×

The local-global property for bitangents of plane quartics. (English) Zbl 1448.14023

Summary: We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans-Jahnel’s results [A.-S. Elsenhans and J. Jahnel, Eur. J. Math. 5, No. 4, 1156–1172 (2019; Zbl 1473.14052)] on the inverse Galois problem for bitangents, we show that, over any global field of characteristic different from \(2\), there exist smooth quartics which have bitangents over every local field, but do not have bitangents over the global field. We give an algorithm to find such quartics explicitly, and give an example over \(\mathbb{Q}\). We also discuss a similar problem concerning symmetric determinantal representations.

MSC:

14H25 Arithmetic ground fields for curves
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties
11G35 Varieties over global fields

Citations:

Zbl 1473.14052

Software:

GAP; SageMath; Maxima
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012. · Zbl 1252.14001
[2] N. Bruin; B. Poonen; M. Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma, 4, 80 pages (2016) · Zbl 1408.11065
[3] A-S. Elsenhans; J. Jahnel, On plane quartics with a Galois invariant Cayley octad, Eur. J. Math., 5, 1156-1172 (2019) · Zbl 1473.14052
[4] A-S. Elsenhans; J. Jahnel, Plane quartics with a Galois-invariant Steiner hexad, Int. J. Number Theory, 15, 1075-1109 (2019) · Zbl 1428.14047
[5] J. Jahnel; D. Loughran, The Hasse principle for lines on del Pezzo surfaces, Int. Math. Res. Not., 2015, 12877-12919 (2015) · Zbl 1331.14030
[6] D. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. Éc. Norm. Supér, 4, 181-192 (1971) · Zbl 0216.05904
[7] Y. Ishitsuka; T. Ito, The local-global principle for symmetric determinantal representations of smooth plane curves, Ramanujan J., 43, 141-162 (2017) · Zbl 1390.14086
[8] J. Harris, Galois groups of enumerative problems, Duke Math. J., 46, 685-724 (1979) · Zbl 0433.14040
[9] T. Shioda, Plane quartics and Mordell-Weil lattices of type \(E_7\), Comment. Math. Univ. St. Pauli, 42, 61-79 (1993) · Zbl 0790.14025
[10] R. Erné, Construction of a del Pezzo surface with maximal Galois action on its Picard group, J. Pure Appl. Algebra, 97, 15-27 (1994) · Zbl 0840.14020
[11] J. Sonn, Polynomials with roots in \(\Q_p\) for all \(p\), Proc. Amer. Math. Soc., 136, 1955-1960 (2008) · Zbl 1195.12007
[12] Y. Ishitsuka; T. Ito; T. Ohshita, On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree, JSIAM Letters, 11, 9-12 (2019) · Zbl 1409.14098
[13] Y. Ishitsuka and T. Ito, The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two, J. Pure Appl. Algebra, 221 (2017), 1316-1321. · Zbl 1368.14046
[14] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.10.2, 2019, https://www.gap-system.org.
[15] Maxima.sourceforge.net, Maxima, a Computer Algebra System (Version 5.41.0), 2017, http://maxima.sourceforge.net.
[16] SageMath, the Sage Mathematics Software System (Version 8.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.