Mapping class groups of trigonal loci

Abstract

In this paper, we study the topology of the stack $\mathcal {T}_g$ of smooth trigonal curves of genus g over the complex field. We make use of a construction by the first named author and Vistoli, which describes $\mathcal {T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal {T}_g$, and of its substrata with prescribed Maroni invariant, and describe their relation with the mapping class group $\mathcal {M}ap_g$ of Riemann surfaces of genus g.

Appendix A: The construction of the monodromy map

We include here a preliminary discussion of the monodromy map as this notion needs to be defined carefully to get our statements right.

The families of trigonal curves are locally trivial in the complex topology. Hence, in our topological analysis, we want to associate some topological datum to it.

Suppose $p:E\rightarrow B$ is a G-bundle on a smooth variety B with respect to an action of the group G on the fiber F. So B is covered by open trivialization patches U with chart diagrams

such that a change in trivialization is given by a diagram

where the map on top is given as

$$\begin{aligned} (e,u) \mapsto (g_{U\cap V}(u)\cdot e, u) \end{aligned}$$

for $u\in U\cap V$ and some continuous $g:U\cap V \rightarrow G$.

Let I be the unit interval. A map $\gamma :I\rightarrow B$ gives rise to a pullback bundle

There is a natural bundle map lifting $\gamma$

Since I is contractible, the pullback bundle can be trivialized

For a given choice of $\gamma$ and $\phi _\gamma$, we get a group element in G

where the third map is the trivialization along I. To define the monodromy map, it thus suffices to show that the component of G to which $g_{\gamma ,\phi _\gamma }$ belongs is independent of $\phi _\gamma$ and depends only on the homotopy class of $\gamma$.

Independence of the trivialization $\phi _\gamma$ follows directly from formula valid in case of a change in trivialization. The two group elements differ by a group element which is in the path-connected component of the identity.

Independence of the representative $\gamma$ of a given homotopy class follows similarly. A homotopy gives rise to a pullback bundle over $I\times I$, which again is trivial by the contractibility of the base. The induced trivializations over the two homotopic paths give the same group element.

The natural consequence of this is the Proposition/Definition that we wanted.

Proposition 4.6

Let $p:E\rightarrow X$ be a G-bundle on a smooth variety X. The following map given on representatives is well-defined

$$\begin{aligned} \begin{matrix} \pi _1(X, x_0 ) &{} \longrightarrow &{} \pi _0 (G, id) \\ [\gamma ] &{} \mapsto &{} [g_{\gamma ,\phi _{\gamma }}] \end{matrix} \end{aligned}$$

and is called the G-monodromy of the G-bundle $p:E\rightarrow X$.

Remark 4.7

• (i) We remark that the monodromy map is in fact part of the long exact homotopy sequence of the corresponding G-bundle.

• (ii) The definition of monodromy map extends harmlessly to the orbifold context.