Du Val curves and the pointed Brill-Noether Theorem

TL;DR

This paper demonstrates that a broad class of explicit Du Val pointed curves satisfy the Brill-Noether Theorem, providing concrete examples of Brill-Noether general curves over Q and exploring properties of pencils of such curves.

Contribution

It establishes that general Du Val pointed curves meet the Brill-Noether criteria and introduces explicit constructions over Q, including for 2-pointed curves on elliptic ruled surfaces.

Findings

Du Val pointed curves satisfy the Brill-Noether Theorem.

A generic pencil of these curves avoids all Brill-Noether divisors.

Explicit examples of Brill-Noether general curves over Q are provided.

Abstract

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.