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Summary: EXTENDED TORELLI MAP TO THE
IGUSA BLOWUP IN GENUS 6, 7, AND 8
VALERY ALEXEEV, RYAN LIVINGSTON, JOSEPH TENINI, MAXIM ARAP, XIAOYAN
HU, LAUREN HUCKABA, PATRICK MCFADDIN, STACY MUSGRAVE, JAEHO SHIN,
AND CATHERINE ULRICH
Abstract. It was conjectured in [Nam73] that the Torelli map Mg Ag
associating to a curve its jacobian extends to a regular map from the Deligne-
Mumford moduli space of stable curves Mg to the (normalization of the) Igusa
blowup A
cent
g . A counterexample in genus g = 9 was found in [AB11]. Here, we
prove that the extended map is regular for all g 8, thus completely solving
the problem in every genus.
1. Introduction
The Torelli map Mg Ag associates to a smooth curve C its jacobian JC, a
principally polarized abelian variety. Does it extend to a regular map Mg Ag,
where Mg is Deligne-Mumford's moduli space of stable curves, and Ag is a toroidal
compactification of Ag?
This question was first asked in a pioneering paper of Namikawa [Nam73] in the
case when Ag = A
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