Stacks of trigonal curves

Abstract

In this paper we study the stack Tg of smooth triple covers of a conic; when g ≥ 5 this stack is embedded Mg as the locus of trigonal curves. We show that Tg is a quotient [Ug/Γg ], where Γg is a certain algebraic group and Ug is an open subscheme of a Γg-equivariant vector bundle over an open subscheme of a representation of Γg . Using this, we compute the integral Picard group of Tg 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.

The Chow ring of the stack of cyclic covers of the projective line

“…We can now put together the previous two Lemmas to get the following expression for p N (t) ∈ Z[c 2 , c 3 , t]/(2c 3 ). …”

Mapping class groups of trigonal loci

“…Let us denote by 2,g+2 the open subscheme of Mat 2,g+2 parametrizing matrices (l i j ) with the property that the matrix (l i j ( p)) has corank 2 at all points p ∈ P 1 . As remarked in [4,Proposition 4.2],…”

“…Let us recall from [23] that the datum of a trigonal curve t : C → P 1 of genus g is equivalent to the datum of a rank two vector bundle E on P 1 (actually obtained as t * O C /O P 1 , and known as Tschirnhausen module) with a few properties and a section of Sym 3 E ⊗ det E * . Notably, the splitting type (m, n) of E should be such that m + n = g + 2 and, if C is integral, then m, n ≥ g+2 3 (see also [4,Proposition 2.2]). The stack of smooth trigonal curves is constructed starting from this datum.…”

“…This in turn follows from the fact that the g -orbit consists exactly of all different ways to describe one trigonal curve in a ruled surface by putting coordinate systems on the base and on the surface (see Sections 4 and 5 of [4] for details on the g action).…”

“…This stack has been constructed in [4] for curves of genus g > 0 (and, accordingly, we will assume this throughout the paper, except where explicitly stated differently), and it has a presentation as a quotient stack [X / g ], where g is a certain algebraic group and X is an open set inside the total space of a vector bundle over an open subset of a representation of g (see Sect. 1.2 for more details).…”