# birational geometry of moduli spaces: why work on the coarse space?

In studying the birational geometry of $$overline{mathcal{M}}_g$$, it seems standard to work with the coarse space $$overline{M}_g$$ rather than the smooth stack $$overline{mathcal{M}}_g$$. Why is this?

More precisely:

(1) In Harris-Mumford’s “On the Kodaira Dimension of the Moduli Space of Curves” (1982), it is proven that $$overline{M}_g$$ is of general type for odd $$gge25$$ (of course generalized to all $$gge24$$ later by Harris and Eisenbud-Harris). A lot of work is done in the first section showing that differential forms can be extended from the smooth locus of $$overline{M}_g$$ to a desingularization of the whole space, so that Kodaira dimension makes sense in the first place, but can one not work with canonical sheaf on the stack $$overline{mathcal{M}}_g$$, which is already smooth? It seems to me that the rest of the proof ought to go through verbatim on the moduli stack, avoiding all the issues with automorphisms.

(2) In Bruno-Verra’s “$$mathcal{M}_{15}$$ is rationally connected,” the strategy of proof of the theorem in question is as follows. Let $$Delta_0subsetoverline{mathcal{M}}_{15}$$ be closure of the locus of irreducible nodal curves, or equivalently the image of the clutching map from $$overline{mathcal{M}}_{14,2}$$. Given general points $$x_iinmathcal{M}_{15}$$, $$i=1,2$$ it is shown that there exist general points $$y_iin Delta_0$$ and rational curves connecting $$x_i$$ to $$y_i$$. It is also shown that $$M_{14,2}$$ is unirational, hence rationally connected, so $$y_1$$ and $$y_2$$ are also connected by a rational curve. Thus $$x_1$$ and $$x_2$$ are connected by a chain of rational curves.

Some effort is needed to show that the $$y_i$$ can be taken to be general, in particular, that they are smooth points of $$overline{M}_{15}$$, and they correspond to curves with exactly one node (i.e. they don’t lie on a self-intersection of $$Delta_0$$). But does the argument not go through if the $$y_i$$ are arbitrary and we work with the moduli stacks? At least on a smooth, projective variety, the condition that a general pair of points are connected by a rational curve implies that an arbitrary pair of points is too. The self-intersections of $$Delta_0$$ don’t seem to me to be a problem, because a rational curve between arbitrary lifts of $$y_1$$ and $$y_2$$ on the smooth stack $$overline{mathcal{M}}_{14,2}$$ maps to a rational curve between $$y_1$$ and $$y_2$$ on $$Delta_0$$. Is the issue perhaps that the various notions of rational connectedness (rational chain-connectedness, general vs. arbitrary pairs of points being connected, …) don’t all coincide on smooth proper DM stacks?