Résumé
In this paper we study the stack Tg\mathcal {T}_g of smooth triple covers of a conic; when g≥5g \geq 5 this stack is embedded Mg\mathcal {M}_{g} as the locus of trigonal curves. We show that T\mathcal {T} is a quotient [Ug/Γg][U_{g}/\Gamma _{g}], where Γg\Gamma _g is a certain algebraic group and UgU_g is an open subscheme of a Γg\Gamma _g-equivariant vector bundle over an open subscheme of a representation of Γg\Gamma _g. Using this, we compute the integral Picard group of Tg\mathcal {T}_g when g>1g > 1. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme SS as given by a locally free sheaf EE of rank two on SS, with a section of Sym3E⊗detE∨\mathrm {Sym}^{3}E\otimes \mathrm {det}\,E^\vee, 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.