# Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of vertex algebras as a home for central charge zero conformal blocks (and probably in other fields for reasons I don't know). Formally speaking, the objects in the category are diagrams $$S \rightrightarrows Frame(\Omega_{C/S}) \to C \to S,$$ where

• $C \to S$ is a smooth proper morphism with one dimensional connected genus $g$ geometric fibers,
• $Frame(\Omega_{C/S}) = \underline{Isom}(\Omega_{C/S}, \mathcal{O}_C) \to C$ is the canonical $\mathbb{G}_m$-torsor of nowhere vanishing relative tangents, and
• $S \rightrightarrows \underline{Isom}(\Omega_{C/S}, \mathcal{O}_C)$ are $n$ sections of the composite map to $S$, whose images in $C$ are pairwise disjoint.

It seems to be a folklore theorem that when $n(g+1) > 1$, this moduli problem is representable (in fact by a quasi-projective object), but I have been unable to locate a proof in the literature. For example, the book Lectures on tensor categories and modular functors by Bakalov and Kirillov only justifies it with the claim that these objects have trivial automorphism group.

It does not appear to be extremely difficult to prove - one may take a suitable principal torus bundle over Knudsen's scheme $H_{g,n}$ of tricanonically embedded pointed curves, and show that the corresponding action of $PGL(5g+3n-5)$ is free with suitably small orbits. The part about orbits seems like it might be a bit delicate, so I am curious:

Question: Is there a full proof of representability somewhere in the literature?

-
The map to the moduli of pointed curves (without tangent vectors) seems to be trivially representable -- am I missing something? –  Moosbrugger Jan 10 '12 at 14:56
The moduli stack of pointed curves is not in general representable (i.e., for $n$ small compared to $g$), so this doesn't always help for representability of the total stack. –  S. Carnahan Jan 10 '12 at 15:11
@S. Carnahan -- The stack is always algebraic. So, as Moosbrugger suggests, you automatically get that your stack is algebraic. Once you know that your stack has finite, reduced stabilizer groups, then it follows that it is a Deligne-Mumford stack. Quite possibly your particular stack has never been proved to be DM in the literature. But this approach to proving a stack such as your is DM is developed, for instance, in Behrend's article "Gromov-Witten Invariants in Algebraic Geometry". –  Jason Starr Jan 10 '12 at 16:16
Right, and if the automorphism groups are trivial then it's an algebraic space. –  Moosbrugger Jan 10 '12 at 17:28
Thanks. I should have mentioned that freeness of the $PGL(5g+3n-5)$ action on the scheme of tricanonically embedded curves with framed points implies the stack is an algebraic space. In order to get a scheme, some more work is needed. Possibilities that come to mind include showing that all orbits lie in affine opens of $H_{g,n}$, giving an explicit Zariski cover, or writing down a compactification together with an ample line bundle. –  S. Carnahan Jan 10 '12 at 22:30