Trigonal Gorenstein curves and special linear systems

Let Y be a Gorenstein trigonal curve with g := p(a)(Y) greater than or equa l to 0. Here we study the theory of special linear systems on Y, extending; the classical case of a smooth Y given by Maroni in 1946. As in the classi cal case, to study it we use the minimal degree surface scroll containing t he canonical model of Y. The answer is different if the degree 3 pencil on Y is associated to a line bundle or not. We also give the easier case of sp ecial linear series on hyperelliptic curves. The unique hyperelliptic curve of genus g which is not Gorenstein has no special spanned line bundle.