Stacks of trigonal curves (Q2839937)

Trigonal curve is a smooth projective curve with \(3:1\) map onto a curve which is isomorphic to \({\mathbb P}^1\). The isomorphism is not part of the data. The stack \({\mathcal T}_g\) of genus \(g\) smooth trigonal curves over fixed field \(k\) of characteristic not equal 2 or 3 is considered. The objects over \(k\)-scheme \(S\) are families \(C \to P \to S\) where \(P\to S\) is a smooth conic bundle, \(C\to P\) smooth \(3:1\) cover and their composite \(C\to S\) is a family of genus \(g\) curves. The forgetful morphism \((C \to P \to S) \mapsto (C\to S)\) takes \({\mathcal T}_g\) to the stack \({\mathcal M}_g\) of smooth genus \(g\) curves. One of central results of the paper is that this morphism is a locally closed immersion whenever \(g\geq 5\). Another central result is a description of the stack \({\mathcal T}_g\) as a quotient \([X_g/\Gamma_g]\) of an appropriate scheme \(X_g\) by the action of a certain algebraic group \(\Gamma_g\). The third central result is a computation of the Picard group of \({\mathcal T}_g\) for \(g\neq 1\). The authors give a description of the stack of vector bundles over a conic as a quotient stack what is of independent interest besides of being one of main tools of the work together with the result of \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)] ``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 \(\text{Sym}^3 E \otimes E^{\vee}\)'' (cited from the abstract). Also the stack \(\hat{\mathcal T}_g\) of triple covers which contains \({\mathcal T}_g\) as an open substack, is examined. In particular, the locus of singular curves in \(\hat{\mathcal T}_g\) is analyzed.