Experimental Mathematics

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

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It was conjectured in Yukihiko Namikawa, “On the Canonical Holomorphic Map from the Moduli Space of Stable Curves to the Igusa Monoidal Transform,” 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 $\bar{A}^{\rm cent}_g$. A counterexample in genus $g = 9$ was found in Valery Alexeev and Adrian Brunyate, “Extending Torelli Map to Toroidal Compactifications of Siegel Space.” Here, we prove that the extended map is regular for all $g \le 8$, thus completely solving the problem in every genus.

Article information

Experiment. Math., Volume 21, Issue 2 (2012), 193-203.

First available in Project Euclid: 31 May 2012

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Zentralblatt MATH identifier

Primary: 14D22: Fine and coarse moduli spaces 14H10: Families, moduli (algebraic) 14K10: Algebraic moduli, classification [See also 11G15] 14M25: Toric varieties, Newton polyhedra [See also 52B20] 05C75: Structural characterization of families of graphs 11E12: Quadratic forms over global rings and fields 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20]

Torelli map abelian varieties curves compactified moduli spaces graphs toroidal


Alexeev, Valery; Livingston, Ryan; Tenini, Joseph; Arap, Maxim; Hu, Xiaoyan; Huckaba, Lauren; McFaddin, Patrick; Musgrave, Stacy; Shin, Jaeho; Ulrich, Catherine. Extended Torelli Map to the Igusa Blowup in Genus 6, 7, and 8. Experiment. Math. 21 (2012), no. 2, 193--203. https://projecteuclid.org/euclid.em/1338430830

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