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LINEAR SYSTEMS OF GEOMETRICALLY IRREDUCIBLE PLANE CUBICS OVER FINITE FIELDS

Part of: Arithmetic problems. Diophantine geometry Curves Cycles and subschemes Projective and enumerative geometry

Published online by Cambridge University Press:  27 February 2025

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 and Computer Science, Santa Clara University, 500 El Camino Real, CA 95053, USA e-mail: sasgarli@scu.edu
DRAGOS GHIOCA*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
e-mail: dghioca@math.ubc.ca
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Abstract

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb {F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a three-dimensional linear system $\mathcal {L}$ with at most one geometrically reducible $\mathbb {F}_q$-member.

Keywords

linear systemcubic curvegeometric irreducibilityfinite field

MSC classification

Primary: 14N05: Projective techniques
Secondary: 14C21: Pencils, nets, webs 14H50: Plane and space curves 14G15: Finite ground fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 9
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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