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Eigenperiods and the moduli of points in the line

Haohua Deng, Patricio Gallardo

TL;DR

This work analyzes eigenperiod maps arising from $d$-fold cyclic covers branched at $n$ marked points on $\mathbb{P}^1$, focusing on how the Hodge-theoretic structure behaves under degenerations. By exploiting eigenspace decompositions, LMHS theory, and eigenspectra, the authors describe precisely when the eigenperiod map remains pure (finite monodromy) and compute the codimension of the non-pure locus within GIT and Hassett compactifications. They then establish a Kato–Usui toroidal-type extension of the eigenperiod map to codimension-1 boundary strata and interpret boundary data via Abel–Jacobi-type constructions on degenerations, proving a local Torelli-type result along Hassett divisors for certain cases. Overall, the paper extends Deligne–Mostow-type phenomena to a broader class of point configurations, clarifying how degenerations of point configurations influence Hodge-theoretic period maps and their extensions.

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.

Eigenperiods and the moduli of points in the line

TL;DR

This work analyzes eigenperiod maps arising from -fold cyclic covers branched at marked points on , focusing on how the Hodge-theoretic structure behaves under degenerations. By exploiting eigenspace decompositions, LMHS theory, and eigenspectra, the authors describe precisely when the eigenperiod map remains pure (finite monodromy) and compute the codimension of the non-pure locus within GIT and Hassett compactifications. They then establish a Kato–Usui toroidal-type extension of the eigenperiod map to codimension-1 boundary strata and interpret boundary data via Abel–Jacobi-type constructions on degenerations, proving a local Torelli-type result along Hassett divisors for certain cases. Overall, the paper extends Deligne–Mostow-type phenomena to a broader class of point configurations, clarifying how degenerations of point configurations influence Hodge-theoretic period maps and their extensions.

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.
Paper Structure (13 sections, 21 theorems, 82 equations, 3 figures, 1 table)

This paper contains 13 sections, 21 theorems, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric compactifications
  4. Eigenperiod map for $n$ points in the line
  5. Eigenspectra of isolated hypersurface singularities
  6. Describing the locus with finite monodromy
  7. Lemmas on eigenspectra
  8. Calculation of Limiting Mixed Hodge Structures
  9. Proof of Proposition \ref{['prop:extension']} and Theorem \ref{['thm:codimensions']}
  10. Hodge-theoretic interpretation of exceptional boundary components
  11. Preliminaries and Notation
  12. Proof of Theorem \ref{['thm:KU']}(i)
  13. Proof of Theorem \ref{['thm:KU']}(ii)

Key Result

Proposition 1.1

Let $n \geq 5$ and $k/d$ be as specified in Table tab:ancestorCases, and suppose that $\mathrm{gcd}(d,n-1)=1$ holds. There exists an eigenperiod map coming from the family of cyclic cover of $\mathbb{P}^1$ branched over marked points, see Proposition prop:EigenPeriodM0n, such that

Figures (3)

  • Figure 1: Hodge-Deligne diagram for boundary type zero (left), type I (center) and the adjoint Hodge-Deligne diagram for type I degeneration (right)
  • Figure 2: Branched cover $C$ and its normalization
  • Figure 3: Exact diagram of LMHS associated to $C$

Theorems & Definitions (40)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 30 more