Extended Torelli map to the Igusa blowup in genus 6, 7, and 8

TL;DR

This paper proves that the Torelli map extends regularly to the Igusa blowup for all genera up to 8, resolving a longstanding conjecture and completing the picture for these cases.

Contribution

It confirms the regularity of the extended Torelli map to the Igusa blowup for all genera up to 8, filling the gap left by previous counterexamples in genus 9.

Findings

The extended Torelli map is regular for all g ≤ 8.

Counterexample exists only at genus 9.

Complete solution to the extension problem for low genera.

Abstract

It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.