Filling curves for ${\mathbb{P}}^1\times {\mathbb{P}}^1$

Abstract

We determine the minimal bi-degree(s) of an irreducible filling curve over ${\mathbb{F}}_q$ for ${\mathbb{P}}^1\times {\mathbb{P}}^1$. It is $(q+1,q+1)$ if $q\ne 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$.

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