Moduli of Trigonal Curves

Abstract

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 $X\to B$ of stable curves with smooth general member is the ratio ${\delta }_B/{\lambda }_B$ of the restrictions of the boundary class $\delta$ and the Hodge class $\lambda$ on the moduli space ${\bar{\mathfrak{M}}}_g$ 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 ${\delta }_B/{\lambda }_B\le 7+6/g$. We further describe the rational Picard group of the {trigonal} locus ${\bar{\mathfrak{T}}}_g$ in the moduli space ${\bar{\mathfrak{M}}}_g$ 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 ${\bar{\mathfrak{T}}}_g$.

arXiv comment69 pages, 34 figures, Latex2e

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