×

Polynomial curves on trinomial hypersurfaces. (English) Zbl 1470.14099

Summary: We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwarz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwarz-Halphen’s Theorem for Pham-Brieskorn surfaces.

MSC:

14M20 Rational and unirational varieties
14R20 Group actions on affine varieties
11D41 Higher degree equations; Fermat’s equation
14J50 Automorphisms of surfaces and higher-dimensional varieties
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox Rings, Cambridge Stud. Adv. Math. 144, Cambridge Univ. Press, New York, 2015. [4] I. Arzhantsev and S. Gaifullin, The automorphism group of a rigid affine variety, Math. Nachr. 290 (2017), 662-671.
[2] I. Arzhantsev and M. Zaidenberg, Acyclic curves and group actions on affine toric surfaces, in: Affine Algebraic Geometry (Osaka, 2011), K. Masuda, et al. (eds.), World Sci., 2013, 1-41. · Zbl 1319.14037
[3] G. Barthel et L. Kaup, Topologie des surfaces complexes compactes singuli‘eres, in: Sur la topologie des surfaces complexes compactes, in: S´em. Math. Sup. 80, Presses Univ. Montr´eal, Qu´e., 1982, 61-297. [7] A. Evyatar, On polynomial equations, Israel J. Math. 10 (1971), 321-326. [8] H. Flenner and M. Zaidenberg, Rational curves and rational singularities, Math. Z. 244 (2003), 549-575.
[4] G. H. Halphen, Sur la r´eduction des ´equations diff´erentielles lin´eaires aux formes int´egrables, M´emoires pr´esent´es par divers savants ‘a l’Academie des sciences de l’Institut National de France, T. XXVIII, N. 1, Paris, F. Krantz, 1883; Oeuvres, Vol. 3, Paris, 1921, 1-260.
[5] J. Hausen and E. Herppich, Factorially graded rings of complexity 1, in: Torsors, Etale Homotopy and Applications to Rational Points, London Math. Soc. Lecture´ Note Ser. 405, Cambridge Univ. Press, 2013, 414-428. · Zbl 1290.13001
[6] J. Hausen, E. Herppich, and H. S¨uß, Multigraded factorial rings and Fano varieties with torus action, Doc. Math. 16 (2011), 71-109 [12] J. Hausen and H. S¨uß, The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), 977-1012. [13] J. Hausen and M. Wrobel, Non-complete rational T-varieties of complexity one, Math. Nachr. 290 (2017), 815-826. [14] J. Hausen and M. Wrobel, On iteration of Cox rings, J. Pure Appl. Algebra 222 (2018), 2737-2745. [15] S. Kaliman and M. Zaidenberg, Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions, Ann. Inst. Fourier (Grenoble) 50 (2000), 1649-1669.
[7] F. Klein, Vorlesungen ¨uber das Ikosaeder und die Aufl¨osung der Gleichungen vom f¨unften Grade, Teubner, Leipzig, 1884; English transl.: Lectures on the Icosahedron and the Solution of Equations of Fifth Degree, Dover, 1956. [17] R. C. Mason, Diophantine Equations over Function Fields, London Math. Soc. Lecture Note Ser. 96, Cambridge Univ. Press, Cambridge, 1984.
[8] V. Popov and E. Vinberg, Invariant theory, in: Algebraic Geometry IV, Encyclopaedia Math. Sci. 55, Springer, Berlin, 1994, 123-284. [19] V. Prasolov, Polynomials, Algorithms Comput. Math. 11, Springer, Berlin, 2004.
[9] H. Schwarz, Ueber diejenigen F¨alle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292-335. [21] W. W. Stothers, Polynomial identities and Hauptmoduln, Quart. J. Math. 32 (1981), 349-370.
[10] V. P. Vel’min, Solutions of the indeterminate equation um+ vn= wk, Mat. Sbornik 24 (1904), 633-661 (in Russian). Ivan Arzhantsev National Research University Higher School of Economics Faculty of Computer Science Kochnovskiy Proezd 3 Moscow, 125319 Russia E-mail: arjantsev@hse.ru
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.