A remark on the Brill–Noether theory of curves of fixed gonality

Abstract

Recently the Brill–Noether theory of curves C of both fixed genus and gonality was established. In particular, in this theory (now called the Hurwitz–Brill–Noether theory), all irreducible components of the variety of complete linear series of a fixed degree and dimension on C are obtained from the closures of certain so-called “Brill–Noether splitting loci” (loci which have a rather succinct description). In this paper, a method previously invented for the construction of some of these irreducible components is applied to get simply designed varieties inside the difference between these splitting loci and their closures, i.e., inside the boundary of the splitting loci.