STACKS OF TRIGONAL CURVES

In this paper we study the stack T-g of smooth triple covers of a conic; when g >= 5 this stack is embedded M-g as the locus of trigonal curves. We show that T is a quotient [U-g/Gamma(g)], where Gamma(g) is a certain algebraic group and U-g is an open subscheme of a Gamma(g)-equivariant vector bundle over an open subscheme of a representation of Gamma(g). Using this, we compute the integral Picard group of T-g when g > 1. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym(3) E circle times det E-V, and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack.