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 $$(-2,3),(-1,-4),(2,5),(4,9),(52,375),(5234,37866),(8,-23),(43,282),(\frac{1}{4},-\frac{33}{8})$$ 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 is a Brill-Noether general curve of genus $g$, while a general curve in that $g$-dimensional linear system is a Brill-Noether-Petri general curve of genus $g$.

New section added containing an explicit example of a 9-tuple of points in P^2(Q) that are of 3g-general for every g. Added a second proof of the fact that a du Val curve is BN general. Improved exposition