Genus three curves and 56 nodal sextic surfaces
Authors:
Bert van Geemen and Yan Zhao
Journal:
J. Algebraic Geom. 27 (2018), 583-592
DOI:
https://doi.org/10.1090/jag/694
Published electronically:
March 30, 2018
MathSciNet review:
3846548
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Abstract |
References |
Additional Information
Abstract: Catanese and Tonoli showed that the maximal cardinality for an even set of nodes on a sextic surface is 56 and they constructed such nodal surfaces. In this paper we give an alternative, rather simple, construction for such surfaces starting from non-hyperelliptic genus three curves. We illustrate our method by giving explicitly the equation of such a sextic surface starting from the Klein curve.
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. Computational algebra and number theory (London, 1993). MR 1484478, DOI https://doi.org/10.1006/jsco.1996.0125
- Sonia Brivio and Alessandro Verra, The theta divisor of ${\rm SU}_C(2,2d)^s$ is very ample if $C$ is not hyperelliptic, Duke Math. J. 82 (1996), no. 3, 503–552. MR 1387683, DOI https://doi.org/10.1215/S0012-7094-96-08222-8
- G. Casnati and F. Catanese, Even sets of nodes are bundle symmetric, J. Differential Geom. 47 (1997), no. 2, 237–256. MR 1601608
- Fabrizio Catanese and Fabio Tonoli, Even sets of nodes on sextic surfaces, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 705–737. MR 2341829, DOI https://doi.org/10.4171/JEMS/94
- Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
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. Computational algebra and number theory (London, 1993). MR 1484478, DOI https://doi.org/10.1006/jsco.1996.0125
- Sonia Brivio and Alessandro Verra, The theta divisor of $\textrm {SU}_C(2,2d)^s$ is very ample if $C$ is not hyperelliptic, Duke Math. J. 82 (1996), no. 3, 503–552. MR 1387683, DOI https://doi.org/10.1215/S0012-7094-96-08222-8
- G. Casnati and F. Catanese, Even sets of nodes are bundle symmetric, J. Differential Geom. 47 (1997), no. 2, 237–256. MR 1601608
- Fabrizio Catanese and Fabio Tonoli, Even sets of nodes on sextic surfaces, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 705–737. MR 2341829, DOI https://doi.org/10.4171/JEMS/94
- Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
Additional Information
Bert van Geemen
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italia
MR Author ID:
214021
Email:
lambertus.vangeemen@unimi.it
Yan Zhao
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italia; and Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333CA Leiden, The Netherlands
Received by editor(s):
March 29, 2016
Received by editor(s) in revised form:
June 20, 2016
Published electronically:
March 30, 2018
Article copyright:
© Copyright 2018
University Press, Inc.
