Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

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Abstract

[EN] We consider rational surfaces Z defined by divisorial valuations ¿ of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when ¿ is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given ; Partially supported by the Spanish Government Ministerio de Economia, Industria y Competitividad (MINECO), Grants MTM2015-65764-C3-2-P, MTM2016-81735-REDT, PGC2018-096446-B-C22 and BES-2016-076314, as well as by Universitat Jaume I, Grant UJI-B2018-10. ; Galindo, C.; Monserrat Delpalillo, FJ.; Moreno-Ávila, C. (2020). Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces. Revista Matemática Complutense. 33(2):349-372. https://doi.org/10.1007/s13163-019-00319-w ; S ; 349 ; 372 ; 33 ; 2 ; Abhyankar, S.S., Moh, T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II. J. Reine Angew. Math. 260, 47–83 (1973). ibid. 261 (1973), 29–54 ; Beauville, A.: Complex Algebraic Surfaces. London Mathematical Society Student Texts, vol. 34, 2nd edn. Cambridge University Press, Cambridge (1996) ; Campillo, A.: Algebroid Curves in Positive Characteristic. Lecture Notes in Mathematics, vol. 613. Springer, Berlin (1980) ; Campillo, A., Piltant, O., Reguera, A.: Curves and divisors on surfaces associated to plane curves with one place at infinity. Proc. Lond. Math. Soc. 84, 559–580 (2002) ; Casas-Alvero, E.: Singularities of Plane Curves. London Mathematical Society Lecture Note Series, vol. 276. Cambridge University Press, Cambridge (2000) ; Ciliberto, C., Farnik, M., Küronya, A., Lozovanu, V., Roé, J., Shramov, C.: Newton–Okounkov bodies sprouting on the valuative tree. Rend. Circ. Mat. Palermo 2(66), 161–194 (2017) ; Cutkosky, S.D., Ein, L., Lazarsfeld, R.: Positivity and complexity of ideal sheaves. Math. Ann. 321(2), 213–234 (2001) ; de la Rosa-Navarro, B.L., Frías-Medina, J.B., ...