Eigenperiods and the moduli of points in the line

TL;DR

This paper investigates the period map of point configurations on the projective line, analyzing its Hodge structure decomposition, extension properties, and Torelli map behavior within a specific moduli space.

Contribution

It introduces a detailed analysis of the eigenperiod map for cyclic covers of points on the projective line and its extension to a moduli space of weighted stable rational curves.

Findings

Determines the codimension of loci where the eigenperiod map remains pure.

Shows the period map extends to divisors of a moduli space of weighted stable rational curves.

Establishes the local Torelli property along fibers of the extended period map.

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.