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LINEAR SYSTEMS OF GEOMETRICALLY IRREDUCIBLE PLANE CUBICS OVER FINITE FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 27 February 2025, pp. 1-9
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- Article
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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.
