Moduli of Trigonal Curves

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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 the ratio δB/λB of the restrictions of the boundary class δ and the Hodge class λ on the moduli space Mg 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 δB/λB≤ 7+6/g. We further describe the rational Picard group of the trigonal locus Tg in the moduli space Mg of genus g curves. In the even genus case, we interpret the above Bogomolov semistability condition in terms of the so-called Maroni divisor in Tg.