2.1 Geometric compactifications

We describe two families of compactifications for the moduli space $M_{0,n}$ of n labeled points in the projective line: The one constructed via Geometric Invariant Theory (GIT) and the one parametrizing stable pairs as defined in the Minimal Model Program (MMP). 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 [Reference Mumford, Fogarty and Kirwan23] for the general theory.

Theorem 2.1 [Reference Mumford, Fogarty and Kirwan23, 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

$$ \begin{align*} \overline{M}_{0,\mathbf{w}}^{\mathrm{GIT}} := (\mathbb{P}^1)^n / \!\! /_{\mathbf{w}} \operatorname{\mathrm{SL}}_2 \end{align*} $$

known as the GIT compactification that satisfies the following properties:

• • It has a Zariski open subset known as the stable locus, that contains $M_{0,n}$ , and parametrizes configurations $(p_1, \ldots , p_n)$ of n labeled 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 with a degeneration where $p_{i_1} = \cdots = p_{i_{\frac {n}{2}}}$ and $p_{i_{\frac {n}{2}+1}}= \cdots = p_{i_n}$ .

We need the following moduli of partially labeled points to use results from [Reference McMullen22, Section 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-labeled points in the complex line up to $\mathrm {Aut}(\mathbb C)$ is given by

(2.1) $$ \begin{align} M_{0,n}^* := \mathbb{C}^{\langle n \rangle} / \mathrm{Aut}(\mathbb C) && \mathrm{Aut}(\mathbb C):= \{ (a,b) \in \mathbb{G}_m \rtimes \mathbb{C} \; | \; z \mapsto az +b\}. \end{align} $$

A labeled version of this space is known as $T_{1,n}$ and it is was introduced for arbitrary dimension in [Reference Chen, Gibney and Krashen4]. Let $T_{1,n}^{\circ } \subset T_{1,n}$ be the open parametrizing configuration of n distinct labeled points. By [Reference Chen, Gibney and Krashen4, Proposition 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

$$ \begin{align*}M_{0,n}^* \cong T_{1,n}^{\circ}/S_n \cong M_{0,n+1}/S_n \end{align*} $$

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 $v_1 + \cdots + v_{n+1}=2$ . If we suppose that $v_1=\cdots = v_n$ , then we have the diagram:

The conditions on the weights imply they can be written as

$$ \begin{align*} v_1 = \cdots = v_n = a && v_{n+1} = 2-na \text{ with } a> 0 \end{align*} $$

From all possible choices, we highlight the case $v_{n+1}=1-n\epsilon $ and $v_1=\cdots =v_n=\frac {1}{n} + \epsilon $ with $\frac {1}{n(n-1)}> \epsilon $ . For the same argument that in [Reference Gallardo and Routis11, Lemma 4.13], we have that $\overline {M}^{\mathrm {GIT}}_{\mathbf {v}} \cong \mathbb {P}^{n-2}$ 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 [Reference Hassett13] 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 $0 < b_i \leq 1$ .

Proposition 2.5 [Reference Hassett13].

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

$$ \begin{align*}\rho_{\mathbf{a},\mathbf{c}}: \overline{M}_{0, \mathbf{a}} \longrightarrow \overline{M}_{0, \mathbf{c}} \end{align*} $$

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 Theorems 2.1 and 2.5.

We remark that we only state the result that we used. It was proved in [Reference Hoon and Bom27] and [Reference Moon24] that the morphism in Proposition 2.6 is a blow-up of the ideal sheaf given by Kirwan’s partial desingularization, see [Reference Kirwan16].

For the partially labeled case described in Equation (2.1), we should be careful in using the same weight at the non-labeled points, and keeping the labeled and non-labeled 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 $b_1 = \cdots = b_n$ , $b_{n+1} + b_{i}> 1$ $b_1 + \cdots +b_{n+1}> 2$ 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 labeled points.

2.2 Eigenperiod map for n points in the line

Let $\mu := (p_1, \ldots , p_n)$ be a collection of n distinct labeled 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 associate with our collection of points $\mu $ a smooth curve $C_{\mu ,d}(z_0,z_1,y)$ of weighted degree $nd$ in the weighted projective plane $\mathbb {P}[d:d:n]$ with $\deg (z_0) = \deg (z_1) = d, \; \deg (y)=n$ defined by the closure of the affine curve

(2.2) $$ \begin{align} C_{\mu,d} &:= \overline{\{ y^d-(z_0-x_1z_1)...(z_0-x_nz_1) =0\}} \end{align} $$

if $d|n$ , and

(2.3) $$ \begin{align} C_{\mu,d}:= \overline{\{ y^d-z_1(z_0-x_1z_1)...(z_0-x_{n-1}z_1) =0\}} \end{align} $$

if $d\nmid n$ and $\mathrm {gcd}(d,n-1)=1$ . In the second case, $C_{\mu ,d}$ admits a branched point of degree d at the infinity.

Geometrically, $C_{\mu ,d}$ can be realized as the $d^{th}$ -cyclic cover of $\mathbb {P}^1$ branched along $(p_1, \ldots , p_n )$ . Then, the isomorphism class of the curve is defined up an action of $\operatorname {\mathrm {SL}}_2 \times S_n$ 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 has been studied in the context of the moduli of curves, see [Reference Fedorchuk9], and it 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 dth 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 $\rho ^*: A_d \longrightarrow \operatorname {\mathrm {Aut}}(H^{1}(C_{\mu ,d}),Q_{\mu })$ . The action of $A_d$ induces an eigenspace decomposition

$$ \begin{align*} H^{1}(C_{\mu,d}, \mathbb{C})_{\chi} = \lbrace x \in H^{1}(C_{\mu,d}, \mathbb{C}) \; | \; g \cdot x = \chi(g) x, \; \forall g \in \rho^*(A_d) \rbrace, && \chi \in \operatorname{\mathrm{Hom}}\left(A_d, \mathbb{C} \right). \end{align*} $$

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 eigenspaces with respect to the eigenvalue $\lambda ^k$ and $\overline {\lambda ^k}$ respectively. We have the eigenspace decomposition

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

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

$$ \begin{align*} H^{p,q}_k(C_{\mu,d}, \mathbb{C}) &:= H^{p,q}(C_{\mu,d}, \mathbb{C}) \cap H^{1}_k(C_{\mu,d}, \mathbb{C}) \end{align*} $$

with eigen-Hodge numbers defined as

(2.4) $$ \begin{align} \mathbf{h}_k=(h^{p,q}_k), && h^{p,q}_k = \dim H^{p,q}_k(C_{\mu,d}, \mathbb{C}). \end{align} $$

To define our eigenperiod space, we consider the flag

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

and denote as $\mathbf {f}_k= (\dim F_k^{p} \;| \; p=0, 1)$ its dimension vector. Now fix a collection of n distinct points $\mu _0$ in $\mathbb {P}^1$ , the Grassmannian $ \mathrm {Grass}( \mathbf {f}_k, H^{1}_k(C_{\mu _0,d}, \mathbb {C})) $ contains as a non-compact locus defined as

(2.5) $$ \begin{align} D_{n,d,k}:= \lbrace F^{\bullet}_{k} \in \mathrm{Grass}( \mathbf{f}_k, H^{1}_k(C_{\mu_0,d}, \mathbb{C}) ) \; \big| \; Q(F_k^1,F_k^1)=0, \;\; Q(Cv, \overline{v})> 0 \; \rbrace \end{align} $$

where C is the Weil operator and acts on $H^{p,q}_k(C_{\mu _0,d}, \mathbb {C})$ via $i^{p-q}\cdot \mathrm {Id}$ . 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

$$ \begin{align*} \widehat{\phi}_{n,d,k}:\widehat{M}_{0,n} \longrightarrow D_{n,d,k} \end{align*} $$

where $\widehat {M}_{0,n}$ is the universal cover of $M_{0,n}$ , $D_{n,d,k}$ is defined by Equation (2.5), 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 [Reference McMullen22, Chapter 3].

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

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

Moreover, the eigen-Hodge numbers, defined in Equation 2.4, are given by

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

To describe the eigenperiod map, let $\mathbb {C}^{ \langle n \rangle }$ be the moduli space of n unlabeled distinct points on $\mathbb {C}$ . We need to discuss some aspects on the monodromy representation of the fundamental group of $\mathbb {C}^{ \langle n \rangle }$ 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

(2.7) $$ \begin{align} \mathrm{U}(\Lambda_{n,d,k}) \subset \mathrm{U}\left( H^1_k(C_{\mu, d}) \right) \cong \mathrm{U}(r,s). \end{align} $$

where $\mathrm {U}(r,s)$ is the unitary group with mixed signature $(r,s)$ . The fundamental groups of the moduli spaces introduced in the previous sections are related by the following result.

Lemma 2.9. Let $d \geq 2$ and $n \geq 5$ be integers such that gcd $(d,n-1)=1$ , then the following diagram commutes

If moreover $d | n$ , we also have the commutative diagram:

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

$$ \begin{align*}\pi_k: B_n \longrightarrow \mathrm{U}(\Lambda_{n,d,k}) \subset \mathrm{U}(r,s) \end{align*} $$

whose behavior we describe next.

The behavior of the monodromy representation described in Lemma 2.9 and Theorem 2.10 implies that we should consider two cases when defining the period map. If d divides n, then by any dth 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 dth 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.

Proposition 2.11. Given $n,d,k$ as in Table 1, the eigenperiod map is defined as

$$ \begin{align*} M_{0,n}/S_n \xrightarrow{\phi_{n,d,k}} \Gamma_{n,d,k} \backslash D_{n,d,k} && \Gamma_{n,d,k}:= \pi_{d,k}(\mathrm{Mod}_{0,n}) \end{align*} $$

if $d|n$ , and as

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

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

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 [Reference Dolgachev and Kondō6, Section 7]. For that purpose, we may also consider the eigenperiod map $\phi _{n,d,k,\overline {k}}$ defined by the Hodge decomposition within

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

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

By [Reference McMullen22, (2.5)], we have that $D_{n,d,k} \cong \text {II}(r,s)\cong \mathrm {Sp}(2r+2s)/\mathrm {U}(r+s)$ , and $\Gamma _{n,d,k}$ 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 $\gcd (d, n-1) = 1$ , we have the following commutative diagram

obtained by composing the period map from Proposition 2.11 with the relevant finite quotient by a symmetric group $S_m$ , $m=n$ or $n-1$ .

Question 2.14. By the work of Deligne–Mostow [Reference Deligne and Mostow7], we know that if $n,d,k$ is equal to either $(8,4,1)$ or $(12,6,1)$ . Then, the map $\Phi _{n,d,k}$ is finite and dominant. What are all the other values of $(n,d,k)$ in Table 1 such that

$$\begin{align*}\dim \left( \Phi_{n,d,k} \left( M_{0,n} \right) \right) = (n-3)? \end{align*}$$

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

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

with $\mathbf {w}\cdot \mathbf {m}$ being 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 $f(\mathbf {z})=0$ .

Denote T as the generator of monodromy group around $0\in \Delta $ acting on cohomology of the fibers. By Borel’s monodromy theorem, T is quasi-unipotent and admits a decomposition $T=T^{\text {ss}}e^N$ where $T^{\text {ss}}$ is the finite-order semisimple factor and N is the monodromy logarithm. To describe our degeneration, Hodge-theoretically, we need some definitions and notations. The following theorems are well-known and we refer the reader to [Reference Kerr and Laza17] and [Reference Peters and Steenbrink25] for details.

Theorem 2.16. Let $\psi _f$ be the nearby cycle functor, and $\phi _f$ be the vanishing cycle functor. If we denote

$$ \begin{align*} H^k_{\mathrm{lim}}(X_t):=\mathbb{H}^k(X_0, \psi_f\mathbb{Q}_{\mathcal{X}}) && H^k_{\mathrm{van}}(X_t):=\mathbb{H}^k(X_0, \phi_f\mathbb{Q}_{\mathcal{X}}), \end{align*} $$

then it holds that

1. 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. 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

(2.8) $$ \begin{align} \cdots\rightarrow H^k(X_0)\xrightarrow{}{}H^k_{\mathrm{lim}}(X_t)\xrightarrow{}\bigoplus V_f\xrightarrow{\delta}H^{k+1}(X_0)\rightarrow H^{k+1}_{\mathrm{lim}}(X_t)\rightarrow \cdots \end{align} $$

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

(2.9) $$ \begin{align} H^k_{\mathrm{ph}}(X_0):=\mathrm{Ker}\{H^k(X_0)\xrightarrow{\mathrm{}}{}H^k_{\mathrm{lim}}(X_t)\} \end{align} $$

is known as the Phantom cohomology, see [Reference Kerr and Laza17].

Theorem 2.19. There is a sequence known as the Clemens–Schmid sequence and given by

(2.10) $$ \begin{align} \cdots\rightarrow H_{2s+2-k}(X_0)\rightarrow H^k(X_0)\xrightarrow{\mathrm{}}{}H^k_{\mathrm{lim}}(X_t)\xrightarrow{N}H^k_{\mathrm{lim}}(X_t) \xrightarrow{}H_{2s-k}(X_0)\rightarrow \cdots \end{align} $$

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 $V_f=\bigoplus _{p,q}V^{p,q}$ . Suppose there is a finite-order automorphism $\rho $ on $V_f$ compatible with $T^{\text {ss}}$ and $e^N$ , then we have a finer splitting of $V_f$ into their eigenspaces:

(2.11) $$ \begin{align} V_f=\bigoplus_{p,q, \lambda, \eta}V_{f,\lambda, \eta}^{p,q}, \end{align} $$

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

Definition 2.20. The eigenspectra associated with 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

$$ \begin{align*} {\sigma_E(f, \rho)}:= \sum_{p,q,\lambda, \eta}m_{p+\lambda, \eta, p+q}[(p+\lambda,\eta, p+q)], && m_{p+\lambda,\eta, p+q}:=\textrm{dim}(V_{f,\lambda, \eta}^{p,q}). \end{align*} $$

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

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 [Reference Steenbrink26, Theorem 1], the only weights appear in the mixed Hodge structure of vanishing cohomology are $p+q=s$ or $s+1$ . Hence the eigenspectra given in Definition 2.20 can be also written as:

$$ \begin{align*} \sigma_E(f,\rho):= \sum_{p+q=s,\lambda, \eta}m_{p+\lambda, \eta, s}[(p+\lambda,\eta, s)]+\sum_{p+q=s+1,\lambda, \eta}m_{p+\lambda,\eta, s+1}[(p+\lambda, \eta, s+1)]. \end{align*} $$

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

$$ \begin{align*} \frac{\mathbb{C}[[\mathbf{z}]]}{\mathcal{J}(f)} := \frac{ \mathbb{C}[[\mathbf{z}]]}{<f, \frac{\partial f}{\partial z_0},...,\frac{\partial f}{\partial z_s}>} \cong \frac{\mathbb{C}[\mathbf{z}]}{\mathcal{J}(f)} \end{align*} $$

This is a finite dimensional $\mathbb {C}$ -vector space of 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

(3.1) $$ \begin{align} l(\beta):= \sum_{0\leq j\leq s}(\beta_j+1)w_j \in \mathbb{Q}. \end{align} $$

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

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

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

Lemma 3.1. There is an isomorphism:

$$ \begin{align*} H^{s+1}(\mathbb{C}^{s+1}\backslash Z_f)\xrightarrow[\cong]{\mathrm{Res}}H^s(Z_f)\cong V_f \end{align*} $$

and each $\eta _{\beta }:=\mathrm {Res}_{Z_f}(\omega _{\beta })$ satisfies:

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

Proof. By definition of $V_f$ and Lemma 3.1

$$ \begin{align*} \dim \left( \mathrm{Gr}_{s+1}^{W}(V_f) \right) = \sum_{p,q, p+q = s+1} \dim(V^{p,q}_f). \end{align*} $$

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

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

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 with the vanishing cohomology of the local degeneration to the singularity $f(x,y)=y^d+x^l$ is:

$$ \begin{align*} \sigma_E(f,\rho) = \sum_{\frac{a+1}{l}+\frac{k}{d}\in \mathbb{Z}}[(\frac{a+1}{l}+\frac{k}{d}, 2-\frac{a+1}{l}-\frac{k}{d}, 2)] + \sum_{\frac{a+1}{l}+\frac{k}{d}\notin \mathbb{Z}}[(\frac{a+1}{l}+\frac{k}{d}, 2-\frac{a+1}{l}-\frac{k}{d}, 1)] \end{align*} $$

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-homogeneous with weight $\mathbf {w}=(\frac {1}{l},\frac {1}{d})$ . Using Equation (3.1), we compute $l(\beta )$ for every $\beta $ in the Jacobian algebra

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

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 dimensions of these graded pieces define the eigenspectra.

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:

$$ \begin{align*} h^{1,1}_k\left( V_f \right) &= \begin{cases} 1 & \text{ if } \; d \text{ divides }kl \\ 0 & \text{ if } \; \text{ otherwise}. \end{cases} \end{align*} $$

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

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

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

$$ \begin{align*} \lbrace a \in \mathbb{N}_{+} \; | \ l-2 \geq a \geq 0 \;\; \text{ and } \;\; \frac{a+1}{l}+\frac{k}{d}=1 \rbrace. \end{align*} $$

The equation implies

$$\begin{align*}a = l-1-\frac{kl}{d}, \end{align*}$$

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 $\mathcal {I}$ be the set indexing irreducible boundary divisors of $\overline {M}_{0,n}$ . For $I\in \mathcal {I}$ , let $\left ( \mathcal {Y}_{I}, \mathcal {P}_I \right ) \to \Delta $ be a 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 $R: \overline {M}_{0,n} \to \overline {M}^{\mathrm {GIT}}_{0,\mathbf {\frac {2}{n}}}$ be the reduction morphism. We obtain a morphism $R\circ f_{\mathcal {Y}}:\Delta \to \overline {M}^{\mathrm {GIT}}_{0,\mathbf {\frac {2}{n}}}$ which has associated a 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 a one-dimensional family of smooth curves $\mathcal {C} \to \Delta $ such that a generic fiber is a smooth dth cover of $\mathbb {P}^1$ . The central fiber $\mathcal {C}|_0$ has singularities that are locally analytically 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 local vanishing cohomology group around the singular point $ \mathfrak {s} \in S$ . Then, the following inequality holds:

(3.3) $$ \begin{align} \mathrm{Max}\{\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1})| \; \mathfrak{s}\in S\} \leq h^{1,1}_{\mathrm{lim}}(C_t)\leq \sum_{\mathfrak{s} \in S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1}). \end{align} $$

Moreover, the inequality (3.3) stays true after restricting to each $\lambda ^k$ -eigenspace of $H^{1,1}(V_f)$ and $H^{1,1}_{\mathrm {lim}}(C_t)$ .

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 local vanishing cohomology around the singular point $ \mathfrak {s}\in S$ (and defined in Theorem 2.16.) In our case, the vanishing cycle sequence (2.17) is equal to:

(3.4) $$ \begin{align} 0\rightarrow H^1(C_0)\xrightarrow{}{} H^1_{\mathrm{lim}}(C_t) \xrightarrow{} \bigoplus_{\mathfrak{s} \in S} V_{f,\mathfrak{s}}\xrightarrow{\delta}H^2_{\mathrm{ph}}(C_0)\rightarrow 0. \end{align} $$

To show the first inequality on the left of (3.3), 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

(3.5) $$ \begin{align} \tilde{\mathcal{C}}\rightarrow \Delta^{|S|}, \ z=(z_1,...,z_{|S|})\in \Delta^{|S|} \end{align} $$

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 a family is given by collapsing $|S|$ tuples of marked points, and the local monodromy group is isomorphic to $\mathbb {Z}^{|S|}$ . By [Reference Kerr, Pearlstein and Robles19, Section 3] we obtain the left hand side of (3.3) due to the polarized relation of LMHS’s summarized in [Reference Kerr, Pearlstein and Robles19, Section 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 (3.3), we recall that sequence (3.4) 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

(3.6) $$ \begin{align} h^{1,1}(C_0) + \sum_{\mathfrak{s} \in S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1}) = h^{1,1}_{\mathrm{lim}}(C_t) + h^{1,1}_{\mathrm{ph}}(C_0). \end{align} $$

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

$$ \begin{align*} 0 \xrightarrow[]{} H^1(C_0) \xrightarrow{\mathrm{sp}}{} H^1_{\mathrm{lim}}(C_t) \xrightarrow{N}H^1_{\mathrm{lim}}(C_t). \end{align*} $$

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

$$ \begin{align*} \sum_{\mathfrak{s} \in S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1}) = h^{1,1}_{\mathrm{lim}}(C_t) + h^{1,1}_{\mathrm{ph}}(C_0) \end{align*} $$

so our desired inequality in (3.3) follows. Notice that when the singular curve $C_0$ has a connected normalization,

$$ \begin{align*} H^{2}_{\mathrm{ph}}(C_0)^{\vee}\cong \mathrm{coker}\{H_2(C_t)_{\mathrm{lim}}\rightarrow H_2(C_0)\}. \end{align*} $$

is trivial, hence in this case, we have

$$ \begin{align*} \sum_{\mathfrak{s}\in S}\mathrm{dim}(V_{f,\mathfrak{s}}^{1,1}) = h^{1,1}_{\mathrm{lim}}(C_t). \end{align*} $$

To conclude the proof, we recall that the sequence (3.4) 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 (3.3) also holds for every $\lambda ^k$ -eigenspace.

Proof. By Lemma 3.5, we can suppose that the central fiber of the family $\mathcal {C} \to \Delta $ has a 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

(3.7) $$ \begin{align} 0 \xrightarrow[]{} H^1_k(C_0) \xrightarrow{\mathrm{}}{} H^1_{\mathrm{lim}}(C_t)_k \xrightarrow{\mathrm{}} (V_f)_k \end{align} $$

which can be visualized via Hodge-Deligne diagrams as follows. Therefore, 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.

3.3 Proof of Proposition 1.1 and Theorem 1.2

Proof of Proposition 1.1

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 $D_{I}$ with $I\subset \{1, \ldots ,n\}$ , $2 \leq |I| \leq \lfloor \frac {n}{2} \rfloor $ . For any of such divisors, we consider its interior

$$\begin{align*}D_{I}^* := D_{I} \setminus \bigcup_{I \neq J} D_{J} \end{align*}$$

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

$$\begin{align*}\mathcal{I}^{\mathrm{pr}} := \{ I \; | \; N_{I} \; \text{ has finite order} \}. \end{align*}$$

Since $\overline {M}_{0,n}$ is a smooth and normal crossing compactification of $M_{0,n}$ , it is a classical result by Griffiths, see [Reference Griffiths12, Theorem 9.5], that $\Phi _{n,d,k}$ can be extended uniquely and holomorphically to

$$\begin{align*}M_{n,d,k}^{\mathrm{pr}}:=M_{0,n} \bigcup (\cup_{I \in \mathcal{I}^{pr}} D_{}^{*}) \subset \overline{M}_{0,n}. \end{align*}$$

This shows Proposition 1.1.

Proof of Theorem 1.2

Let $\Delta $ be a $1$ -dimensional subfamily of $\overline {M}_{ 0, \mathbf {w} }^{\mathrm {GIT}}$ 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 $\overline {M}_{0,n}$ centered at some $\tilde {p}\in \pi ^{-1}(p)$ 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 $\tilde {p}$ is also finite. Therefore, the space $\pi (M_{n,d,k}^{\mathrm {pr}})\subset \overline {M}_{ 0, \mathbf {w} }^{\mathrm {GIT}}$ contains exactly all points $p\in \overline {M}_{ 0, \mathbf {w} }^{\mathrm {GIT}}$ 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. Next, we compute separately compute the two functions in Theorem 1.2 separately.

Case 1: $d|n$ : For $\frac {n}{2} \geq |I| \geq 2$ , the morphism $\pi : \overline {M}_{0,n} \to \overline {M}_{0,\mathbf {w}}^{\mathrm {GIT}}$ contracts all divisors $D_{I,I^c}$ with $|I|=l$ , and their image $\pi \left ( D_I \right )$ is of codimension $|I|-1 = (l-1)$ in $\overline {M}_{0,\mathbf {w}}^{\mathrm {GIT}}$ for $l \leq \lfloor \frac {n-1}{2}\rfloor $ . Particularly, if $l=\frac {n}{2}$ , this implies $D_I$ is contracted to $0$ -dimensional boundary strata. By Lemma 3.6 and Corollary 3.4, the LMHS of our family is pure if and only if d does not divide $kl$ . 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 $\pi (M_{n,d,k}^{\mathrm {pr}})=\overline {M}_{0,\mathbf {w}}^{\mathrm {GIT}}$ (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 $(\overline {M}_{0,\mathbf {w}}^{\mathrm {GIT}}\backslash \pi (M_{n,d,k}^{\mathrm {pr}}))=\frac {d}{\mathrm {gcd}(k,d)}-1$ .

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

Case 2: $d\nmid n$ and $\mathrm {gcd}(d,n-1)=1$ : Using the weight

$$\begin{align*}(w_1,...,w_{n-1},w_{n})= \left( \frac{1}{n-1}+\epsilon,...,\frac{1}{n-1}+\epsilon, 1-(n -1) \epsilon \right) \end{align*}$$

with $0< \epsilon \ll \frac {1}{n^2}$ , the complement of the image of $M_{0,n}$ in $\overline {M}^{\mathrm {GIT}}_{\mathbf {w}}/S_{n-1}$ can be stratified by the partition of $[n-1]$ , namely $S_1\sqcup S_2\sqcup ... \sqcup S_r=\{1,2,...,n-1\}$ with $\mathrm {Max}\{|S_1|,...,|S_r|\}\leq n-2$ . More precisely, we have the following analog results:

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

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

3. 3. When gcd $(k,d)=\frac {d}{n-2}$ , for $l=(n-2)$ it holds that $d | kl$ . Then, the only divisors where the LMHS may not be pure is $D_{I}$ with $I^c = \{ i,n \}$ . Then, $\pi (D_I)$ parametrizes configurations where the $(n-2)$ points $\{ p_i | i \notin I \}$ collide. This has codimension $(n-3)$ .

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 with N, see [Reference Kato and Usui20]. More precisely,

(4.1)

where we denote $\check {D}_{n,d,k}$ as the compact dual of $D_{n,d,k}$ , and

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

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

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

Where H is the underlying $\mathbb {Q}$ -vector space of the Hodge structure.

To interpret such an 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 an 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 as 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 is 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 [Reference Kerr and Pearlstein18, Chapter 7], the boundary components parametrizing our LMHS are

(4.2) $$ \begin{align} B(N) &\cong e^{\langle N\rangle_{\mathbb{C}}} \backslash Z(N)\cap(U_W(\mathbb{C})\rtimes P_W(\mathbb{R}))/Z(N)\cap \mathrm{Stab}_{F^{\bullet}}(\mathbb{C}) \\ B^0(N) & \cong B(N)/Z(N)\cap U_W(\mathbb{C}). \notag \end{align} $$

Figure 4.1

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(i)

The case when the domain $D_{n,d,k}$ is a complex ball is proved in [Reference Gallardo, Kerr and Schaffler10, Section 3.3]. Their arguments still hold 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 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 $\overline {M}_{0,n}$ 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 [Reference Kato and Usui20, Theorem B], we have a unique extension to the partial compactification

$$ \begin{align*} \Phi_{n,d,k}^{ \mathrm{nm} }: \overline{M}^{\mathrm{nm}}_{0,n} \dashrightarrow \left( \Gamma^{\mathrm{nm}}_{n,d,k}\backslash D_{n,d,k} \right) \bigsqcup_{ N\in \Gamma^{\mathrm{nm}}_{n,d,k} \backslash \Sigma}(\Gamma_N\backslash B(N)) \end{align*} $$

where the fan $\Sigma $ is constructed 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 gives rise to a Kato–Usui type extension to codimension $1$ strata.

4.3 Proof of Theorem 1.3(ii)

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:

(4.3) $$ \begin{align} \mathcal{R}:& \mathcal{D}^{\circ}\rightarrow M_{0,r}/S_r\times M_{0,r}/S_r, && \mathcal{D}^{\circ} \subset \overline{M}_{0,\frac{2}{n}+\epsilon}^{\circ} \end{align} $$

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

(4.4) $$ \begin{align} \left( X, \sum_{i=1}^{2r} \left( \frac{1}{2} + \epsilon \right)x_i \right) \mapsto \left( \mathbb{P}^1, \sum_{i=1}^{r} x_i \right) , \left( \mathbb{P}^1, \sum_{i=r+1}^{2r} x_i \right). \end{align} $$

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.

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 $q_i:=q_{1,i}+\cdots +q_{d,i}$ in $C_i$ is $\rho $ -invariant - recall $\rho $ is the automorphism of C associated with the finite cover $\mathcal {C} \mapsto X$ , see Figure 2.

Figure 2

Normalization $C_1 \sqcup C_2$ of the branched cover C. The dashed lines indicate the points identified at C.

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 [Reference Carlson2, Section 3.b], we obtain the exact diagram of mixed Hodge structures given by Figure 3.

Figure 3

Exact diagram of LMHS associated with C.

The diagram in Figure 3 implies that

(4.5) $$ \begin{align} H_1(C_1)_k\bigoplus H_1(C_2)_k\cong \mathrm{Gr}_1H^1_{\mathrm{lim}}(C_t) \end{align} $$

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 that 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 $\mathcal {D}$ . It suffices to check a local $2$ -parameter family:

(4.6) $$ \begin{align} \Phi: \Delta\times \Delta\rightarrow B(N_0) \end{align} $$

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:

(4.7) $$ \begin{align} \Phi^0: \Delta\times \Delta\rightarrow B^0(N_0) \end{align} $$

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

(4.8) $$ \begin{align} (W_{N_0}=:W_0, F^{\bullet}, N_0)\in B(N_0)\twoheadrightarrow (\mathrm{Gr}(W_{0}), F^{\bullet}, N_0)\in B^0(N_0). \end{align} $$

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 4.2, we have:

(4.9) $$ \begin{align} \mathrm{Img}(\Phi)\subset e^{\langle N_0\rangle_{\mathbb{C}}} \backslash Z(N_0)\cap U_{W_0}(\mathbb{C}) /Z(N_0)\cap P_{F_0^{\bullet}}(\mathbb{C}) \end{align} $$

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 split 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 does not change the local behavior, we consider the local extended period map of the ordered case, that is, 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 $H_1(C_1,\{q_{i,1}\};\mathbb {Z})$ :

$$\begin{align*}\gamma^1_+:=\gamma^1_{q_1}+\zeta_d^{-1}\rho \gamma^1_{q_1}+\cdots+\zeta_d^{-(d-1)}\rho^{d-1} \gamma^1_{q_1}\\ \gamma^1_-:=\gamma^1_{q_1}+\zeta_d^{}\rho \gamma^1_{q_1}+\cdots+\zeta_d^{(d-1)}\rho^{d-1} \gamma^1_{q_1}. \end{align*}$$

It is clear $\rho (\gamma ^1_{\pm })=\zeta _d^{\pm 1}\gamma ^1_{\pm }$ . If we do a 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 2.6, we have:

(4.10) $$ \begin{align} H^1(C_1)_k \xrightarrow[]{\cong} \Omega^1(C_1)_{\pm k}, \end{align} $$

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

(4.11) $$ \begin{align} q_{1,1}\xrightarrow[]{\gamma^1_{q_1}} \gamma^1_{\pm} \xrightarrow[]{\tau} \Omega^1(C_1)_{k=\pm 1}^{\vee}. \end{align} $$

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:

$$ \begin{align*} f_{\omega}: U_1 \longrightarrow \mathbb{C}, && f_{\omega}(q_{1,1}):=\int_{\gamma^1_{q_1}}\omega \end{align*} $$

is locally injective. This is because

(4.12) $$ \begin{align} \int_{\gamma_+^1}\omega = \sum_{j=0}^{d-1}\zeta_d^{-j}\int_{\rho^{j-1}\gamma_{q_{1}}^1}\omega = \sum_{j=0}^{d-1}\zeta_d^{-j}\zeta_{d}^{j}\int_{\gamma_{q_1}^1}w = d\int_{\gamma_{q_1}^1}\omega. \end{align} $$

In other words, we need to show for any point $q_{1,1}$ there exists some $\omega \in \Omega ^1(C_1)_{k=\pm 1}$ such that $\omega (q_{1,1})\neq 0$ . This will imply $f_{\omega }$ is conformal around any $q_{1,1}$ .

By Lemma 2.8, $\Omega ^{1}(C_1)_{-1}$ is popularized by:

(4.13) $$ \begin{align} \frac{z^{j-1}dz}{y^{d-1}} \ , \ 0 <j/r\neq (d-1)/d<1. \end{align} $$

In our case where we assume $d|r$ , $r\geq 3$ and $k=1$ , by checking the formula [Reference McMullen22, (3.2)], the only possible zero for $dz/y^{d-1}\in \Omega ^{1}(C_1)_{-1}$ is $\widetilde {\infty }$ , the preimage of $\infty $ under the branched covering map $C_1\rightarrow \mathbb {P}^1$ , while $\widetilde {\infty }$ is not a zero of the form $z^{\frac {n}{d}-1}dz/y^{d-1}\in \Omega ^{1}(C_1)_{-1}$ . Therefore, $\Omega ^1(C_1)_{k=\pm 1}$ is basepoint-free. We finish the proof of the theorem by showing the same statement on the component $C_2$ .