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 into M_g as the locus of trigonal curves. We show that T_g is a quotient [U_g/Γ_g], where Γ_g is a certain algebraic group and U_g is an open subscheme of a Γ_g-equivariant vector bundle over an open subscheme of a representation of Γ_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 ⊗ det E^∨, 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.