Unbounded negativity on rational surfaces in positive characteristic
-
Raymond Cheng
and
Remy van Dobben de Bruyn
Abstract
We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.
Funding statement: Remy van Dobben de Bruyn was partly supported by the Oswald Veblen Fund at the Institute for Advanced Study.
Acknowledgements
We thank Johan de Jong, Joaquín Moraga, Takumi Murayama, Will Sawin, and John Sheridan for helpful discussions, and we thank the referee for suggestions to streamline the exposition.
References
[1] T. Bauer, C. Bocci, S. Cooper, S. Di Rocco, M. Dumnicki, B. Harbourne, K. Jabbusch, A. L. Knutsen, A. Küronya, R. Miranda, J. Roé, H. Schenck, T. Szemberg and Z. Teitler, Recent developments and open problems in linear series, Contributions to algebraic geometry, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2012), 93–140. 10.4171/114-1/4Search in Google Scholar
[2] T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora and T. Szemberg, Bounded negativity and arrangements of lines, Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9456–9471. 10.1093/imrn/rnu236Search in Google Scholar
[3] T. Bauer, B. Harbourne, A. L. Knutsen, A. Küronya, S. Müller-Stach, X. Roulleau and T. Szemberg, Negative curves on algebraic surfaces, Duke Math. J. 162 (2013), no. 10, 1877–1894. 10.1215/00127094-2335368Search in Google Scholar
[4] H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser, Basel 1992. 10.1007/978-3-0348-8600-0Search in Google Scholar
[5] D. Ghioca, The dynamical Mordell–Lang conjecture in positive characteristic, Trans. Amer. Math. Soc. 371 (2019), no. 2, 1151–1167. 10.1090/tran/7261Search in Google Scholar
[6] F. Hao, Weak bounded negativity conjecture, Proc. Amer. Math. Soc. 147 (2019), no. 8, 3233–3238. 10.1090/proc/14376Search in Google Scholar
[7] B. Harbourne, Global aspects of the geometry of surfaces, Ann. Univ. Paedagog. Crac. Stud. Math. 9 (2010), 5–41. Search in Google Scholar
[8] B. Harbourne, Asymptotics of linear systems, with connections to line arrangements, Phenomenological approach to algebraic geometry, Banach Center Publ. 116, Institute of Mathematics of the Polish Academy of Sciences, Warsaw (2018), 87–135. 10.4064/bc116-6Search in Google Scholar
[9] A. Langer, The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic, Duke Math. J. 165 (2016), no. 14, 2737–2769. 10.1215/00127094-3627203Search in Google Scholar
[10] F. Rodríguez Villegas, J. F. Voloch and D. Zagier, Constructions of plane curves with many points, Acta Arith. 99 (2001), no. 1, 85–96. 10.4064/aa99-1-8Search in Google Scholar
[11] T. Shioda, An example of unirational surfaces in characteristic p, Math. Ann. 211 (1974), 233–236. 10.1007/BF01350715Search in Google Scholar
[12] T. Shioda and T. Katsura, On Fermat varieties, Tohoku Math. J. (2) 31 (1979), no. 1, 97–115. 10.2748/tmj/1178229881Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field
- Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)
- Prescribing Ricci curvature on homogeneous spaces
- Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
- Ricci flow on manifolds with boundary with arbitrary initial metric
- Unbounded negativity on rational surfaces in positive characteristic
- Extending torsors on the punctured Spec(Ainf)
- Non-isomorphic 2-groups with isomorphic modular group algebras
- Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)
Articles in the same Issue
- Frontmatter
- An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field
- Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)
- Prescribing Ricci curvature on homogeneous spaces
- Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
- Ricci flow on manifolds with boundary with arbitrary initial metric
- Unbounded negativity on rational surfaces in positive characteristic
- Extending torsors on the punctured Spec(Ainf)
- Non-isomorphic 2-groups with isomorphic modular group algebras
- Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)
