Moduli of trigonal curves

We study the moduli of trigonal curves. We establish the exact upper bound of 36(g + 1)/(5g + 1) for the slope of trigonal fibrations. Here, the slope of any fibration X --> B of stable curves with smooth general member is th e ratio delta(B)/lambda(B) Of the restrictions of the boundary class delta and the Hedge class lambda on the moduli space (M) over bar(g) to the base B. We associate to a trigonal family X a canonical rank two vector bundle V , and show that for Bogomolov-semistable V the slope satisfies the stronger inequality delta(B)/lambda(B) less than or equal to 7 + 6/g. We further de scribe the rational Picard group of the trigonal locus (I) over bar(g) in ( M) over bar(g). in the even genus case, we interpret the above Bogomolov se mistability condition in terms of the so-called Maroni divisor in (I) over bar(g).