Degree formalism for line bundles on Deligne-Mumford stacks

Question

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper "Les schemas de modules des courbes elliptiques" by P. Deligne and M. Rapoport, I understand that there is a notion of degree for line bundles $\mathscr{L}$ on $\mathcal{C}$. My question is about some of the compatibility claims left to the reader in that reference and has several subquestions listed below.

The degree of $\mathscr{L}$ is defined starting from a "rational section" $f$ of $\mathscr{L}$. My first question is:

1. What precisely is $f$? How is it defined?

The only reasonable candidate in my mind is a rational section of the pushforward to $\mathscr{L}$ to the coarse space of $\mathcal{C}$, but does this really work? For instance, why can't it happen that this pushforward is zero?

Once we have $f$, we look at strict Henselizations of $\mathcal{C}$ at closed points $x$ and, letting $R_x$ be the coordinate algebra of such a Henselization, we look at the "order" $\mathrm{deg}_x(f)$ with respect to $R_x$ of the pullback of $f$ to the total ring of fractions of $R_x$. In the text it is implicitly claimed that either the pullback of $f$ or its inverse is actually in $R_x$; I cannot see why this is so, but the order makes good sense nevertheless by expressing the pullback as a fraction (presumably at this stage one sees that the Cohen-Macaulayness condition cannot actually be dropped?). One finally defines $$ \mathrm{deg}(\mathscr{L}) = \sum_x \frac{1}{|\mathrm{Aut}(x)|} \mathrm{deg}_x(f).$$

My other subquestions are:

2. Why is $\mathrm{\deg}(\mathscr{L})$ independent of $f$?
3. Why is $\mathrm{\deg}(\mathscr{L})$ invariant under base change to an algebraically closed overfield (I bet this one is easy, but I cannot quite see it)?
4. Why is $\mathrm{\deg}(\mathscr{L})$ "invariant under specialization", as claimed in $\gamma)$ of the reference? In other words, if $S$ is a spectrum of a DVR and $\widetilde{\mathcal{C}}$ and $\widetilde{\mathscr{L}}$ are over $S$, why is $\mathrm{deg}(\mathscr{\widetilde{L}}_\eta) = \mathrm{deg}(\mathscr{\widetilde{L}}_s)$?

I apologize for so many multiple parts, but comments on any one of them would help.

Answer 1

Let us do the example where $\mathscr{M}$ is a proper Deligne-Mumford moduli stack of varieties. Note that if $\mathscr{M}$ is one-dimensional, then the Cohen-Macauleyness of $\mathscr{M}$ is equivalent to the absence of embedded points in $M$; this assumption together with the properness is to insure that the degree map $\mathrm{Pic}(M) \to \mathbf{Z}$ is well defined.

A line bundle $\mathscr{L}$ over $\mathscr{M}$ in this case is an assignment of each family $f : \mathscr{X} \to S$ of varieties parameterized by $\mathscr{M}$ to a line bundle $\mathscr{L}_f$ over $S$ such that whenever we have a fibered product

\begin{array}{c} \mathscr{X'} & \to & \mathscr{X} & \\ \downarrow& & \downarrow \\ S' & \xrightarrow{p} & S \end{array} the line bundles $\mathscr{L}_{f'}$ and $p^*\mathscr{L}_f$ coincide. The first Chern class of $\mathscr{L}$ is defined by picking a rational section $s_f$ of $\mathscr{L}_{f}$ for each $f$ such that $\mathrm{div}(s_{f'}) = p^*\mathrm{div}(s_{f})$ in $\mathrm{Pic}(S')$ for all $p$ as above.

One can also define a rational section $s$ of $\mathscr{L}$ by picking a rational section $s_f$ of $\mathscr{L}_{f}$ for each $f$ such that the induced map $S \to \mathscr{M}$ is étale, compatible with $\mathscr{L}_{f'} = p^*\mathscr{L}_f$ for all $p$ as above which are necessarily étale. For such an $f$, let $q : S \to M$ be the map induced by the family $f: \mathscr{X} \to S$. The $\mathbf{Q}$-divisor class

$$\frac{1}{\deg(q)} q_*\mathrm{div}(s_f) \in \mathrm{Pic}(M)_\mathbf{Q}$$

is independent of the choices of $s$ and $f$, which is precisely the image of $\mathscr{L}$ under the pushforward map

$$\pi_* : \mathrm{Pic}(\mathscr{M})_\mathbf{Q} \to \mathrm{Pic}(M)_\mathbf{Q}$$ as Niels already mentioned in his comment.

When $\mathscr{M}$ is one-dimensional, the degree of $\mathscr{L} \in \mathrm{Pic}(\mathscr{M})_\mathbf{Q}$ that Deligne and Rapoport define is the degree of $\pi_* \mathscr{L}$, and your questions 2), 3) and 4) follow from the same statements for line bundles over schemes.