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On the irreducible components of the singular locus of $A_{g}$. II


Authors: V. González-Aguilera, J. M. Munoz-Porras and A. G. Zamora
Journal: Proc. Amer. Math. Soc. 140 (2012), 479-492
MSC (2010): Primary 14K10, 14K22; Secondary 14D15
DOI: https://doi.org/10.1090/S0002-9939-2011-10933-X
Published electronically: June 16, 2011
MathSciNet review: 2846316
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Abstract: In our earlier paper it was proved that the singular locus of $A_{g}$ (coarse moduli space of principally polarized abelian varieties over $\mathbb {C}$) is expressed as the union of irreducible varieties $A_{g}(p,\alpha )$ representing abelian varieties with an order $p$ automorphism of fixed entire representation. In this paper we prove that $A_{g}(p,\alpha )$ is an irreducible component of $\text {Sing} A_{g}$ if and only if for a general element of this variety its automorphism group modulo $\{\pm 1\}$, $G_{+}$, satisfies the equivalent conditions: $G_{+}=\langle \alpha \rangle$ or $N_{G_{+}}(\langle \alpha \rangle )=\langle \alpha \rangle$. We illustrate how these results can be used by studying the case $g=4$ and $p=5$.


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Additional Information

V. González-Aguilera
Affiliation: Departamento de Matemáticas, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
Email: victor.gonzalez@usm.cl

J. M. Munoz-Porras
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, Salamanca 37008, Spain
Email: jmp@usal.es

A. G. Zamora
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Zacatecas, Camino a la Bufa y Calzada Solidaridad, C.P. 98000, Zacatecas, Zac., México
MR Author ID: 627620
Email: alexiszamora06@gmail.com

Received by editor(s): May 4, 2010
Received by editor(s) in revised form: September 17, 2010, and November 30, 2010
Published electronically: June 16, 2011
Additional Notes: The first author was partially supported by Fondecyt Grant 1080030 and UTFSM’s DGIP
The third author was partially supported by CoNaCyT Grant 25811
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society