Main result

We want to show some basic facts on the homotopy type of the moduli stacks of algebraic curves. The notion of stack is now becoming very popular and for some purposes it is indispensable. However, unfortunately, sometimes the claims on stacks are used without proofs. Such manner of “string-math” causes disharmony in the sound universe of mathematics.

We shall write a proof on the problem of the title. As a bonus of writing a proof, we find some new problems on Teichmüller groups. A motivation of this paper is found in a plan of Grothendieck [5]. Our notion of algebraic stack is that of Deligne-Mumford [3,§4] throughout this paper.

Let ℳg,n be the moduli stack of proper smooth curves of genus g with n distinct (ordered) points. For each scheme S, the category of sections of ℳg,n over S is the category of objects:

{p : C → S, a proper smooth morphism whose fibers are geometrically connected curves of genus g, plus n sections t1, …, tn: S → C of p such that ti(s) ≠ tj(s) for any s ϵ S, if i ≠ j (1 ≤ i, j ≤ n)},

the morphisms being the isomorphisms of the above data over S.

It is known that ℳg,n is an algebraic stack in the sense of Deligne-Mumford [3]. In fact, if n = 0, our ℳg,0 is of [3], and if n = 1, ℳg,1 is the “universal family” of curves of genus g over ℳg,0.