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arXiv · 2407.18104

Linear system of geometrically irreducible plane cubics over finite fields

Shamil Asgarli·Dragos Ghioca
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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 $3$-dimensional linear system $\mathcal{L}$ with at most one geometrically reducible $\mathbb{F}_q$-member.

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Shamil Asgarli, Dragos Ghioca. 2024-07-25. Linear system of geometrically irreducible plane cubics over finite fields. https://arxiv.org/abs/2407.18104

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