On the irreducible components of the singular locus of $A_{g}$. II
HTML articles powered by AMS MathViewer
- 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 $p=5$.References
- H. Bennama and J. Bertin, Remarques sur les varietés abéliennes avec un automorphisme d’ordre premier, Manuscripta Math. 94 (1997), no. 4, 409–425 (French). MR 1484635, DOI 10.1007/BF02677863
- Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673, DOI 10.1007/978-3-662-06307-1
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- V. Gonzalez-Aguilera, J. M. Muñoz-Porras, and A. G. Zamora, On the irreducible components of the singular locus of $A_g$, J. Algebra 240 (2001), no. 1, 230–250. MR 1830552, DOI 10.1006/jabr.2000.8707
- V. González-Aguilera, J. M. Muñoz-Porras, and Alexis G. Zamora, On the 0-dimensional irreducible components of the singular locus of $\scr A_g$, Arch. Math. (Basel) 84 (2005), no. 4, 298–303. MR 2135039, DOI 10.1007/s00013-004-1193-x
- V. González-Aguilera, J. M. Muñoz-Porras, and Alexis G. Zamora, Some recent results on the irreducible components of the singular locus of $A_g$, The geometry of Riemann surfaces and abelian varieties, Contemp. Math., vol. 397, Amer. Math. Soc., Providence, RI, 2006, pp. 119–125. MR 2218002, DOI 10.1090/conm/397/07466
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- Frans Oort, Finite group schemes, local moduli for abelian varieties, and lifting problems, Compositio Math. 23 (1971), 265–296. MR 301026
- Frans Oort, Singularities of coarse moduli schemes, Séminaire d’Algèbre Paul Dubreil, 29ème année (Paris, 1975–1976), Lecture Notes in Math., vol. 586, Springer, Berlin, 1977, pp. 61–76. MR 518011
- Irving Reiner, Integral representations of cyclic groups of prime order, Proc. Amer. Math. Soc. 8 (1957), 142–146. MR 83493, DOI 10.1090/S0002-9939-1957-0083493-6
- Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449, DOI 10.1515/9781400883943
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
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