# Degree formalism for line bundles on Deligne-Mumford stacks

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:

1. Why is $\mathrm{\deg}(\mathscr{L})$ independent of $f$?
2. 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)?
3. 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.

• See B.2.2 of uni-due.de/~ade847f/bdp5-hl for a development of degree for proper 1-dimensional Artin stacks over a field that admit a finite flat scheme cover (as for the curves of interest to you). This avoids needing such an $f$. Briefly, one transports certain definitions from a finite flat scheme cover down to the stack by appropriate division and has to prove independence of the choice of that cover (and various properties); ultimately #4 (when the stack over the dvr has a finite flat scheme cover) is deduced from the known analogue for proper flat schemes over a dvr. – nfdc23 Feb 9 '16 at 7:06
• A quick way to define the degree is : let $\pi:\mathcal C \to C$ be the morphism to the moduli space. Then $\pi^*: \operatorname{Pic}(C)_{\mathbb Q} \to \operatorname{Pic}(\mathcal C)_{\mathbb Q}$ induces an isomorphism of rational Picard groups, with inverse $\pi_*$. So $\deg\pi_*(\mathcal L)$ is a well defined rational number. This works more generally for Chern classes of vector bundles. – Niels Feb 9 '16 at 10:30
• Thank you both. I am aware of the approaches you are suggesting, but I would like to understand the reasoning that the authors of the reference I've mentioned had in mind. – O-Ren Ishii Feb 9 '16 at 17:08
• Hi @O-RenIshii - are you aware of Totaro-type models for quotient stacks? I am not an expert on the paper you mentioned, but one can perhaps address your questions by using the fact that DM stacks are etale locally quotient stacks, and $\mathbb{A}^1$-invariant questions on quotient stacks (like their $Pic$) may be addressed using a concrete model as an ind-scheme; see arxiv.org/pdf/alg-geom/9609018v3.pdf. – Elden Elmanto Feb 19 '16 at 23:24
• @EldenElmanto: Thanks, but I think one cannot work etale locally on the coarse space for my question because the degree formalism is something that only makes good sense in the case of a proper curve. – O-Ren Ishii Feb 20 '16 at 18:21

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.

• Thank you for your answer. Why does a compatible system of $s_f$ exist? The way you phrase it, it seems that every $s_f$ needs to be everywhere defined (take $S'$ to be a variable point of $S$). To avoid this problem you may restrict to $S$ that are etale over $\mathscr{M}$, but then I still do not see how to get the $s_f$ (this is related to the pushforward of $\mathscr{L}$ to $M$ being generically a line bundle). – O-Ren Ishii Feb 20 '16 at 18:19
• You're right, what we can only define if we take all possible $\mathscr{X} \to S$ into consideration is the class of $\mathscr{L}$ inside $\mathrm{Pic}(\mathscr{M})_\mathbf{Q}$, but this will be enough for us to define the degree of $\mathscr{L}$. To define what rational sections are, as you said we need to restrict to those $\mathscr{X} \to S$ such that are the induced map $S \to \mathcal{M}$ is étale. In this case, the morphism $p : S' \to S$ in every base change diagram above is étale as well, so the pullback of a rational section on $S$ to $S'$ is well-defined. – HYL Feb 20 '16 at 19:26
• I agree that the pullback of a rational section is then well-defined. However, the existence of a single nonzero rational section is not clear due to the required compatibility with $p$. – O-Ren Ishii Feb 22 '16 at 17:58
• A nonzero rational section need not exist. Take for example a finite group $G$ acting trivially on $X$ but non-trivially on its trivial bundle. In this case, any rational section of the quotient bundle over $[X/G]$ is $0$ since it is identified with a $G$-invariant rational section of $\mathcal{O}_X$. – HYL Feb 23 '16 at 18:11