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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the irreducible components of the singular locus of $A_{g}$. II
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by V. González-Aguilera, J. M. Munoz-Porras and A. G. Zamora PDF
Proc. Amer. Math. Soc. 140 (2012), 479-492 Request permission

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
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2846316