Surfaces with canonical map of maximum degree
Author:
Carlos Rito
Journal:
J. Algebraic Geom. 31 (2022), 127-135
DOI:
https://doi.org/10.1090/jag/761
Published electronically:
September 13, 2021
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the $\mathbb {Z}/3$-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.
References
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- Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, DOI 10.1007/BF01390086
- L. A. Borisov and J. Keum, Explicit equations of a fake projective plane, Duke Math J. 169 (2020), no. 6, 1135–1162.
- L. A. Borisov and S.-K. Yeung, Explicit equations of the Cartwright-Steger surface, Épijournal Géom. Algébrique 4 (2020), Art. 10, 13 pp.
- Donald I. Cartwright, Vincent Koziarz, and Sai-Kee Yeung, On the Cartwright-Steger surface, J. Algebraic Geom. 26 (2017), no. 4, 655–689. MR 3683423, DOI 10.1090/jag/696
- Donald I. Cartwright and Tim Steger, Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 11–13 (English, with English and French summaries). MR 2586735, DOI 10.1016/j.crma.2009.11.016
- C. Gleissner, R. Pignatelli, and C. Rito, New surfaces with canonical map of high degree, Commun. Anal. Geom. (to appear).
- J. Keum, The bicanonical map of the Cartwright-Steger surface, arXiv:1801.00733 (2018).
- Rita Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213. MR 1103912, DOI 10.1515/crll.1991.417.191
- Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321–370. MR 2289867, DOI 10.1007/s00222-007-0034-5
- Gopal Prasad and Sai-Kee Yeung, Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007) [MR2289867], Invent. Math. 182 (2010), no. 1, 213–227. MR 2672284, DOI 10.1007/s00222-010-0259-6
- Carlos Rito, A surface with canonical map of degree 24, Internat. J. Math. 28 (2017), no. 6, 1750041, 10. MR 3663791, DOI 10.1142/S0129167X17500410
- Sai-Kee Yeung, A surface of maximal canonical degree, Math. Ann. 368 (2017), no. 3-4, 1171–1189. MR 3673651, DOI 10.1007/s00208-016-1450-x
References
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. MR 1484478
- Janko Böhm, Wolfram Decker, Claus Fieker, and Gerhard Pfister, The use of bad primes in rational reconstruction, Math. Comp. 84 (2015), no. 296, 3013–3027. MR 3378860, DOI 10.1090/mcom/2951
- Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, DOI 10.1007/BF01390086
- L. A. Borisov and J. Keum, Explicit equations of a fake projective plane, Duke Math J. 169 (2020), no. 6, 1135–1162.
- L. A. Borisov and S.-K. Yeung, Explicit equations of the Cartwright-Steger surface, Épijournal Géom. Algébrique 4 (2020), Art. 10, 13 pp.
- Donald I. Cartwright, Vincent Koziarz, and Sai-Kee Yeung, On the Cartwright-Steger surface, J. Algebraic Geom. 26 (2017), no. 4, 655–689. MR 3683423, DOI 10.1090/jag/696
- Donald I. Cartwright and Tim Steger, Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 11–13 (English, with English and French summaries). MR 2586735, DOI 10.1016/j.crma.2009.11.016
- C. Gleissner, R. Pignatelli, and C. Rito, New surfaces with canonical map of high degree, Commun. Anal. Geom. (to appear).
- J. Keum, The bicanonical map of the Cartwright-Steger surface, arXiv:1801.00733 (2018).
- Rita Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213. MR 1103912, DOI 10.1515/crll.1991.417.191
- Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321–370. MR 2289867, DOI 10.1007/s00222-007-0034-5
- Gopal Prasad and Sai-Kee Yeung, Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007) [MR2289867], Invent. Math. 182 (2010), no. 1, 213–227. MR 2672284, DOI 10.1007/s00222-010-0259-6
- Carlos Rito, A surface with canonical map of degree 24, Internat. J. Math. 28 (2017), no. 6, 1750041, 10. MR 3663791, DOI 10.1142/S0129167X17500410
- Sai-Kee Yeung, A surface of maximal canonical degree, Math. Ann. 368 (2017), no. 3-4, 1171–1189. MR 3673651, DOI 10.1007/s00208-016-1450-x
Additional Information
Carlos Rito
Affiliation:
Universidade de Trás-os-Montes e Alto Douro, UTAD, Quinta de Prados, 5000-801 Vila Real, Portugal; and Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
MR Author ID:
744585
Email:
crito@utad.pt; and crito@fc.up.pt
Received by editor(s):
September 24, 2019
Received by editor(s) in revised form:
December 19, 2019
Published electronically:
September 13, 2021
Additional Notes:
This research was supported by FCT (Portugal) under the project PTDC/MAT-GEO/2823/2014, the fellowship SFRH/BPD/111131/2015 and by CMUP (UIDB/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
Article copyright:
© Copyright 2021
University Press, Inc.
