Du Val curves and the pointed Brill-Noether theorem (Q2362850)

Brill-Noether theory governs the behavior of linear series on algebraic curves. In this paper the authors study Brill-Noether theory for Du Val curves of genus \(g\) which are suitably general singular plane curves of degree \(3g\) having multiplicity \(g\) at eight points in \(\mathbb P^2\) and multiplicity \(g-1\) at a further ninth point. The main result shows that a general pointed Du Val curve satisfies the pointed Brill-Noether dimension theorem. Moreover, the authors show that a general pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. As a consequence, such Du Val curves provide explicit examples of smooth pointed Brill-Noether general curves of arbitrary genus defined over \(\mathbb Q\). The authors also establish a similar result for \(2\)-pointed curves by using curves on elliptic ruled surfaces.