insightful
discussions and comments. We are grateful for the working environments at the
Department of Mathematics in Washington University in St. Louis, University of
California, Riverside and Duke University where this research was conducted.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Geometric compactifications
    2. 2.2 Eigenperiod map for $n$ points in the line
    3. 2.3 Eigenspectra of isolated hypersurface singularities
  3. 3  Describing the locus with finite monodromy
    1. 3.1 Lemmas on eigenspectra
    2. 3.2 Calculation of limiting mixed Hodge structures
    3. 3.3 Proof of Theorem 1.1 and Proposition 1.2
  4. 4 Hodge-theoretic intepretation of exceptional boundary components
    1. 4.1 Preliminaries and Notation
    2. 4.2 Proof of Theorem 1.3

##  2\. Preliminaries

###  2.1. Geometric compactifications

We describe two families of compactifications for the moduli space $M_{0,n}$
of $n$ labelled points in the projective line: The one constructed via
Geometric Invariant Theory (GIT) and the ones constructed via the Minimal
Model Program. We refer to them as geometric compactifications because every
point of such compactifications parametrizes configurations of points in a
curve. We will start with the GIT compactification. We refer the reader to
[mumford1994geometric] for the general theory.

######  Theorem 2.1.

[mumford1994geometric, Chapter 4.4] Given a weight vector
$\mathbf{w}:=(w_{1},\ldots,w_{n})\in\mathbb{Q}^{n}$ such that $1\geq w_{i}>0$
and $w_{1}+\cdots+w_{n}=2$, there is an irreducible projective variety

$\displaystyle\overline{M}_{0,\mathbf{w}}^{git}:=(\mathbb{P}^{1})^{n}/\\!\\!/_{\mathbf{w}}\operatorname{SL}_{2}$

known as the GIT compactification that satisfies the following properties:

  * •

It has a Zarisky open subset known as the stable locus, that contains
$M_{0,n}$, and parametrizes configurations $(p_{1},\ldots,p_{n})$ of $n$
labelled points in $\mathbb{P}^{1}$ such that $\sum_{p_{i}=p}w_{i}<1$ for all
$p\in\mathbb{P}^{1}$.

  * •

For even $n$ and weights $w_{i}=\frac{2}{n}$ for all $i$, the complement of
the stable locus is a finite collection of points. Each of such points is
associated to a degeneration where $p_{i_{1}}=\cdots=p_{i_{\frac{n}{2}}}$ and
$p_{i_{\frac{n}{2}+1}}=\cdots=p_{i_{n}}$.

######  Remark 2.2.

In general, the complement of the stable locus in the GIT quotient is known as
the GIT boundary. Every point of the stable locus is associated with a unique
$\operatorname{SL}_{2}$-orbit (that is a configuration of points up to
isomorphism.) This is not true for the GIT boundary where several non-
isomorphic orbits maybe associated with the same point in the quotient.

We need the following moduli of partially labeled points to use results from
[mcmullen2013braid, Sec 2]. If $\mathbb{C}^{\langle n\rangle}$ is the space of
monic complex polynomials of degree $n$ with non-vanishing discriminant, then
the moduli space of $n$ distinct non-labelled points in the complex line up to
$\mathrm{Aut}(\mathbb{C})$ is given by

(1) $\displaystyle M_{0,n}^{*}:=\mathbb{C}^{\langle
n\rangle}/\mathrm{Aut}(\mathbb{C})$
$\displaystyle\mathrm{Aut}(\mathbb{C}):=\\{(a,b)\in\mathbb{G}_{m}\rtimes\mathbb{C}\;|\;z\mapsto
az+b\\}.$

A labelled version of this space is known as $T_{1,n}$ and it is was
introduced for arbitrary dimension in [chen2009pointed]. Let
$T_{1,n}^{\circ}\subset T_{1,n}$ be the open parametrizing configuration of
$n$ distinct labelled points. By [chen2009pointed, Prop 3.4.3], we know that
$T_{1,n}^{\circ}\cong M_{0,n+1}$ with the map $\mathbb{C}\to\mathbb{P}^{1}$
induced by $z_{i}\mapsto[1:z_{i}]$ and the $(n+1)$ point being $[0:1]$.
Therefore, by their definition we have

$M_{0,n}^{*}\cong T_{1,n}^{\circ}/S_{n}\cong M_{0,n+1}/S_{n}$

To obtain a GIT compactification, let
$\mathbf{v}=(v_{1},\ldots,v_{n},v_{n+1})\in\mathbb{Q}^{n+1}$ be a vector of
weights such that $1\geq v_{i}>0$ and $v_{1}+\ldots+v_{n+1}=2$. If we suppose
that $v_{1}=\ldots=v_{n}$, then we have the diagram:

The conditions on the weights imply they can be written as

$\displaystyle v_{1}=\cdots=v_{n}=a$ $\displaystyle v_{n+1}=2-na\text{ with
$a>0$ }$

From all possible choices, we highlight the case $v_{n+1}=1-n\epsilon$ and
$v_{1}=...=v_{n}=\frac{1}{n}+\epsilon$ with $\frac{1}{n(n-1)}>\epsilon$. For
the same argument that in [gallardo2017wonderful, Lemma 4.13], we have that
$\overline{M}^{git}_{\mathbf{v}}\cong\mathbb{P}^{n-2}$ and the GIT stable
locus with respect to this choice of weights will parametrize configurations
where any proper subset of the first $n$ points can collide within each other
but never with the point at infinity.

Next, we describe the stable pair compactifications. They were introduced by
Hassett in [hassett2003moduli] for arbitrary genus. However, we will focus in
the genus zero case. Let $n$ be a positive integer and let
$\mathbf{b}\in\mathbb{Q}^{n}$ be a rational weight vector such that $2<\sum
b_{i}$, and $0<b_{i}\leq 1$.

######  Definition 2.3.

[hassett2003moduli] A weighted stable rational curve for the weights
$\mathbf{b}=(b_{1},\ldots,b_{n})$ is a pair
$\left(C,\sum_{i=1}^{n}b_{i}x_{i}\right)$ such that

  * •

$C$ is a nodal connected curve of arithmetic genus zero,

  * •

$x_{i}$ are smooth points of $C$,

  * •

If $x_{i_{1}}=\cdots=x_{i_{k}}$, then $b_{i_{1}}+\cdots b_{i_{k}}\leq 1$

  * •

The divisor $K_{C}+\sum_{i=1}^{n}b_{i}x_{i}$ is ample.

######  Theorem 2.4.

[hassett2003moduli, Theorem 2.1] There exist a finite smooth projective moduli
space $\overline{M}_{0,\mathbf{b}}$ parametrizing weighted stable rational
curves of weight $\mathbf{b}$, and containing $M_{0,n}$ as a Zariski open
locus. For $\mathbf{b}=(1,\ldots,1)$, this space is the simple normal crossing
compactification of $M_{0,n}$ known as the Deligne-Mumford-Knudsen
compactification $\overline{M}_{0,n}$.

######  Proposition 2.5.

[hassett2003moduli] Let $\mathbf{a}=(a_{1},\ldots,a_{n})$ and
$\mathbf{c}=(c_{1},\ldots,c_{n})$ such that $a_{i}\geq c_{i}$ for all $n\geq
i\geq 1$. Then, there exist a birational morphism

$\rho_{\mathbf{a},\mathbf{c}}:\overline{M}_{0,\mathbf{a}}\longrightarrow\overline{M}_{0,\mathbf{c}}$

which is called the reduction morphism. This morphism is an isomorphism in the
locus within $\overline{M}_{0,\mathbf{c}}$ parametrizing stable curves that
are stable with respect to both weights $\mathbf{a}$ and $\mathbf{c}$.

Next, we recall the relationship between the compactifications introduced in
Theorem 2.1 and Theorem 2.5.

######  Proposition 2.6.

[hassett2003moduli, Sec 8] Given
$\mathbf{w}+\epsilon:=(w_{1}+\epsilon,\ldots,w_{n}+\epsilon)$ were
$\epsilon\ll 1$ is a positive rational number and $\mathbf{w}$ as in Theorem
2.1. Then, there is a morphism
$\varphi_{\mathbf{w}}:\overline{M}_{0,\mathbf{w}+\epsilon}\longrightarrow\overline{M}_{0,\mathbf{w}}^{git}$
such that:

  * 1.

If there are subsets of indexes $I\subsetneq\\{1,\cdots,n\\}$ with $\sum_{i\in
I}w_{i}=1$, then $\varphi_{\mathbf{w}}$ is a blow up of the GIT
compactification at the GIT boundary $p_{I,I^{c}}$.

  * 2.

If there is not such subset, then $\varphi_{\mathbf{w}}$ is an isomorphism.

For the partially labeled case described in Equation (1), we should be careful
in using the same weight at the non-labelled points, and keeping the labelled
and non-labelled points separated at the degenerations. That is, we consider
the rational weights $(b_{1},\ldots,b_{n+1})\in\mathbb{Q}^{n+1}$ such that
$b_{1}=\ldots=b_{n}$, $b_{n+1}+b_{i}>1$ $b_{1}+\ldots+b_{n+1}>2$ and $1\geq
b_{1}>0$. We have $M_{0,n}^{*}\cong
M_{0,n+1}/S_{n}\subset\overline{M}_{0,\mathbf{b}}/S_{n}$ with the permutation
group $S_{n}$ acting on the first $n$ labelled points.

###  2.2. Eigenperiod map for $n$ points in the line

Let $\mu:=(p_{1},\ldots,p_{n})$ be a collection of $n$ labelled points in
$\mathbb{P}^{1}$ and let us choose an affine chart such that $p_{i}=[1:x_{i}]$
for all $i$. Given a fixed integer $d\geq 2$, we can associated to our
collection of points $\mu$ a smooth curve $C_{\mu,d}$ of weighted degree $nd$
in $\mathbb{WP}[d:d:n]$ defined by

(2) $\displaystyle
C_{\mu,d}:=\overline{\\{y^{d}-(z_{0}-x_{1}z_{1})...(z_{0}-x_{n}z_{1})=0\\}},$
$\displaystyle\deg(z_{0})=\deg(z_{1})=d,\;\deg(y)=n.$

Geometrically, $C_{\mu,d}$ can be realized as the $d^{th}$-cyclic cover of
$\mathbb{P}^{1}$ branched along $(p_{1},\cdots,p_{n})$. Then, the isomorphism
class of the curve is defined up an action of $\operatorname{SL}_{2}\times
S_{n}$ on the points, and by varying the points $(p_{1},\ldots,p_{n})$ we
obtain a flat family of smooth curves
$\mathcal{C}_{\mu,d}\xrightarrow{\pi}M_{0,n}$. This family defines a polarized
variation of Hodge structure (PVHS) over $M_{0,n}$ such that for a fixed
$\mu_{0}\in M_{0,n}$, we have
$\left(H^{1}(C_{\mu_{0},d}),F^{\bullet}_{\mu_{0}},Q_{\mu_{0}}\right)$.

Let $A_{d}$ be the cyclic group generated by the $d$-th primitive root of
unity $\lambda:=e^{\frac{2\pi i}{d}}$. This group acts on $C_{\mu,d}$ by
$\rho:y\rightarrow\lambda y$ and on $(H^{1}(C_{\mu,d}),Q_{\mu})$ by Hodge
isometries via a linear representation
$\rho^{*}:A_{d}\longrightarrow\operatorname{Aut}(H^{1}(C_{\mu,d}),Q_{\mu})$.
The action of $A_{d}$ induces an eigenspace decomposition

$\displaystyle H^{1}(C_{\mu,d},\mathbb{C})_{\chi}=\\{x\in
H^{1}(C_{\mu,d},\mathbb{C})\;|\;g\cdot x=\chi(g)x,\;\forall
g\in\rho^{*}(A_{d})\\},$
$\displaystyle\chi\in\operatorname{Hom}\left(A_{d},\mathbb{C}\right)$

Since we can identify the character $\chi$ of $A_{d}$ with either
$\lambda^{k}$ or its complex conjugated, then we labeled them by $k$ within
$1\leq k\leq\lfloor\frac{d}{2}\rfloor$. We denote as $H^{1}_{k}(C_{\mu,d})$
and $H^{1}_{\overline{k}}(C_{\mu,d})$ the eigen-spaces with respect to the
eigen-value $\lambda^{k}$ and $\overline{\lambda^{k}}$ respectively. We have
the eigenspace decomposition

$\displaystyle H^{1}(C_{\mu,d},\mathbb{C})=\bigoplus_{1\leq
k\leq\big{\lfloor}\frac{d}{2}\big{\rfloor}}\left(H^{1}_{k}(C_{\mu,d},\mathbb{C})\bigoplus
H^{1}_{\bar{k}}(C_{\mu,d},\mathbb{C})\right).$

By considering the summands $H^{p,q}(C_{\mu,d},\mathbb{C})$ with $(p,q)=(1,0)$
or $(0,1)$ in the Hodge decomposition of $H^{1}(C_{\mu,d},\mathbb{C})$, we
obtain the eigenpieces

$\displaystyle H^{p,q}_{k}(C_{\mu,d},\mathbb{C})$
$\displaystyle:=H^{p,q}(C_{\mu,d},\mathbb{C})\cap
H^{1}_{k}(C_{\mu,d},\mathbb{C})$

with eigenHodge numbers defined as

(3) $\displaystyle\mathbf{h}_{k}=(h^{p,q}_{k}),$ $\displaystyle
h^{p,q}_{k}=\dim H^{p,q}_{k}(C_{\mu,d},\mathbb{C})$

To define our eigenperiod space, we consider the flag

$F^{p}_{k}:=\bigoplus_{s\geq p}H^{s,p}_{k}(C_{\mu,d},\mathbb{C})$

and denote as $\mathbf{f}_{k}=(\dim F_{k}^{p}\;|\;0\leq p)$ its dimension
vector. Now fix a collection of $n$ distinct points $\mu_{0}$ in
$\mathbb{P}^{1}$, the flag variety
$\mathrm{Flag}(\mathbf{f}_{k},H^{1}_{k}(C_{\mu_{0},d},\mathbb{C}))$ contains
as a non-compact locus defined as

(4) $\displaystyle
D_{n,d,k}:=\\{F^{\bullet}_{k}\in\mathrm{Flag}(\mathbf{f}_{k},H^{1}_{k}(C_{\mu_{0},d},\mathbb{C}))\;\big{|}\;F_{k}^{p}\cap\overline{F_{k}^{j-p+1}}=\\{0\\},\;\;(-1)^{\frac{j(j-1)}{2}}Q(Cv,\overline{v})>0\;\\}$

where $C$ acts on $H^{p,q}_{k}(C_{\mu_{0},d},\mathbb{C})$ via $i^{p-q}$. An
entirely analogous definition works for $D_{n,d,\overline{k}}$.

######  Definition 2.7.

For fixed $k$ and $d$ as above, the (universal) eigenperiod map of $n$ points
in the line is the holomorphic map

$\displaystyle\widehat{\phi}_{n,d,k}:\widehat{M}_{0,n}\longrightarrow
D_{n,d,k}$

where $\widehat{M}_{0,n}$ is the universal cover of $M_{0,n}$, $D_{n,d,k}$ is
defined by Equation (4), and the map $\widehat{\phi}_{n,d,k}$ is obtained from
analytic continuation. The values of $k$ and $d$ are known as the cover data
and denoted as $k/d$.

Finally, we need the following explicit basis of
$H^{1}(C_{\mu,d},\mathbb{C})$.

######  Lemma 2.8.

[mcmullen2013braid, Chap. 3] For $n$ and $d$ fixed, the following holomorphic
forms as well as their complex conjugates give a basis of
$H^{1}(C_{\mu,d},\mathbb{C})$:

(5) $\left\\{\frac{z^{j-1}dz}{y^{k}}\ \bigg{|}\
0<\frac{j}{n}<\frac{k}{d}<1,\;j,k\in\mathbb{N}\right\\}$

Moreover, the eigenHodge numbers, defined in Equation 3, are given by

$\displaystyle h^{1,0}_{k}=h^{0,1}_{\bar{k}}=\lceil n(k/d)-1\rceil,$
$\displaystyle h^{0,1}_{k}=h^{1,0}_{\bar{k}}=\lceil n(1-k/d)-1\rceil,$
$\displaystyle 1\leq k\leq\bigg{\lfloor}\frac{d}{2}\bigg{\rfloor}.$

To describe the eigenperiod map, we need to discuss some aspects on the
monodromy representation of the fundamental group of $\mathbb{C}^{\langle
n\rangle}\cong\mathbb{C}^{n}/S_{n}$ which is known as the braid group $B_{n}$.
Let fix $\lambda^{k}$ with $\frac{1}{n}<\frac{k}{d}<1$ where
$\lambda:=e^{\frac{2\pi i}{d}}$ is the $d$-th primitive root of unity, and
let’s consider the $\mathbb{Z}[\lambda^{k}]$\- module defined as
$\Lambda_{n,d,k}:=H^{1}_{k}(C_{\mu,d},\mathbb{Z}[\lambda^{k}]).$ The unitary
automorphisms preserving this module form the group

(6)
$\displaystyle\mathrm{U}(\Lambda_{n,d,k})\subset\mathrm{U}\left(H^{1}_{k}(C_{\mu,d})\right)\cong\mathrm{U}(r,s).$

The fundamental groups of the moduli spaces introduced in the previous
sections are related by the following result.

######  Lemma 2.9.

[mcmullen2013braid, 2.5] Let $d\geq 2$ and $n\geq 5$ be integers such that
$d|n$, then the following diagram commutes:

${B_{n}:=\pi_{1}\left(\mathbb{C}^{n}/S_{n}\right)}$${\mathrm{U}(r,s)}$${\mathrm{Mod}_{0,n}:=\pi_{1}\left(M_{0,n}/S_{n}\right)}$${\mathrm{PU}(r,s)}$

where the map $B_{n}\rightarrow\mathrm{Mod}_{0,n}$ is surjective. On the other
hand, if $gcd(d,n-1)=1$, then the following diagram commutes

$\textstyle{B_{n}:=\pi_{1}\left(\mathbb{C}^{n}/S_{n}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{U}(r,s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}^{*}_{0,n-1}:=\pi_{1}\left(M_{0,n}/S_{n-1}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{PU}(r,s)}$

Therefore, the period map described in Definition 2.7 induces the monodromy
representation
$\pi:B_{n}\longrightarrow\operatorname{Aut}(H^{1}(C_{\mu_{0},d}))$ which
factors through either the mapping class group $\mathrm{Mod}^{*}_{0,n-1}$ or
$\mathrm{Mod}_{0,n}$. By considering the eigenspace $H^{1}_{k}(C_{\mu_{0},d})$
we obtain the monodromy representation

$\pi_{k}:B_{n}\longrightarrow\mathrm{U}(\Lambda_{n,d,k})\subset\mathrm{U}(r,s)$

whose behavior we describe next.

######  Theorem 2.10.

[mcmullen2013braid, Theorem 7.1] The values of $n\geq 5$ and $\lambda^{k}\neq
1$ such that $\mathrm{U}(\Lambda_{n,d,k})$ is a discrete subgroup
$\mathrm{U}\left(H^{1}_{k}(C_{\mu,d})\right)\cong\mathrm{U}(r,s)$ are those
given in Table 1 and their complex conjugates. In these cases, either:

  * •

$\mathrm{U}(\Lambda_{n,d,k})$ is an arithmetic lattice in $\mathrm{U}(r,s)$,
or

  * •

We have $k=d/2$ and
$\mathrm{U}(\Lambda_{n,d,k})\cong\mathrm{Sp}_{2g}(\mathbb{Z})$ is a lattice in
$\mathrm{Sp}_{2g}(\mathbb{R})\subset\mathrm{U}(g,g)$

Moreover, the image $\pi_{k}(B_{n})$ of the braid group is discrete in all
these cases.

The behavior of the monodromy representation described in Lemma 2.9 and
Theorem 2.10 implies that we should consider two cases for defining the period
map. If $d$ divides $n$, then by any $d$-th branched cover of $\mathbb{C}$
branching over $n$ points can be completed to a branched cover of
$\mathbb{P}^{1}$ over the same points, as the infinity is unramified. On the
other hand, if $\gcd(d,n-1)=1$, then a $d$-th branched cover of $\mathbb{C}$
branching along $(n-1)$ points can be completed to a branched cover of
$\mathbb{P}^{1}$ over $n$ points, as the ramification index of infinity is
$d$.

Table 1.  Cases when the image of the braid group $\pi_{q}(B_{n})$ is discrete
in $\mathrm{U}(\Lambda_{n,d,k})$ for $n\geq 5$, see Theorem 2.10. Here, $n$
denotes the number of points. $k/d$ cover data, We only include the cases with
$r,s\geq 1$.

$n$ |  k/d |  (r,s) |  Cases
---|---|---|---
5 |  $1/4$ |  (1,3) |
5 |  $3/10$ |  (1,3) |
5 |  $1/3$ |  (1,3) |
5 |  $3/8$ |  (1,3) |
5 |  $2/5$ |  (1,2) |
5 |  $5/12$ |  (2,2) |
5 |  $1/2$ |  (2,2) |
6 |  $1/4$ |  (1,4) |
6 |  $3/10$ |  (1,4) |
6 |  $1/3$ |  (1,3) |
6 |  $3/8$ |  (2,3) |
6 |  $5/12$ |  (2,3) |
6 |  $1/2$ |  (2,2) |
7 |  $1/6$ |  (1,5) |
7 |  $1/4$ |  (1,5) |
7 |  $3/10$ |  (2,4) |
7 |  $1/3$ |  (2,4) |
7 |  $3/8$ |  (2,4) |
7 |  $5/12$ |  (2,4) |
7 |  $1/2$ |  (3,3) |
8 |  $1/6$ |  (1,6) |
8 |  $1/4$ |  (1,5) |
8 |  $3/10$ |  (2,5) |
8 |  $1/3$ |  (2,5) |
8 |  $3/8$ |  (2,4) |
8 |  $5/12$ |  (3,4) |
8 |  $1/2$ |  (3,3) |

n |  k/d |  (r,s) |  Cases
---|---|---|---
9 |  $1/6$ |  (1,7) |
9 |  $1/4$ |  (2,6) |
9 |  $3/10$ |  (2,6) |
9 |  $1/3$ |  (2,5) |
9 |  $5/12$ |  (3,5) |
9 |  $1/2$ |  (4,4) |
10 |  $1/6$ |  (1,8) |
10 |  $1/4$ |  (2,7) |
10 |  $3/10$ |  (2,6) |
10 |  $1/3$ |  (3,6) |
10 |  $5/12$ |  (4,5) |
10 |  $1/2$ |  (4,4) |
11 |  $1/6$ |  (1,9) |
11 |  $1/4$ |  (2,8) |
11 |  $1/3$ |  (3,7) |
11 |  $5/12$ |  (4,6) |
11 |  $1/2$ |  (5,5) |
12 |  $1/6$ |  (1,9) |
12 |  $1/4$ |  (2,8) |
12 |  $1/3$ |  (3,7) |
12 |  $5/12$ |  (4,6) |
12 |  $1/2$ |  (5,5) |
|  $1/6$ |  $\left(\lceil\frac{n}{6}-1\rceil,\lceil\frac{5n}{6}-1\rceil\right)$ |
$n>12$ |  $1/4$ |  $\left(\lceil\frac{n}{4}-1\rceil,\lceil\frac{3n}{4}-1\rceil\right)$ |
|  $1/3$ |  $\left(\lceil\frac{n}{3}-1\rceil,\lceil\frac{2n}{3}-1\rceil\right)$ |
|  $1/2$ |  $\left(\lceil\frac{n}{2}-1\rceil,\lceil\frac{n}{2}-1\rceil\right)$ |
|  |  |

######  Proposition 2.11.

Given $n,k/d$ as in Table 1, the eigenperiod map defined as

$\displaystyle M_{0,n}/S_{n}\xrightarrow{\phi_{n,d,k}}\Gamma_{n,d,k}\backslash
D_{n,d,k}$ $\displaystyle\Gamma_{n,d,k}:=\pi_{d,k}(\mathrm{Mod}_{0,n})$

if $d|n$, and as

$\displaystyle
M_{0,n}/S_{n-1}\xrightarrow{\phi_{n,d,k}}\Gamma_{n,d,k}\backslash D_{n,d,k}$
$\displaystyle\Gamma_{n,d,k}:=\pi_{d,k}(\mathrm{Mod}^{*}_{0,n-1})$

if $\gcd(d,n-1)=1$, factors through the quotient of a Mumford-Tate subdomain
by $\Gamma_{n,d,k}$. In other words, the eigenperiod map is not Hodge-generic.

###### Proof.

We follow the argument in [gallardo2021geometric, Sec 3.1].
$V_{\mathbb{C}}:=H^{1}_{\chi}\oplus H^{1}_{\bar{\chi}}$.
$G:=\mathrm{Res}_{\mathbb{Q}(q)/\mathbb{Q}}(\mathrm{SL}(V))\cap\mathrm{Sp}(\tilde{V},Q)$
is the Hodge group of the period map which is a rational form of
$\mathrm{U}(r,s)$, therefore, the image of the period map in
$\Gamma\backslash\mathrm{Sp}(V)/\mathrm{U}(r+s)$ factors through the
(generalized) ball quotient
$\Gamma\backslash\mathrm{U}(r,s)/\mathrm{U}(r)\times\mathrm{U}(s)$. The case
$\gcd(d,n-1)=1$ follows by the discussion within [mcmullen2013braid, Sec 2]. ∎

######  Remark 2.12.

The period map from Proposition 2.11 is coming from a $\mathbb{C}$-VHS, and in
occasion is convenient to have a period map define over a number field, for
more details see [dolgachev2007moduli, Sec 7]. For that purpose, we may also
consider the eigenperiod map $\phi_{n,d,k,\overline{k}}$ defined by the Hodge
decomposition within

$H^{1}_{k}(C_{\mu,d})\bigoplus H^{1}_{\bar{k}}(C_{\mu,d}).$

In this case, we obtain a $\mathbb{Q}(\mathrm{cos}(\frac{2\pi}{d}))$ \- VHS of
type $(2d-2,2d-2)$.

By [mcmullen2013braid, (2.5)], we have that
$D_{n,d,k}\cong\text{II}(r,s)\cong\mathrm{Sp}(2r+2s)/\mathrm{U}(r+s)$, and
$\Gamma_{d,n,k}$ is a discrete lattice in $\mathrm{PU}(r,s)$. Moreover, we can
compose the map $\phi_{n,d,k}$ with the natural finite map $M_{0,n}\to
M_{0,n}/S_{m}$ to obtain the following result.

######  Proposition 2.13.

Let $n$, $k$ and $d$ be integers as listed in Table 1. If either
$\gcd(d,n-1)=1$ or $d|n$, we have the following commutative diagram

obtained by composing the period map 2.11 with the relevant finite quotient by
a symmetric group.

######  Question 2.14.

By the work of Deligne-Mostow [DM86], we know that if $n,k,d$ is equal to
either $(8,1,4)$ or $(12,1,6)$. Then, the map $\Phi_{n,d,k}$ is finite and
dominant. What are all the other values of $(n,k,d)$ in Table 1 such that

$\dim\left(\Phi_{n,d,k}\left(M_{0,n}\right)\right)=(n-3).$

###  2.3. Eigenspectra of isolated hypersurface singularities

Let $\mathbf{z}:=(z_{0},\cdots,z_{s})$, we recall that a polynomial
$f(\mathbf{z})$ is called quasi-homogenuous if there exists a weight
$\mathbf{w}:=(w_{0},...,w_{s})\in\mathbb{Q}^{s+1}$ such that

$\displaystyle f(\mathbf{z})=\sum a_{\mathbf{m}}\mathbf{z}^{\mathbf{m}},$
$\displaystyle\mathbf{m}\in\mathbb{Z}_{\geq 0}^{s+1}$

with only finitely many $a_{\mathbf{m}}\neq 0$, and
$\mathbf{w}\cdot\mathbf{m}$ is a constant for all $\mathbf{m}\neq 0$.
Therefore, for any $t\in\Delta$, the graph of $(f(\mathbf{z})-t=0)$ defines a
hypersurface $\mathcal{X}$ in $\mathbb{C}^{s+2}$ which is a family of
algebraic varieties $\mathcal{X}\to\Delta$ via the map
$f:\mathbb{C}^{s+1}\rightarrow\mathbb{C}$, We further assume
$X_{t}:=f^{-1}(t)$ is smooth for any $t\in\Delta^{*}$, and $X_{0}$ has an
isolated singularity given by $f(\mathbf{z})=0$.

######  Definition 2.15.

The Milnor fiber $\mathcal{F}_{f}$ associated to the family
$\mathcal{X}\rightarrow\Delta$ is defined to be $f_{1}^{-1}(t_{0})\cap
B_{\epsilon}(0)$ for $0<\epsilon\ll t_{0}\ll 1$. Its reduced cohomology
$V_{f}:=\tilde{H}^{s}(\mathcal{F}_{f},\mathbb{Q})$ is called the vanishing
cohomology of the degeneration.

Denote $T=T^{\text{ss}}e^{N}$ be the generator of monodromy group around
$0\in\Delta$ which admits natural actions on cohomology groups. To describe
our degeneration Hodge-theoretically, we need some definitions and notations.
The following theorems are well-known and we refer the reader to [KL20] and
[PS08] for details.

######  Theorem 2.16.

Let $\psi_{f}$ be the nearby cycle functor, and $\phi_{f}$ be the vanishing
cycle functor. If we denote

$\displaystyle
H^{k}_{\mathrm{lim}}(X_{t}):=\mathbb{H}^{k}(X_{0},\psi_{f}\mathbb{Q}_{\mathcal{X}})$
$\displaystyle
H^{k}_{\mathrm{van}}(X_{t}):=\mathbb{H}^{k}(X_{0},\phi_{f}\mathbb{Q}_{\mathcal{X}})$

Then, it holds that

  * 1.

The nearby and vanishing cycle functors associate both
$H^{k}_{\mathrm{lim}}(X_{t})$ and $H^{k}_{\mathrm{van}}(X_{t})$ natural mixed
Hodge structures compatible with $T$, in the sense $T^{\mathrm{ss}}$ acts
finitely and $N$ acts as an $(-1,-1)$-morphism.

  * 2.

$H^{k}_{\mathrm{van}}(X_{t})\cong V_{f}$ and $(V_{f})^{p,q}=0$ for $p+q<k$ and
$k+1<p+q$.

######  Theorem 2.17.

There is a sequence known as the Vanishing cycle sequence and given by

(7) $...\rightarrow
H^{k}(X_{0})\xrightarrow{}{}H^{k}_{\mathrm{lim}}(X_{t})\xrightarrow{}V_{f}\xrightarrow{\delta}H^{k+1}(X_{0})\rightarrow
H^{k+1}_{\mathrm{lim}}\rightarrow...$

which is an exact sequence of mixed Hodge structures. Moreover, the monodromy
$T=T^{\text{ss}}e^{N}$ acts on each term and is compatible with all maps in
the sequence. The kernel

(8)
$H^{k}_{\mathrm{ph}}(X_{0}):=\mathrm{Ker}\\{H^{k}(X_{0})\xrightarrow{\mathrm{}}{}H^{k}_{\mathrm{lim}}(X_{t})\\}$

is known as the Phantom cohomology, see [KL20].

######  Remark 2.18.

The phantom cohomology $H^{k}_{\mathrm{ph}}$ describes the global linear
dependence relations of local vanishing cycles.

######  Theorem 2.19.

There is a sequence known as the Clemens-Schmid sequence and given by

(9) $...\rightarrow H_{2s+2-k}(X_{0})\rightarrow
H^{k}(X_{0})\xrightarrow{\mathrm{}}{}H^{k}_{\mathrm{lim}}(X_{t})\xrightarrow{N}H^{k}_{\mathrm{lim}}(X_{t})\xrightarrow{}H_{2s-k}(X_{0})\rightarrow...$

which is an exact sequence of mixed Hodge structures, and the monodromy
$T=T^{\text{ss}}e^{N}$ acts on each term and is compatible with all maps in
the sequence.

To investigate the Mixed Hodge structure on the vanishing cohomology $V_{f}$,
we write its Deligne splitting as $V_{f}=\bigoplus_{p,q}V^{p,q}$. Since the
finite-order automorphisms $T^{\text{ss}}$, $e^{N}$ and $\rho$ are mutually
compatible, we have a finer splitting into their eigenspaces:

(10) $V_{f}=\bigoplus_{p,q,\alpha,\eta}V_{f,\alpha,\eta}^{p,q},$

where $e^{2\pi i\alpha}$ and $e^{2\pi i\eta}$. are eigenvalues correspond to
$T^{\text{ss}}$ and $\rho$, respectively. Notice that we can always choose
$\alpha,\eta\in[0,1)\cap\mathbb{Q}$ because the eigenvalues of $T^{\text{ss}}$
and $\rho$ are roots of unity.

######  Definition 2.20.

The eigenspectra associated to the vanishing cohomology $V_{f}$ and the finite
automorphism $\rho$ is equal to the element in the group algebra
$\mathbb{Z}[\mathbb{Q}\times\mathbb{Q}\times\mathbb{Z}]$ defined as

$\displaystyle{\sigma_{E}(f,\rho)}:=\sum_{p,q,\lambda,\eta}m_{p+\lambda,\eta,p+q}[(p+\lambda,\eta,p+q)],$
$\displaystyle
m_{p+\lambda,\eta,p+q}:=\textrm{dim}(V_{f,\lambda,\eta}^{p,q}).$

This eigenspectra is a natural generalization of Steenbrink’s mixed-spectrum
which only contains information about the monodromy operator.

##  3\.  Describing the locus with finite monodromy

In this section, we proof Theorems 1.1.(i) and Corollary 1.2. We start with
some require results on the eigenspectra of the degenerations.

###  3.1. Lemmas on eigenspectra

Let $\mathbf{z}\in\mathbb{C}^{s+1}$ and $f(\mathbf{z})$ be a quasi-homogenuous
polynomial of weight $\mathbf{w}\in\mathbb{Q}^{s+1}_{>0}$ and degree $1$. By
[Ste77, Thm. 1], the only weights appear in the mixed Hodge structure of
vanishing cohomology are $p+q=s$ or $s+1$. Therefore the eigenspectra given in
Definition 2.20 can be also written as:

$\sigma_{E}(f,\rho):=\sum_{p+q=s,\lambda,\eta}m_{p+\lambda,\eta,s}[(p+\lambda,\eta,s)]+\sum_{p+q=s+1,\lambda,\eta}m_{p+\lambda,\eta,s+1}[(p+\lambda,\eta,s+1)]$

Since, $f(\mathbf{z})$ is quasi-homogeneous. The Jacobian algebra satisfies

$\frac{\mathbb{C}[[\mathbf{z}]]}{\mathcal{J}(f)}:=\frac{\mathbb{C}[[\mathbf{z}]]}{<f,\frac{\partial
f}{\partial z_{0}},...,\frac{\partial f}{\partial
z_{s}}>}\cong\frac{\mathbb{C}[\mathbf{z}]}{\mathcal{J}(f)}$

This is a finite dimensional $\mathbb{C}$-vector space which basis is a set of
monomials that we denote $\operatorname{B}(\mathbf{f})$. Let
$\mathcal{B}=\\{\beta\in\mathbb{N}^{s+1}_{\geq
0}\;|\;\mathbf{z}^{\beta}\in\operatorname{B}(\mathbf{f})\\}$ and set

(11) $l(\beta):=\sum_{0\leq j\leq s}(\beta_{j}+1)w_{j}\in\mathbb{Q}.$

We need the holomorphic $(s+1)$-forms

(12)
$\omega_{\beta}:=\frac{\mathbf{z}^{\beta}d\mathbf{z}}{(f(\mathbf{z})-1)^{\lceil
l(\beta)\rceil}},\;\;\beta\in\mathcal{B}$

defined on $\mathbb{C}^{s+1}\backslash\\{f(\mathbf{z})-1=0\\}$ for our
following isomorphism.

######  Lemma 3.1.

There is an isomorphism:

$H^{s+1}(\mathbb{C}^{s+1}\backslash
Z_{f})\xrightarrow[\cong]{\mathrm{Res}}H^{s}(Z_{f})\cong V_{f}$

and each $\eta_{\beta}:=\mathrm{Res}_{Z_{f}}(\omega_{\beta})$ satisfies:

$\eta_{\beta}\in\bigoplus_{\mu}V^{\lfloor s+1-l(\vec{\beta})\rfloor,\lfloor
l(\vec{\beta})\rfloor}_{f,\\{s+1-l(\vec{\beta})\\},\eta},\beta\in\mathcal{B}$

###### Proof.

The isomorphism is given by the residue map and by [KL20, Chap. 2] with the
later generated by $\mathrm{Res}_{Z_{f}}([\omega_{\beta}])$. ∎

######  Corollary 3.2.

The dimension of $\mathrm{Gr}_{s+1}^{W}(V_{f})$ equals to the number of
$\beta\in\mathcal{B}$ such that $l(\beta)\in\mathbb{Z}$.

###### Proof.

By definition of $V_{f}$ and Lemma 3.1

$\displaystyle\dim\left(\mathrm{Gr}_{s+1}^{W}(V_{f})\right)=\sum_{p,q,p+q=s+1}\dim(V^{p,q}_{f}).$

For any $\beta\in\mathcal{B}$, $\eta_{\beta}\in\mathrm{Gr}_{\lfloor
s+1-l(\vec{\beta})\rfloor+\lfloor l(\vec{\beta})\rfloor}^{W}(V_{f})$, so
$\dim\left(\mathrm{Gr}_{s+1}^{W}(V_{f})\right)$ is exactly the number of
$\beta\in\mathcal{B}$ such that

$s+1=\lfloor s+1-l(\vec{\beta})\rfloor+\lfloor l(\vec{\beta})\rfloor.$

We know that either $\lfloor s+1-l(\vec{\beta})\rfloor+\lfloor
l(\vec{\beta})\rfloor=s$ or $s+1$, and the latter happens if and only if
$l(\beta)\in\mathbb{Z}$. ∎

######  Lemma 3.3.

The eigenspectra associated to the vanishing cohomology of the local
degeneration to the singularity $f(x,y)=y^{d}+x^{l}$ is:

$\displaystyle\sigma_{E}(f,\rho)=\sum_{\frac{a+1}{l}+\frac{k}{d}\in\mathbb{Z}}[(\frac{a+1}{l}+\frac{k}{d},2-\frac{a+1}{l}-\frac{k}{d},2)]+\sum_{\frac{a+1}{l}+\frac{k}{d}\notin\mathbb{Z}}[(\frac{a+1}{l}+\frac{k}{d},2-\frac{a+1}{l}-\frac{k}{d},1)]$

where $0\leq a\leq l-2$ and $0\leq k\leq d-1$.

###### Proof.

The monodromy logarithm and its induced LMHS on the vanishing cohomology can
be computed by eigenspectra via the local degenerating equation
$f(x,y)=y^{d}+x^{l}=0$ which is quasi-homogenuous with weight
$\mathbf{w}=(\frac{1}{l},\frac{1}{d})$. Using Equation (11), we compute
$l(\beta)$ for every $\beta$ in the Jacobian algebra

$\frac{\mathbb{C}[\mathbf{z}]}{\mathcal{J}(f)}=\\{x^{a}y^{r}|\;0\leq a\leq
l-2,0\leq r\leq d-2\\}.$

By using Lemma 3.1, we can use the value of $l(\beta)$ to calculate the
dimension of $V_{f}^{p,q}$ as in the proof of Corollary 3.2. The dimension of
these graded pieces define the eigespectra. ∎

######  Corollary 3.4.

Let $d,k,l$ be fixed positive integers such that $2\leq l$, $1\leq k\leq d-1$,
then we have:

$\displaystyle h^{1,1}_{k}\left(V_{f}\right)$
$\displaystyle=\begin{cases}1&\text{ if }\;\text{ $d$ divides $kl$}\\\
0&\text{ if }\;\text{ otherwise }\end{cases}$

###### Proof.

The $h^{1,1}(V_{f})$ of the vanishing cohomology $V_{f}$ is equal to the
cardinality of the set:

$\\{\beta:=(a,k)\in\mathbb{N}_{+}\;|\;l(\beta)=\frac{a+1}{l}+\frac{k}{d}\in\mathbb{Z}\\}.$

by Corollary 3.2 and the fact that
$\mathrm{Gr}^{W}_{2}(V_{f})=H^{1,1}(V_{f})$. By definition the weighted degree
of the monomials in $\mathcal{B}$ is less than or equal to one, so
$l(\beta)<2$. For a fixed $k$, $h^{1,1}_{k}(V_{f})$ is equal to the
cardinality of the set

$\displaystyle\\{a\in\mathbb{N}_{+}\;|\ l-2\geq a\geq 0\;\;\text{ and
}\;\;\frac{a+1}{l}+\frac{k}{d}=1\\}.$

The equation implies

$a=l-1-\frac{kl}{d},$

so the corresponding $\rho$-eigenperiod satisfies $h_{k}^{1,1}=1$ or $0$,
depends on whether $d|kl$ or not. ∎

###  3.2. Calculation of limiting mixed Hodge structures

We recall our setting. Let
$\left(\mathcal{Y}_{I},\mathcal{P}_{I}\right)\to\Delta$ be an one dimensional
family of stable rational curves which central fiber corresponds to a point in
the boundary divisor $D_{I}\subset\overline{M}_{0,n}$, and which generic point
is contained in $M_{0,n}$. To define such family is equivalent to give a
morphism $f_{\mathcal{Y}}:\Delta\to\overline{M}_{0,n}$ because
$\overline{M}_{0,n}$ is a fine moduli space. Let
$R:\overline{M}_{0,n}\to\overline{M}^{git}_{0,\mathbf{\frac{2}{n}}}$ be the
reduction morphism. We obtain a morphism $R\circ
f_{\mathcal{Y}}:\Delta\to\overline{M}^{git}_{0,\mathbf{\frac{2}{n}}}$ which
has associated an unique family
$\left(\mathcal{Y}^{\prime}_{I},\mathcal{P}^{\prime}_{I}\right)\to\Delta$ of
GIT semistable closed orbits. By Section 2.2, we have an one dimensional
family of smooth curves $\mathcal{C}\to\Delta$ such that all fibers are a
smooth $d$-th cover of $\mathbb{P}^{1}$. The central fiber $\mathcal{C}|_{0}$
has singularities which are locally analytical isomorphic to one defined by
the equation $f(x,y)=y^{d}+x^{l}$.

######  Lemma 3.5.

Let $\mathcal{C}\to\Delta$ be a family of curves as above. Suppose that the
central fiber has a non-empty set $S$ of multiple singularities
$\mathfrak{s}\in S$ and let $V_{f,\mathfrak{s}}$ be the vanishing cohomology
of the singular point $\mathfrak{s}\in S$. Then, the following inequality
holds:

(13) $\mathrm{Max}\\{\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1})|\;\mathfrak{s}\in
S\\}\leq h^{1,1}_{\mathrm{lim}}(C_{t})\leq\sum_{\mathfrak{s}\in
S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1}).$

###### Proof.

By construction the central fiber of $\mathcal{C}\to\Delta$ has singularities
of form $y^{d}+x^{l_{i}}$ for some $l_{i}\geq 2$. They appear as the cover of
a configuration of points where the marked points
$\\{p_{j},...,p_{j+l_{i}-1}\\}$ overlap in $\mathbb{P}^{1}$. Let $V_{f,s}$ be
the vanishing cohomology of the singular point $\mathfrak{s}\in S$.  Can we be
more precise? The vanishing cohomology of the curve $C$? In our case, the
vanishing cycle sequence (2.17) is equal to:

(14) $0\rightarrow
H^{1}(C_{0})\xrightarrow{}{}H^{1}_{\mathrm{lim}}(C_{t})\xrightarrow{}\bigoplus_{\mathfrak{s}\in
S}V_{f,\mathfrak{s}}\xrightarrow{\delta}H^{2}_{\mathrm{ph}}(C_{0})\rightarrow
0.$

To show the first inequality on the left of (13), we use Schmid’s nilpotent
orbit theorem, polarized relations on nilpotent orbits and associated limiting
mixed Hodge structures. Indeed, we consider the following $|S|$-dimensional
family

(15) $\mathcal{\tilde{C}}\rightarrow\Delta^{|S|},\
z=(z_{1},...,z_{|S|})\in\Delta^{|S|}$

with smooth fibers for $z\in(\Delta^{*})^{|S|}$. For $i=1,...,|S|$, the
divisor $\\{z_{i}=0\\}$ is given by collapsing the $l_{i}$ points
$\\{p_{j},...,p_{j+l_{i}-1}\\}$ to a single point. By construction, the
central fiber of such family is given by collapsing $|S|$ tuples of marked
points, and the local monodromy group is isomorphic to $\mathbb{Z}^{|S|}$. By
[KPR19, Sec. 3] we obtain the left hand side of (13) due to the polarized
relation of LMHS’s summarized in [KPR19, Sec. 1]. Note that in our weight-$1$
case, the polarized relation on all possible LMHS’s is just induced by the
values of the $h^{1,1}\in\mathbb{Z}$ and the natural linear relation.

For the inequality on the right of (13), we recall that sequence (14) is a
sequence of MHS, so we count dimensions for the (1,1)-pieces by using
additivity of the dimension with respect to that exact sequence. We obtain

(16) $h^{1,1}(C_{0})+\sum_{\mathfrak{s}\in
S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1})=h^{1,1}_{\mathrm{lim}}(C_{t})+h^{1,1}_{\mathrm{ph}}(C_{0}).$

By Clemens-Schmid in the weight $1$ case, see (2.19), and the fact that
$\dim(C_{0})=1$ implies $H_{3}(C_{0})=0$, we obtain

$0\xrightarrow[]{}H^{1}(C_{0})\xrightarrow{\mathrm{sp}}{}H^{1}_{\mathrm{lim}}(C_{t})\xrightarrow{N}H^{1}_{\mathrm{lim}}(C_{t}).$

The fact that $\mathrm{sp}\left(H^{1}(C_{0})\right)=\mathrm{Ker}(N)$ with
$N:H^{1,1}_{\mathrm{lim}}(C_{t})\to H^{0,0}_{\mathrm{lim}}(C_{t})$ implies the
natural MHS on $H^{1}(C_{0})$ is of Hodge type $(1,0)$, $(0,1)$ and $(0,0)$.
Therefore, $h^{1,1}(C_{0})=0$ and we obtain that

$\displaystyle\sum_{\mathfrak{s}\in
S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1})=h^{1,1}_{\mathrm{lim}}(C_{t})+h^{1,1}_{\mathrm{ph}}(C_{0})$

so our desired inequality in (13) follows. Notice that when the singular curve
$C_{0}$ has a connected normalization,

$H^{2}_{\mathrm{ph}}(C_{0})^{\vee}\cong\mathrm{coker}\\{H_{2}(C_{t})_{\mathrm{lim}}\rightarrow
H_{2}(C_{0})\\}.$

is trivial, hence in this case we have

$\sum_{\mathfrak{s}\in
S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1})=h^{1,1}_{\mathrm{lim}}(C_{t}).$

To conclude this part of the proof we recall that the sequence (14) is an
exact sequence of Mixed Hodge Structures, and it holds whenever we restricted
to any eigenspace of the character $\lambda^{k}$. Therefore the inequality
(13) also holds for every $\lambda^{k}$-eigenspace. ∎

######  Lemma 3.6.

Notation as on Lemma 3.5, $h_{k}^{1,1}(V_{f})=0$ if and only the LMHS on
$H^{1}_{\mathrm{lim}}(C_{t})_{k}$ is pure.

###### Proof.

By Lemma 3.5, we can suppose that the central fiber of the family
$\mathcal{C}\to\Delta$ has an unique singularity which is locally modelled by
$y^{d}+x^{l}=0$. By Theorem 2.16, we have that $V_{f}^{0,0}=0$. We recall that
the vanishing cycle sequence within Theorem 2.17 is compatible with the
eigenspace decomposition, so we can restrict them to a fixed eigenspace. We
obtain

(17)
$0\xrightarrow[]{}H^{1}_{k}(C_{0})\xrightarrow{\mathrm{}}{}H^{1}_{\mathrm{lim}}(C_{t})_{k}\xrightarrow{\mathrm{}}(V_{f})_{k}$

which can be visualized via Hodge-Deligne diagrams as

$0$

The symmetries of a LMHS imply that $H^{1,1}_{\mathrm{lim}}(C_{t})_{k}=0$ if
and only if the LMHS on $H^{1}_{\mathrm{lim}}(C_{t})_{k}$ is pure. Therefore,
the exact sequence (17) implies that $h_{k}^{1,1}(V_{f})=0$ if and only the
LMHS on $H^{1}_{\mathrm{lim}}(C_{t})_{k}$ is pure. ∎

###  3.3. Proof of Theorem 1.1 and Proposition 1.2

###### Proof Theorem 1.1.

We start with the proof of Theorem 1.1(i). Let
$\Phi_{n,d,k}:M_{0,n}\to\Gamma_{n,d,k}\backslash D_{n,d,k}$ be the period map
defined on Proposition 2.13. The complement of $M_{0,n}$ in
$\overline{M}_{0,n}$ is the union of divisors denoted as $D_{I,I^{c}}$ with
$I\subset\\{1,\ldots,n\\}$, $2\leq|I|\leq(n-2)$. For any of such divisors, we
consider its interior

$D_{I,I^{c}}^{*}:=D_{I,I^{c}}\setminus\bigcup_{I,I^{c}\neq
J,J^{c}}D_{J,J^{c}}$

Let $N_{I,I^{c}}$ be the local monodromy of the lifted period map around
$D_{I,I^{c}}$. We label the set of divisors with finite monodromy, that is

$\mathcal{I}^{pr}:=\\{I\;|\;N_{I,I^{c}}\;\text{ has finite order}\\}.$

Since $\overline{M}_{0,n}$ is a smooth and normal crossing compactification of
$M_{0,n}$, it is a classical result by Griffiths, see [Gri70, Theorem 9.5],
that $\Phi_{n,d,k}$ can be extended uniquely and holomorphically to

$M_{0,n}^{pr}:=M_{0,n}\bigcup(\cup_{I\in\mathcal{I}^{pr}}D_{I,I^{c}}^{*}).$

To obtain Theorem 1.1 (ii), we discuss the extension to divisors associated to
degenerations with non pure LMHS. The case when the domain is $D_{n,d,k}$ is a
complex ball is proved in [gallardo2021geometric, Sec. 3.3]. Their arguments
still holds for general signatures $(r,s)$ if we only try to extend the
eigenperiod map to the codimension $1$ boundary strata. We describe the proof
next. Choose a normal neat subgroup
$\Gamma^{\mathrm{nm}}_{n,d,k}\leq\Gamma_{n,d,k}$ of finite index The choice of
a neat subgroup induces a finite cover of $M_{0,n}$ which we normalize and
denote as $\overline{M}^{\mathrm{nm}}_{0,n}$. By construction,
$\overline{M}^{\mathrm{nm}}_{0,n}$ has a singular locus of codimension larger
than or equal to two. Then, by Kato-Usui’s theory [KU08, Thm. B], we have a
unique extension to the partial compactification

$\displaystyle\Phi_{n,d,k}^{\mathrm{nm}}:\overline{M}^{\mathrm{nm}}_{0,n}\longrightarrow\left(\Gamma^{\mathrm{nm}}_{n,d,k}\backslash
D_{n,d,k}\right)\bigsqcup_{N\in\Gamma^{\mathrm{nm}}_{n,d,k}\backslash\Sigma}(\Gamma_{N}\backslash
B(N))$

where the fan $\Sigma$ is by taking the monodromy logarithm around each
divisor and their $\mathrm{Ad}(\Gamma^{\mathrm{nm}}_{n,d,k})$-conjugates. This
fan is clearly $\Gamma^{\mathrm{nm}}_{n,d,k}$-strongly compatible and thus
give rises to a Kato-Usui type extension to codimension $1$ strata. ∎

######  Remark 3.7.

All the Deligne-Mostow cases can be obtained from $n=8$ and $n=12$ which are
known as the ancestral cases [doran2004hurwitz]. For the case $n=12$, we don’t
need to make a finite base change for the Kato-Usui extension. However, for
$n=8$ we need to lift to the ordered case by [HM22].

###### Proof of Proposition 1.2.

Let $\Delta$ be a $1$-dimensional subfamily of
$\overline{M}_{0,\left(\mathbf{\frac{2}{n}}\right)^{n}}^{git}$ centered at $p$
such that $\Delta^{*}\subset M_{0,n}$. The local period map
$\phi_{n,d,k}|_{\Delta^{*}}$ is obtained from a complex local system
$\mathcal{V}\rightarrow\Delta^{*}$. This induces a $1$-dimensional subfamily
of $\mathcal{M}^{n}$ centered at $\pi^{-1}(p)$ given by
$\pi^{*}\mathcal{V}\rightarrow\pi^{-1}\Delta^{*}$. By assumption these two
($\mathbb{C}$)-variation of Hodge structures are isomorphic via $\pi$, which
means the local monodromy of $\pi^{*}\mathcal{V}\rightarrow\pi^{-1}\Delta^{*}$
around $\pi^{-1}(p)$ is also finite.

Therefore, the space
$\pi(M_{0,n}^{\mathrm{pr}})\subset\overline{M}_{0,\left(\mathbf{\frac{2}{n}}\right)^{n}}^{git}$
contains exactly all points
$p\in\overline{M}_{0,\left(\mathbf{\frac{2}{n}}\right)^{n}}^{git}$ such that
for any one-dimensional family $\Delta$ centered at $p$ with
$\Delta^{*}\subset M_{0,n}$, $\mathcal{V}|_{\Delta^{*}}$ has finite monodromy
around the center $p$. ∎

We should separately compute the two functions in Corollary 1.2 separately.

Case 1: $d|n$: For $(n-2)>|I|\geq 3$, the morphism
$\pi:\overline{M}_{0,n}\to\overline{M}_{0,n}^{git}$ contracts all divisors
$D_{I}$, and their image $\pi\left(D_{I}\right)$ is of codimension
$|I|-1=(l-1)$ in $\overline{M}_{0,n}^{git}$ for
$l\leq\lfloor\frac{n-1}{2}\rfloor$. Particularly, if $l=\frac{n}{2}$, this
implies $D_{I}$ is contracted to $0$-dimensional cusps. The LMHS of our family
is pure if and only if $d$ does not divide $kl$ by Corollary 3.4 and above
discussion. To conclude our argument, we need to find the smallest $l$ such
that $d$ divides $kl$. Notice that we also need $l\leq\frac{n}{2}$ for the
sake of GIT-semistability. We have three possibilities.

  1. (1)

When gcd$(k,d)<\frac{2d}{n}$, which means there does not exist an
$1<l<\frac{n}{2}$ such that $d|kl$. In this case
$M_{0,n}^{\mathrm{pr}}=\overline{M_{0,n}^{git}}$ (and we use the convention
dim$(\emptyset)=-\infty$).

  2. (2)

When gcd$(k,d)>\frac{2d}{n}$, there exists
$0<l=\frac{d}{\mathrm{gcd}(k,d)}<\frac{n}{2}$ such that $d|kl$. In this case
codim$(\overline{M_{0,n}^{git}}\backslash
M_{0,n}^{\mathrm{pr}})=\frac{d}{\mathrm{gcd}(k,d)}-1$.

  3. (3)

When $n$ is even and gcd$(k,d)=\frac{2d}{n}$,
$\overline{M_{0,n}^{git}}\backslash M_{0,n}^{\mathrm{pr}}$ is exactly the set
of cusps and has codimension $n-3$.

Case 2: gcd$(d,n)=1$: Using the weight
$(w_{1},...,w_{n},w_{n+1})=(\frac{1}{n}+\epsilon,...,\frac{1}{n}+\epsilon,1-n\epsilon)$
with $0<\epsilon\ll\frac{1}{n^{2}}$, the semi-stable boundary components of
$\overline{M}^{git}_{\mathbf{w}}/S_{n}$ are described by the partition of
$[n]$, namely $S_{1}\sqcup S_{2}\sqcup...\sqcup S_{r}=\\{1,2,...,n\\}$ with
$\mathrm{Max}\\{|S_{1}|,...,|S_{r}|\\}\leq n-1$. The strictly semi-stable
locus is now described by the existence of some $|S_{k}|=n-1$.

More precisely, we have the following analog results:

  1. (1)

When gcd$(k,d)<\frac{d}{n-1}$, which means there does not exist an $1<l\leq
n-1$ such that $d|kl$. In this case
$M_{0,n}^{\mathrm{pr}}=\overline{M_{0,n}^{git}}$ (and we use the convention
dim$(\emptyset)=-\infty$).

  2. (2)

When gcd$(k,d)>\frac{d}{n-1}$, there exists
$0<l=\frac{d}{\mathrm{gcd}(k,d)}<n-1$ such that $d|kl$. In this case
codim$(\overline{M_{0,n}^{git}}\backslash
M_{0,n}^{\mathrm{pr}})=\frac{d}{\mathrm{gcd}(k,d)}-1$.

  3. (3)

When gcd$(k,d)=\frac{d}{n-1}$, $\overline{M_{0,n}^{git}}\backslash
M_{0,n}^{\mathrm{pr}}$ is exactly the set of cusps and has codimension $n-2$.

This completes the proof of Proposition 1.2.

##  4\. Hodge-theoretic intepretation of exceptional boundary components

In this section we focus on the proof of Theorem 1.3, which describes the
Hodge theoretical extension of the period map introduced in the previous
sections.

###  4.1. Preliminaries and Notation

Let $\phi_{n,d,k}$ be the eigenperiod map defined in Proposition 2.11. Let
$\Delta\subset\overline{M}_{0,n}$ be a one-parameter smoothing of a point in
the boundary. That is $0\in\Delta$ is a point in $\overline{M}_{0,n}\setminus
M_{0,n}$ and $\Delta^{*}\subset M_{0,n}$. By the construction on Section 2.2,
we have a one dimensional family of generically smooth curves
$\mathcal{C}_{d}\to\Delta$ that induces a variation of Hodge Structures.
Denote the monodromy of this local VHS as $\langle
T\rangle\leq\Gamma_{n,d,k}$, and $N:=\mathrm{log}(T)$. After a finite base
change we can assume $\langle T\rangle$ is unipotent. The local period map
$\Delta^{*}\to\langle T\rangle\backslash D_{n,d,k}$ extends to a space of
nilpotent orbits or Limiting Mixed Hodge structures (LMHS) associated to $N$,
see [KU08]. More precisely,

(18)
$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
T\rangle\backslash D_{n,d,k}\sqcup\langle T\rangle\backslash
B(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
T\rangle\backslash D_{n,d,k}\sqcup\langle T\rangle\backslash B^{0}(N)}$

where

$\displaystyle B(N)$ $\displaystyle:=e^{\langle
N\rangle_{\mathbb{C}}}\backslash\\{F^{\bullet}\in\check{D}_{n,d,k}\;|\;(N,F^{\bullet})\text{
is a nilpotent orbit }\\},$ $\displaystyle B^{0}(N)$
$\displaystyle:=\\{F(\mathrm{Gr}(W_{N}))^{\bullet}\in\prod\mathrm{Gr}(D^{W}_{n,d,k})|(N,F^{\bullet})\in
B(N)\\}$

with $W_{N}$ being the Jacobison-Morozov filtration on $\mathrm{Lie}(U(r,s))$
associated to $N$, and $D^{W}_{n,d,k}$ being the period domain of
$N$-polarized pure Hodge structures on graded quotients, and

$F^{p}(\mathrm{Gr}(W_{N}))=F^{p}\cap W_{N}(H)_{\mathbb{C}}/F^{p}\cap
W_{N-1}(H)_{\mathbb{C}}$

where $H_{\mathbb{C}}$ is the underlying complex vector of the Hodge
structure.

To interpret such extension, we recall that $B(N)$ parametrizes all
$N_{\mathbb{C}}$-nilpotent orbits, or say all limiting Mixed Hodge structures
polarized by $N$ up to rescaling, with fixed weight filtration $W_{N}$, and
that the natural map $B(N)\twoheadrightarrow B^{0}(N)$ sends a LMHS’s
parametrized by $B(N)$ to their graded quotients.

Moreover, since we are working with polarized Hodge structures of weight one,
the limiting mixed Hodge structure of the central fiber can be described is
either pure or of type I , see Figure 4.1. If $T$ is not finite, $N$ gives a
type I degeneration. In our cases the type of LMHS are uniquely determined by
the weight filtration. In that case, we denote the $\mathbb{Q}$-parabolic
subgroup corresponds to the weight filtration as $P_{W}$, its unipotent
radical $U_{W}$, the complex parabolic corresponds to a fixed base point
$F^{\bullet}\in D$ as $P_{F^{\bullet}}$, and the centralizer of $N$ as $Z(N)$.
By [KP16, chap. 7], the boundary components parametrizing our LMHS are

(19) $\displaystyle B(N)$ $\displaystyle\cong e^{\langle
N\rangle_{\mathbb{C}}}\backslash Z(N)\cap(U_{W}(\mathbb{C})\rtimes
P_{W}(\mathbb{R}))/Z(N)\cap\mathrm{Stab}_{F^{\bullet}}(\mathbb{C})$
$\displaystyle B^{0}(N)$ $\displaystyle\cong B(N)/Z(N)\cap U_{W}(\mathbb{C})$

######  Remark 4.1.

In [KP16], by $B(N)$ they meant a single connected component, or $Z(N)$-orbit
as $N$ may polarize LMHS of different types. In our weight $1$ case, since
$Z(N)$ acts transitively on $B(N)$, therefore the two definitions agree.
Examples where $B(N)$ contains multiple connected components parametrizing
LMHS with different Hodge numbers appear in some higher weights.

$r$$s$$a$$a$$s-a$$r-a$$N$ Figure 1. Hodge-Deligne diagram for boundary type
zero (left), type I (center) and the adjoint Hodge-Deligne diagram for type I
degeneration (right)

###  4.2. Proof of Theorem 1.3

The extension across codimension one strata follows from Kato-Usui’s theorem
as in Theorem 1.1. This is Theorem 1.3 (1).

Next, we describe the geometry of the map $\mathcal{R}$. Let
$[n]=\\{1,2,...,n=2r\\}$ and fix $I\subset[n]$ such that $|I|=r$. Let’s recall
a generic point $p\in\mathcal{D}^{\circ}\subset
M^{\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}$ parametrizes a stable curve
$(X,\sum_{i=1}^{2r}\left(\frac{1}{2}+\epsilon\right)x_{i})$ such that
$X=X_{1}\cup X_{2}\cong\mathbb{P}^{1}\cup\mathbb{P}^{1}$ with $X_{1}\cap
X_{2}=q$. The points $x_{i}\in X_{1}$ for $i\in I$ and $x_{i}\in X_{2}$ for
$i\in[n]\setminus I$. These points are mutually distinct and away from
$X_{1}\cap X_{2}$. There is a natural reduction map:

(20) $\displaystyle\mathcal{R}:$ $\displaystyle\mathcal{D}^{\circ}\rightarrow
M_{0,r}/S_{r}\times M_{0,r}/S_{r},$
$\displaystyle\mathcal{D}^{\circ}\subset\overline{M}_{0,\frac{2}{n}+\epsilon}^{\circ}$

induced by sending the configuration on each component $X_{i}$ to the
corresponding element in $M_{0,r}/S_{r}$, that is

(21)
$\displaystyle\left(X,\sum_{i=1}^{2r}\left(\frac{1}{2}+\epsilon\right)x_{i}\right)\mapsto\left(\mathbb{P}^{1},\sum_{i=1}^{r}x_{i}\right),\left(\mathbb{P}^{1},\sum_{i=r+1}^{2r}x_{i}\right).$

For any $x\in M_{0,r}/S_{r}\times M_{0,r}/S_{r}$, the fiber
$\mathcal{R}^{-1}(x)$ is a $2$-dimensional subfamily given by fixing all
$x_{i}$ and varying the gluing point on each component.

Figure 2. Branched cover $C$ and its normalization

Let $\Delta\subset\overline{M}_{0,\mathbf{\frac{2}{n}}+\epsilon}$ be a one-
parameter family such that $\Delta^{*}\subset M_{0,n}$ and
$0\in\mathcal{D}^{\circ}$. As described in Section 2.2, when $d|r$ we have a
family of curves $\mathcal{C}\to\Delta$ such that $\mathcal{C}|_{t=0}$ is
cyclic $d$-cover of the nodal curve $X$ branched along
$\\{x_{1},...,x_{n}\\}$. Let $C$ be the normalization of $\mathcal{C}|_{t=0}$.
By construction, $\mathcal{C}|_{t=0}$ is the union of two components as
$C_{1}\cup C_{2}$. Both $C_{i}$ are curves of genus $\frac{(d-1)(r-2)}{2}$
with $C_{1}\cap C_{2}=\\{q_{1},...,q_{d}\\}$ as the set of nodes. Notice that
there are $r$ branched points on each component $X_{i}$ that define the map
$C_{i}\mapsto X_{i}$. We identify the (disconnected) curve $C$ with its
normalization $C_{1}\sqcup C_{2}$ and let $q_{j,1}\in C_{1}$ and $q_{j,2}\in
C_{2}$ be the points identified with $q_{i}$ by the normalization map. The
divisor $q_{i}:=q_{1,i}+...+q_{d,i}$ in $C_{i}$ is $\rho$-invariant - recall
$\rho$ is the automorphism of $C$ associated to the finite cover
$\mathcal{C}\mapsto X$, see Figure 2.

By Theorem 1.1.(ii), there exist a finite cover a
$M^{\mathrm{nm},\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}\longrightarrow
M^{\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}$ such that the period map extends
to it. Let $\mathcal{D}^{\mathrm{nm},\circ}$ be a connected component of the
divisor within $M^{\mathrm{nm},\circ}_{0,\mathbf{\frac{2}{n}}+\epsilon}$ that
maps to $\mathcal{D}^{\circ}$. The extended period map $\phi_{n,d,k}^{\circ}$
is defined within $\mathcal{D}^{\mathrm{nm},\circ}$, so we describe it next.

By using the Clemens-Schmid and normalization sequences together, see [Car80,
Sec 3.b], we obtain the exact diagram of mixed Hodge structures given by
Figure 3.

${0}$${H_{1}(C_{1})_{k}\bigoplus
H_{1}(C_{2})_{k}}$${0}$${H^{1}(C)_{k}}$${H^{1}_{\mathrm{lim}}(C_{t})_{k}}$${H^{1}_{\mathrm{lim}}(C_{t})_{k}}$${H_{1}(C)_{k}}$${0}$${H^{1,1}_{\mathrm{lim}}(C_{t})_{k}}$${0}$$\scriptstyle{N}$
Figure 3. Exact diagram of LMHS associated to $C$

The diagram in Figure 3 implies that

(22) $H_{1}(C_{1})_{k}\bigoplus
H_{1}(C_{2})_{k}\cong\mathrm{Gr}_{1}H^{1}_{\mathrm{lim}}(C_{t})$

and since $\mathcal{C}|_{t=0}$ is nodal, $H^{1,1}_{\mathrm{lim}}(C_{t})_{k}$
is described by the Abel-Jacobi map of certain non-closed paths in its
normalization $C$ mapped to cycles in $\mathcal{C}|_{t=0}$ via the
identification of $q_{j,1}$ and $q_{j,2}$.

Since our result is a local statement which is irrelevant to finite base
change, we can directly work on the original map $\mathcal{R}$ locally -
instead of the lifted map $\mathcal{R}^{\mathrm{nm}}$. Denote $N_{0}$ as the
local monodromy logarithm corresponds to the KSBA exceptional divisor
$\mathcal{D}^{\circ}$. It suffices to check a local $2$-parameter family:

(23) $\Phi:\Delta\times\Delta\rightarrow B(N_{0})$

for $\mathbf{N}:=(N_{1},N_{2})\in\Delta\times\Delta$. Since moving the
intersection points $C_{1}\cap C_{2}$ (i.e., the pinched nodes) does not
change the normalized curve, we have:

(24) $\Phi^{0}:\Delta\times\Delta\rightarrow B^{0}(N_{0})$

is constant, and the period map only varies among the extension data with
fixed graded quotient, i.e a fiber of the natural map.

(25) $(W_{N_{0}}=:W_{0},F^{\bullet},N_{0})\in
B(N_{0})\twoheadrightarrow(\mathrm{Gr}(W_{0}),F^{\bullet},N_{0})\in
B_{0}(N_{0}).$

Take $F_{0}^{\bullet}$ such that $(N_{0},F_{0}^{\bullet})=\Phi(0,0)$. It then
follows that $\mathrm{Img}(\Phi)\subset\pi^{-1}(\Phi^{0}(0,0))$ where
$\pi:B(N)\twoheadrightarrow B^{0}(N)$, or more precisely, combine with formula
19, we have:

(26) $\mathrm{Img}(\Phi)\subset e^{\langle
N_{0}\rangle_{\mathbb{C}}}\backslash Z(N_{0})\cap
U_{W_{0}}(\mathbb{C})/Z(N_{0})\cap P_{F_{0}^{\bullet}}(\mathbb{C})$

This implies the map $\Phi$ is totally determined by its image in $I^{-1,0}$,
where
$\mathrm{Lie(U}(r,s))=\oplus_{p,q}I^{p,q}(=I^{p,q}_{W_{0},F_{0}^{\bullet}})$
is the Deligne splitting of the adjoint diagram 4.1 (recall that $I^{-1,-1}$
is modded out by $N_{0}$. $I^{-1,1}$ lies in the Levi subgroup and is
irrelevant along any $\pi$-fiber.)

Working on the dual homology sequence, the previous discussion shows the
extension class is determined by the image under Abel-Jacobi map of some
element in $H^{0}(\\{q_{j,i}\\})[-1],i=1,2$. Several things to be noticed are:

  1. (1)

The normalized curve $C$ is not path-connected in our case, therefore the path
should be splits into $2$ components with endpoints identified under the
normalization map;

  2. (2)

The ”$\rho$-eigen”-extension data should come from (the Abel-Jacobi image of)
paths coming from normalizing $\rho$-eigen $1$-cycles.

  3. (3)

The standard Abel-Jacobi map only gives a class in the extension data modulo
the lattice of integral homology cycles. To define a class in the affine
space, we have to find a locally canonical construction.

We fix a component $C_{1}$. Since any finite cover do not change the local
behaviour, we consider the local extended period map of the ordered case,
i.e., we associate the points a fixed order
$\\{x_{1},...,x_{r}\\}\subset\mathbb{P}^{1}$ and denote the corresponding
branched points in $C$ as $\\{p_{1},...,p_{r}\\}$. By applying a local
$\mathrm{SL}_{2}$-(constant) section of linear transformation we can assume
$(x_{1},x_{2},x_{3})=(0,1,\infty)\in(\mathbb{P}^{1})^{3}$. For the ordered set
of pinched points $\\{q_{j,1}\\}$ with $\rho^{l}(q_{j,1})=N_{j+l,1}$,
$q_{l+d,1}:=q_{l,1}$, we choose a simple path from $0$ to $q_{1,1}$, denoted
as $\gamma^{1}_{q_{1}}$. If $q_{1,1}$ varies in a contractible neighborhood,
we assume the path $\gamma^{1}_{q_{1}}$ also varies in a contractible tubular
neighborhood. Let $\zeta_{d}=e^{\frac{2\pi i}{d}}$ be the $d$-root of unity
and consider the following two elements in the relative homology group
$H_{1}(C_{1},\\{q_{i,1}\\};\mathbb{Z})$:

$\gamma^{1}_{+}:=\gamma^{1}_{q_{1}}+\zeta_{d}^{-1}\rho\gamma^{1}_{q_{1}}+...+\zeta_{d}^{-(d-1)}\rho^{d-1}\gamma^{1}_{q_{1}}$

$\gamma^{1}_{-}:=\gamma^{1}_{q_{1}}+\zeta_{d}\rho\gamma^{1}_{q_{1}}+...+\zeta_{d}^{(d-1)}\rho^{d-1}\gamma^{1}_{q_{1}}$

It is clear $\rho(\gamma^{1}_{\pm})=\zeta_{d}^{\pm 1}\gamma^{1}_{\pm}$. If we
do the similar procedure on $C_{2}$ to obtain $\gamma^{2}_{\pm}$. Let
$C_{1}\sqcup C_{2}\to C$ be the normalization map, the image of
$\gamma^{1}_{\pm}-\gamma^{2}_{\pm}$ will give a well-defined element in
$H_{1}(C)_{\pm k}$.

Given $\gamma^{1}_{\pm}$ and a differential $1$-form $\omega\in
H^{1}(C_{1})_{k}$, by the symmetry in Lemma 5, we have:

(27) $H^{1}(C_{1})_{k}\xrightarrow[]{\cong}\Omega^{1}(C_{1})_{\pm k},$

where $\omega$ is sent to $\omega$ (resp. $\bar{\omega}$) if $\omega$ is
holomorphic (resp. anti-holomorphic). We have the natural map
$\omega\mapsto\int_{\gamma^{1}_{\pm}}\omega$. In the construction of
$\gamma^{1}_{\pm}$, we choose the point $q_{1,1}$ and a path
$\gamma^{1}_{q_{1}}$, they define maps

(28)
$q_{1,1}\xrightarrow[]{\gamma^{1}_{q_{1}}}\gamma^{1}_{\pm}\xrightarrow[]{\tau}\Omega^{1}(C_{1})_{k=\pm
1}^{\vee}.$

For a fixed $q_{1,1}$, if we select two different paths $\gamma^{1}_{q_{1}}$
and $\hat{\gamma}^{1}_{q_{1}}$ varying in the same contractible tubular
neighborhood, the resulting image of $\tau$ in $\Omega^{1}(C_{1})_{k=\pm
1}^{\vee}$ will not be changed. So to prove the theorem it remains to show
$\tau$ is locally injective, because this is the construction that defines the
Abel-Jacobi map. We only need to prove the following statement: For some
$\omega\in\Omega^{1}(C_{1})_{k=\pm 1}$, when varying $q_{1,1}$ in a
contractible neighborhood $U_{1}\subset C_{1}$, the function:

$\displaystyle f_{\omega}:U_{1}\longrightarrow\mathbb{C},$ $\displaystyle
f_{\omega}(q_{1,1}):=\int_{\gamma^{1}_{q_{1}}}\omega$

is not constant. This is because

(29)
$\int_{\gamma_{+}^{1}}\omega=\sum_{j=0}^{d-1}\zeta_{d}^{-j}\int_{\rho^{j-1}\gamma_{q_{1}}^{1}}\omega=\sum_{j=0}^{d-1}\zeta_{d}^{-j}\zeta_{d}^{j}\int_{\gamma_{q_{1}}^{1}}w=d\int_{\gamma_{q_{1}}^{1}}\omega.$

By Lemma 5, $\Omega^{1}(C_{1})_{\pm 1}$ is popularized by:

(30) $\frac{z^{j-1}dz}{y^{k}}\ ,\ 0<j/r\neq 1/d<1$

In our case where we assume $d|r$ and $r\geq 3$, $\Omega^{1}(C_{1})_{\pm 1}$
contains a non-zero holomorphic differential $1$-form $\omega_{0}$, for which
the function

(31) $f_{\omega_{0}}(q_{1,1}):=\int_{\gamma^{1}_{q_{1}}}\omega_{0}$

cannot have zero germ at any point on $\mathbb{P}^{1}$. Combine with the
similar statements on $C_{2}$, or proceed dually on $\gamma_{-}^{1}$, we
proved the theorem.


,

# Reviewing pigeonhole effect with transversal modes

Marcin Wieśniak  Institute of Theoretical Physics and Astrophysics, Faculty of
Mathematics, Physics, and Informatics,
University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
International Centre for Theory of Quantum Techologies
University of Gdańsk, ul. Wita Stwosza 63, 80-308 Gdańsk, Poland

###### Abstract

We investigate the interference setup claimed to reveal the quantum pigeonhole
effect. It is a claim that it is possible that three parties travel across a
two-path interferometer, but no pair of them share a path. We demonstrate the
inconsistency of the description based on the weak values. We introduce
transversal modes and observe the arise of correlations between path and
traversal degrees of freedom. Also, the rotational symmetry of the output
pattern is studied as a witness of pairwise interaction. It shows no apparent
relation to the probability of observing three particles in the same output.

##  I Introduction

Intuitively we expect that when we send, say, three particles from one place
to the other, each of should take some path from the initial to the final
location. If we restrict possible paths to two possibilities, we expect that
at least two them propagate together. Consequently, if these particles repulse
one another, we would observe the witness of this repulsion in terms of a
change of the observed output pattern.

These beliefs are challenged so-called quantum pigeonhole effect pigeon . The
claim of the original work is that, in a specific situation, three electrons
travel through a two-path interferometer, but, basing on the weak value
description, no pair of particles ever travelled together. This claim was
subsequently generalized for $N$ particles with at most $M$ traveling through
the same arm FOOTPRINTS . In this work we reexamine these claims by
introducing spatial degrees of freedom.

Quantum pigeonhole effect arose as one of the attempts of understanding
interference. Traditional quantum mechanics is extremally successful in
predicting results of experiments, but it forbids the observer to trace a
particle inside an interferometer, leaving the observer with the nonclassical
concept of superposition. Any attempt to localize a particle inside an
inteferometer must result in the decrease of the interferometric visibility
BERGE . An alternative approach includes Bohmian mechanics and the two-state
vector formalism (TSVF) and introduces some description inside an
interferometer, but at the cost of many extra entities, such as nonlocal
potentials, weak traces of presence, etc., and some paradoxes and
counterintuitive claims, such as discountinuous trajectories Danan . The core
of TSVF is the realistic interpretation of weak measurements.

We begin with recalling the original presentation of the quantum pigeohole
effect. We point out the inconsistency of the realistic interpretation of weak
values, and stress the importance of the Hamming distance. Subsequently, we
compare a quantity that witnesses propagation of two particles in the same arm
and the probability of observing all three parties at the same side of the
second beam-splitter. This probability is in a way complementary to the model
of propagation expected in Ref. pigeon . We find no clear relation between
these two quantities.

##  II Quantum pigenhole effect

The effect supposedly appears when we send three interacting particles
(electrons) are sent through a suitably tuned (for the purpose of
demonstrating the effect) two-path Mach-Zehnder interferometer, with arms
denoted as $L$ and $R$. We assume that the interaction is only possible when
two electrons are in the same arm. The state at the input is
$\left|LLL\right\rangle$. The first beam splitter transforms it to
$\left|+++\right\rangle$, where
$\left|+\right\rangle=1/\sqrt{2}(\left|L\right\rangle+\left|R\right\rangle)$.
The second beam splitter makes detectors sensitive to state $\left|\pm
i\right\rangle=1/\sqrt{2}(\left|L\right\rangle\pm i\left|R\right\rangle)$ and
we expect all three electrons to arrive at the same detector (regardless which
one). If the electrons initially form a perfect triangle, we expect, as
pointed out in Ref. pigeon , the output and after the post-selection exhibits
an enlarged triangle (due to all three particles in one arm) and disfigured
individual spots (due to pairs propagating together).

However, if we apply the weak value description, the Authors of the Ref.
pigeon  conclude that no pair of particles has traveled through the same path.
The projector certifying that electrons 1 and 2 travel through the same is
given by

$\displaystyle\Pi^{same}_{12}=$ $\displaystyle\Pi^{LL}_{12}+\Pi^{RR}_{12}$
$\displaystyle=$ $\displaystyle\left(\left|LL\right\rangle\left\langle
LL\right|+\left|RR\right\rangle\left\langle
RR\right|\right)\left(\left|L\right\rangle\left\langle
L\right|+\left|R\right\rangle\left\langle R\right|\right).$

Importantly, this projector does not reveal which path was taken. The weak
value is
$\frac{\left\langle+i+i+i\right|\Pi^{same}_{12}\left|+++\right\rangle}{\left\langle{+i+i+i}|{+++}\right\rangle}=0$.
The same argument applies to other pairs.

Nevertheless, if we study the weak value of the projection
$\Pi^{same}_{123}=\left|LLL\right\rangle\left\langle
LLL\right|+\left|RRR\right\rangle\left\langle RRR\right|$, it turns out to be
-1/2. The weak value of $\Pi^{same}_{12}$ has four terms, which cancel
pairwise and the total. Two of theme enter the weak value of
$\Pi^{same}_{123}$, which raises the inconsistency of this weak value
interpretation.

At this point, it is interesting to consider a Hamming distance between
various components of the state. It tells us, how many particles need to move
around to get to other state, So, for example, the the distance between states
$\left|LRR\right\rangle$ and $\left|RRL\right\rangle$ is 2. Notice that when
we consider the post-selection of all electrons leaving the interferometer in
the same direction, pairs of components having odd Hamming distance interfere
neutrally, i.e., the probability of observing all of the at the same side
behind the second beam splitter is 1/8. If it is 2 modulo 4 – destructively,
and constructively for the Hamming distance equal to 0 mod 4. The weak value
for particles 1 and 2 was found to be 0, since two pairs of contributions
cancel each other. If we consider propagation of three electrons in the same
arm, they would have Hamming distance of 3.

##  III Gaussian profile description

We shall now discuss the optical description of the experiment. The Hilbert
space is divided into two main parts: eight-dimensional Hilbert space
$\\{\left|L\right\rangle,\left|R\right\rangle\\}^{\otimes 3}$ and the spaces
of complex functions of pairs of real parameters, describing transversal modes
of all three particles. We actually need only the product of the amplitudes of
each electron, so effectively our Hilbert space is of eight-dimensional
complex functions of six real arguments.

We adopt the following assumptions:

  1. 1.

Similarly to the original proposal, we consider three distinguishable
particles. In other words, we do not perform the Hartree-Fock
antisymmetrization. Still, we consider electrons as it is necessary for them
to repulse each other.

  2. 2.

Initially the electrons have a very small separation, as compared to the width
of the beam. They form a perfect triangle

  3. 3.

The transverse profile of each electron is Gaussian and rotationally
invariant. It doesn’t change or get resized during the propagation, other than
being displaced accordingly to the interaction.

  4. 4.

The momentum kicks due to the repulsion is very small, thus strictly real
2-dimensional Gaussian functions describe the profiles after the repulsion
adequately. Alternatively, we could consider the displacement in the momentum
space.

At first, these assumption may seem to difficult to fulfilled simultaneously.
However, together they make the calculations feasible, serving for a guide for
our intuition. We will discuss their relaxation after the main result is
presented.

We assume that the electrons form a perfet triangle with vertices on lines of
three vectors, $\vec{v}_{1}=(1,0)$, $\vec{v}_{2}=(-1,\sqrt{3})/2$, and
$\vec{v}_{3}=(-1,-\sqrt{3})/2$, The beams have Gaussian profiles with spread
$\sigma$. Thus, the state behind the first beam splitter is

$\left|\phi_{0}\right\rangle=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{c}1\\\
1\end{array}\right)^{\otimes 3}\times G,$ (2)

where $G=\prod_{i=1}^{3}F(x_{i},y_{i},(0,0))$ and
$F(x,y,\vec{v})=\frac{1}{\sigma\sqrt{\pi}}\exp\left(-\frac{|(x,y)-\vec{v}|^{2}}{2\sigma}\right)$,
and displacement due to traveling through different arms of the interferometer
are already included in each of the components. After the repulsion and the
second beam splitter we have

$\left|\phi_{1}\right\rangle=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{c}F(x_{1},y_{1},a\vec{v}_{1})F(x_{2},y_{2},a\vec{v}_{2})F(x_{3},y_{3},a\vec{v}_{3})\\\
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{3})F(x_{2},y_{2},a(\vec{v}_{2}+\vec{v}_{3}))F(x_{3},y_{3},(0,0))\\\
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{2})F(x_{2},y_{2},(0,0))F(x_{3},y_{3},a(\vec{v}_{2}+\vec{v}_{3})\\\
F(x_{1},y_{1},(0,0))F(x_{2},y_{2},a(\vec{v}_{1}+\vec{v}_{2}))F(x_{3},y_{3},a(\vec{v}_{1}+\vec{v}_{3}))\\\
F(x_{1},y_{1},(0,0))F(x_{2},y_{2},a(\vec{v}_{1}+\vec{v}_{2}))F(x_{3},y_{3},a(\vec{v}_{1}+\vec{v}_{3}))\\\
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{2})F(x_{2},y_{2},(0,0))F(x_{3},y_{3},a(\vec{v}_{2}+\vec{v}_{3})\\\
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{3})F(x_{2},y_{2},a(\vec{v}_{2}+\vec{v}_{3}))F(x_{3},y_{3},(0,0))\\\
F(x_{1},y_{1},a\vec{v}_{1})F(x_{2},y_{2},a\vec{v}_{2})F(x_{3},y_{3},a\vec{v}_{3})\end{array}\right).$
(3)

Finally, we can compute the probability that all three electrons reach the
same detector, $P_{LLL}$.

$\displaystyle P_{LLL}=$ $\displaystyle
K(|(1,i,i,-1,i,-1,-1,-i)\left|\phi_{1}\right\rangle|^{2})/8$ (4)

with

$\displaystyle K(f(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}))$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx_{1}\int_{-\infty}^{\infty}dx_{2}\int_{-\infty}^{\infty}dx_{3}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dy_{1}\int_{-\infty}^{\infty}dy_{2}\int_{-\infty}^{\infty}dy_{3}$
$\displaystyle\times$ $\displaystyle f(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}).$
(5)

This probability is highly relevant, as the the quantum pigeonhole effect is
conditioned on observing all three particles at the same time.

The amplitude in the modulo reads

$\displaystyle(1,i,i,-1,i,-1,-1,-i)\left|\phi_{1}\right\rangle$
$\displaystyle=$ $\displaystyle
e^{-\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+5a^{2}+a(-2x_{1}+x_{2}+x_{3}-\sqrt{3}y_{2}-\sqrt{3}y_{3})}{\sigma^{2}}}$
$\displaystyle\times$
$\displaystyle(1+i)\left(\frac{1}{\sqrt{\pi}\sigma}\right)^{3}\left(-e^{\frac{a^{2}}{\sigma^{2}}}+e^{\frac{3a^{2}+a(x_{1}+x_{2}+x_{3})}{2\sigma^{2}}}\right.$
$\displaystyle+$ $\displaystyle
e^{\frac{a(x_{1}+x_{2}+x_{3}-\sqrt{3}y_{1}+\sqrt{3}y_{2}-\sqrt{3}y_{3})}{\sigma^{2}}}$
$\displaystyle+$
$\displaystyle\left.e^{\frac{a(x_{1}+x_{2}+x_{3}-\sqrt{3}y_{1}-\sqrt{3}y_{2}+\sqrt{3}y_{3})}{\sigma^{2}}}\right).$

After transformations we find that

$P_{LLL}=\frac{1}{8}\left(1+\frac{3}{2}e^{-\frac{5}{8}x^{2}}-\frac{3}{2}e^{-\frac{3}{8}x^{2}}\right),$
(7)

where $x=a/\sigma$. This function reaches its minimum of
$1-\left(\frac{3}{5}\right)^{\frac{5}{2}}$ for $x=2\sqrt{\log{5/3}}$.

Unfortunately, Ref. pigeon  does not offer a constructive explanation of the
quantum pigeonhole effect. That is to say, we do learn from the paper how
three electrons could have traveled without sharing a path. Also, within the
above formalism, the projection onto not having more than one particle in each
arm has no sense. Instead, we can verify a pair of particles have interacted.
We have demonstrated above, that even in the weak value description, we can
have three parties in one arm. We thus devise a test for exactly two particles
propagating together. If such a pair travels together, it breaks the
rotational symmetry of the output pattern. We thus define the probability of
certifying the interaction as

$\displaystyle
P_{INT}=1-\left|\left\langle\phi_{1}\right|U\left|\phi_{1}\right\rangle\right|^{2},$
(8)

and $U$ is composed of a rotation matrix by $2\Pi/3$ and a cyclic permutation
$(x_{1},y_{1})\rightarrow(x_{3},y_{3})\rightarrow(x_{2},y_{2})\rightarrow(x_{1},y_{1})$.
After plugging in the state we get

$\displaystyle P_{INT}$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle-$
$\displaystyle\frac{1}{4}|K(G\times
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{3}))F(x_{2},y_{2},a(\vec{v}_{2}+\vec{v}_{3}))F(x_{3},y_{3},(0,0)))|^{2}$
$\displaystyle-$ $\displaystyle\frac{1}{4}|K(G\times
F(x_{1},y_{1},a(\vec{v}_{1}+\vec{v}_{2}))F(x_{2},y_{2},(0,0))F(x_{3},y_{3},a(\vec{v}_{2}+\vec{v}_{3})h))|^{2}$
$\displaystyle-$ $\displaystyle\frac{1}{4}|K(G\times
F(x_{1},y_{1},(0,0))F(x_{2},y_{2},a(\vec{v}_{1}+\vec{v}_{2}))F(x_{3},y_{3},a(\vec{v}_{1}+\vec{v}_{3})))|^{2}$
$\displaystyle-$ $\displaystyle\frac{1}{4}|K(G\times
F(x_{1},y_{1},a\vec{v}_{1})F(x_{2},y_{2},a\vec{v}_{2})F(x_{3},y_{3},a\vec{v}_{3}))|^{2}$
$\displaystyle=$ $\displaystyle\frac{3}{4}(1-e^{-\frac{5}{4}x^{2}}).$ (9)

Note that $U$ acts on the profile degrees of freedom, whereas the beam
splitters act solely on the discrete degrees. We have thus calculated just in
front of the second beam splitter, rather than behind it. This also means that
it also commutes with the projectors $\Pi^{same}_{1,2}$ and
$\Pi^{same}_{1,2,3}$.

When the interaction is strong, the state detection of even a single electron
with a spatially-resolving detector reveals, which electrons shared a path,
but not which path was more occupied. In other words, the state decoheres to
two-dimensional block given by Hamming distance equal to 3. This lead to
neutral decoherence, and equal probability for any output combinations in
terms of $\left|L\right\rangle$ or $\left|R\right\rangle$ states behind the
second beam splitter. For $x\rightarrow 0$ we do not observe symmetry braking,
as the interaction is simply to weak, rather than being absent.

Let us now discuss possible relaxation of the assumptions. When we do not
assume that the initial separation is equal to zero, we can assume that the
initial wave-function is similar to the one with $x_{0}<x$. In such a case,
some weak interaction has already taken place and we expect to get
$\left.\partial\left(|\left\langle{\phi_{i}}|{\phi_{f}}\right\rangle|^{2}\right)/\partial
x\right|_{x=x_{0}}\neq 0$. Allowing for distortion of the spots and appearance
of imaginary part due to repulsion can only decrease the overlaps. That means
that all the effects are more dramatic: the dip of the probability for
detecting all three particles is more prominent and the rise of probability of
confirming the interaction grows more rapidly. However, the asymptotic
behavior remains unchanged.

We shall now discuss Refs. PNAS ; EXP1  , which experimentally verifies the
pigeonhole effect. Therein, the three particles were three photons propagating
along three paths. Their polarizations are the analogue of the paths inside
the interferometer in the original proposal. A single polarizing beam splitter
(PBS) between any two represents the measurement of $\Pi^{same}_{i,j}$, while
a sequence of two PBSs combining all three paths realizes
$\Pi^{same}_{1,2,3}$. Under the condition that one photon reaches each
detector and given the initial state, the mean of the former has been found to
be 0, while the mean of the latter has been found to be about 0.23. This is
another example of how this effect can be seen as a destructive interference,
much like disappearing (dark) fringes when we close one slit in Young’s
experiment.

It should be stressed that the setup with one or two PBSs are two different
strong measurements. Thus, in the Copenhagen interpretation the results are
fully consistent, as they describe two distinct experiments.h

##  IV Conclusions

In conclusion, we have revisited the the experiment supposedly revealing the
quantum pigeonhole effect by introducing the Gaussian transverse profiles of
the electrons traveling through the Mach-Zehnder interferometer. We have first
observed that the second statement of the effect, namely that the first
derivative of all relevant probabilities with respect to the interaction
strength vanishes, is a typical behavior in quantum mechanics. Second, there
is no concept of an experiment to confirm the the claim that no two particles
have traveled together. This remains an interpretative conclusion of the two-
vector state formalism. As this formalism allows projectors to yield negative
values, it turned out that it is not-self consistent, giving the weak value of
two electrons propagating together equal to zero (which is interpreted as an
impossibility of at least two electrons traveling together), but the weak
value of the projection onto three electrons in one arm is different than
zero. For weak interaction we do not see any effects, but this is due to the
interaction having not enough time to change the state. For mid-range
interaction strengths, we have a dip of probability of observing all three
electrons at the same detector. For strong interaction, we observe the revival
of three fold coincidences, as the repulsion causes a partial decoherence.
Only the components with the Hamming distance of three will interfere.

The result presented above highlight a few things. First, weak values cannot
be interpreted realistically, as this leads to inconsistencies, even within
the weak value description itself. We have experienced it by finding a more
general weak value to be zero, but a more specific one - different. Even the
fact that they can be arbitrary complex number suggests no direct operational
meaning. Only the results of sharp, von Neumann measurements provide us the
knowledge about the system’s state, but one should be careful with the
experimental context of such measurements.

Second, it is logically incorrect to conclude that, since we have not observed
the effects of interaction between particles, there wasn’t any. This mistake
was made, for example, Expression “we have not observed” is subjective and
depends on our perceptive skills. The only way we could reach the conclusion
of absence of the paairwise interaction is by designing a specific test for
absence of such an interaction. Otherwise, it could be present, but simply too
weak to bring in any observable effects. Thus far, claimed experimental
demonstrations did not include such a test. Interestingly, Ref. pigeon
mentions the deformation of the output beam as the method to verify the
effect. The deformation would become clearer for longer interferometers.
However, this test specifically requires electrons as pigeons.

## Acknowledgements

This work is a part of NCN grants 2017/26/E/ST2/01008 and 2015/19/B/ST2/01999,
and joint DFG/NCN grant 2016/23/G/ST2/04273. The Author acknowledges the ICTQT
IRAP project of FNP (Contract No. 2018/MAB/5), financed by structural funds of
EU.

## References

  * (1) Yakir Aharonov, Fabrizio Colombo, Sandu Popescu, Irene Sabadini, Daniele C. Struppa, and Jeff Tollaksen, Quantum violation of the pigeonhole principle and the nature of quantum correlations, PNAS 3, 532-535 (2016).
  * (2) Gregory Reznik, Shrobona Bagchi, Justin Dressel, and Lev Vaidman, Footprints of quantum pigeons Phys. Rev. Res. 2, 023004 (2020).
  * (3) P. Grangier, Thèse d’État, Institut d’Optique et Université Paris (1986);B.-G. Englert, Fringe Visibility and Which-Way Information: An Inequality, Phys Rev. Lett. 77, 2154 (1996).
  * (4) Ming-Cheng Chen, Chang Liu, Yi-Han Luo, He-Liang Huang, Bi-Ying Wang, Xi-Lin Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan, Experimental demonstration of quantum pigeonhole paradox, PNAS 5, 1549-1551 (2019).
  * (5) Mordecai Waegell, Tobias Denkmayr, Hermann Geppert, David Ebner, Tobias Jenke, Yuji Hasegawa, Stephan Sponar, Justin Dressel, and Jeff Tollaksen, Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect, Phys. Rev. A 96, 052131 (2017).
  * (6) A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, Asking photon where they have been, em Phys. Rev. Lett. 111, 240402 (2014).