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), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr), 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 .