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?