Skip to main content
×
×
Home

Sharp Bertini Theorem for Plane Curves over Finite Fields

  • Shamil Asgarli (a1)
Abstract

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$ , then there is an $\mathbb{F}_{q}$ -line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$ .

Copyright
References
Hide All
[1] Ballico, E., An effective Bertini theorem over finite fields . Adv. Geom. 3(2003), no. 4, 361363. https://doi.org/10.1515/advg.2003.020.
[2] Borges, H., On complete (N, d)-arcs derived from plane curves . Finite Fields Appl. 15(2009), no. 1, 8296. https://doi.org/10.1016/j.ffa.2008.08.003.
[3] Giulietti, M., Pambianco, F., Torres, F., and Ughi, E., On complete arcs arising from plane curves . Des. Codes Cryptogr. 25(2002), no. 3, 237246. https://doi.org/10.1023/A:1014979211916.
[4] Hefez, A. and Voloch, J. F., Frobenius nonclassical curves . Arch. Math. 54(1990), no. 3, 263273. https://doi.org/10.1007/BF01188523.
[5] Hirschfeld, J. W. P., Korchmáros, G., and Torres, F., Algebraic curves over a finite field. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2008.
[6] Homma, M. and Kim, S. J., Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: supplements to a work of Tallini . Linear Algebra Appl. 438(2013), no. 3, 969985. https://doi.org/10.1016/j.laa.2012.08.032.
[7] Pardini, R., Some remarks on plane curves over fields of finite characteristic . Compositio Math. 60(1986), no. 1, 317.
[8] Poonen, B., Bertini theorems over finite fields . Ann. of Math. 160(2004), no. 3, 10991127. https://doi.org/10.4007/annals.2004.160.1099.
[9] Wallace, A. H., Tangency and duality over arbitrary fields . Proc. London Math. Soc. 6(1956), 321342. https://doi.org/10.1112/plms/s3-6.3.321.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed