Explicit Brill-Noether-Petri general curves

TL;DR

The paper constructs explicit examples of Brill-Noether-Petri general curves of any genus as plane curves with specific singularities at nine given points, demonstrating their existence and generality.

Contribution

It provides explicit constructions of Brill-Noether-Petri general curves of arbitrary genus using plane curves with prescribed singularities at nine points.

Findings

Existence of plane curves with specified singularities for all genera.

Such curves are Brill-Noether general.

A general curve in the constructed linear system is Brill-Noether-Petri general.

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 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 $g$.