Stacks of trigonal curves

Bolognesi, Michele; Vistoli, Angelo

doi:10.1090/S0002-9947-2012-05370-0

Stacks of trigonal curves
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by Michele Bolognesi and Angelo Vistoli PDF

Trans. Amer. Math. Soc. 364 (2012), 3365-3393 Request permission

Abstract:

In this paper we study the stack $\mathcal {T}_g$ of smooth triple covers of a conic; when $g \geq 5$ this stack is embedded $\mathcal {M}_{g}$ as the locus of trigonal curves. We show that $\mathcal {T}$ 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 $\mathcal {T}_g$ 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 $\mathrm {Sym}^{3}E\otimes \mathrm {det} E^\vee$, 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.

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