Linear series on ribbons

A ribbon is a double structure on $\mathbb{P}^{1}$. The geometry of a ribbon is closely related to that of a smooth curve. In this paper we consider linear series on ribbons. Our main result is an explicit determinantal description for the locus $W^{r}_{2n}$ of degree $2n$ line bundles with at least $(r+1)$-dimensional sections on a ribbon. We also discuss some results of Clifford and Brill-Noether type.