Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-01T15:09:30.240Z Has data issue: false hasContentIssue false
  • English
  • Français

SMOOTHNESS IN PENCILS OF HYPERSURFACES OVER FINITE FIELDS

Part of: Surfaces and higher-dimensional varieties Projective and enumerative geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  25 August 2022

SHAMIL ASGARLI Open the ORCID record for SHAMIL ASGARLI [Opens in a new window]  and
DRAGOS GHIOCA Open the ORCID record for DRAGOS GHIOCA [Opens in a new window]
SHAMIL ASGARLI
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada e-mail: sasgarli@math.ubc.ca
DRAGOS GHIOCA*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
e-mail: dghioca@math.ubc.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We study pencils of hypersurfaces over finite fields $\mathbb {F}_q$ such that each of the $q+1$ members defined over $\mathbb {F}_q$ is smooth.

Keywords

pencils of hypersurfacesfinite fieldssmoothness

MSC classification

Primary: 14N05: Projective techniques
Secondary: 14G15: Finite ground fields 14J70: Hypersurfaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 1 , February 2023 , pp. 85 - 94
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author is supported by a Postdoctoral Research Fellowship and an NSERC PDF award at the University of British Columbia. The second author is supported by an NSERC Discovery grant.

References

Asgarli, S. and Ghioca, D., ‘A Bertini type theorem for pencils over finite fields’, Finite Fields Appl. 77 (2022), Article no. 101936, 13 pages.10.1016/j.ffa.2021.101936CrossRefGoogle Scholar
Aubry, Y. and Perret, M., ‘A Weil theorem for singular curves’, in: Arithmetic, Geometry and Coding Theory (Luminy, 1993) (eds. Pellikaan, R., Perret, M. and Vladut, S. G.) (de Gruyter, Berlin, 1996), 1–7.Google Scholar
Cafure, A. and Matera, G., ‘Improved explicit estimates on the number of solutions of equations over a finite field’, Finite Fields Appl. 12(2) (2006), 155–185.CrossRefGoogle Scholar
Eisenbud, D. and Harris, J., 3264 and All That: A Second Course in Algebraic Geometry (Cambridge University Press, Cambridge, 2016).10.1017/CBO9781139062046CrossRefGoogle Scholar
Entin, A., ‘Monodromy of hyperplane sections of curves and decomposition statistics over finite fields’, Int. Math. Res. Not. IMRN 2021(14) (2014), 10409–10441.10.1093/imrn/rnz120CrossRefGoogle Scholar
Kaltofen, E., ‘Effective Noether irreducibility forms and applications’, J. Comput. System Sci. 50(2) (1995), 274–295. 23rd Symposium on the Theory of Computing (New Orleans, LA, 1991).CrossRefGoogle Scholar
Lang, S. and Weil, A., ‘Number of points of varieties in finite fields’, Amer. J. Math. 76 (1954), 819–827.CrossRefGoogle Scholar