Explicit Brill–Noether–Petri general curves

Abstract

Let $p_1,\dots ,p_9$ be the points in ${\mathbb{A}}^2(\mathbb{Q})\subset {\mathbb{P}}^2(\mathbb{Q})$ with coordinates

respectively. We prove that, for any genus $g$, a plane curve of degree $3g$ having a $g$-tuple point at $p_1,\dots ,p_8$, and a $(g-1)$-tuple point at $p_9$, and no other singularities, exists and that the general plane curve of that degree and with those singularities is a Brill–Noether–Petri general curve of genus $g$.