Trigonal Gorenstein curves

We notice that the Maroni invariant of a trigonal Gorenstein curve of arithmetic genus g larger than four may be equal to zero, and we show that this happens if and only if the g31 admits a non-removable base point, which is necessarily a singularity of the curve. We realize and study trigonal curves on rational scrolls, which in the case, where the g31 admits a base point Q, degenerate to a cone with vertex Q.