Twisted curves are a central object in the theory of {\em twisted stable maps} \cite{Abramovich-Vistoli,Chen-Ruan,Abramovich-Olsson-Vistoli}: in order to have a complete moduli space of stable maps $C \to X$ of type $\Gamma$, where $X$ is a proper tame stack with projective coarse moduli space and $\Gamma=(g,n,\beta)$ are the relevant discrete data, one must allow the curve $C$ itself to be a certain type of stack, called {\em twisted curve}.

The original treatments of twisted  curves relied on ad-hoc methods. The more recent approach of \cite{Abramovich-Olsson-Vistoli} relies on a method introduced in \cite{Olsson-curves}, which uses a construction with logarithmic structures.


\subsection{Twisted curves} 

For simplicity we will stick with the case of Deligne--Mumford stacks.

First consider the geometric objects: fix an algebraically closed field $k$.

\begin{definition}\label{Def:twisted-curve}
A {\em twisted curve} over $k$ is a tame, purely 1-dimensional Deligne-- Mumford stack $\cC /k$, with at most nodes as singularities, satisfying the following conditions:
\begin{enumerate}
\item Let $\pi: \cC \to C$ be the morphism to the coarse moduli space. Then $\cC^{sm} = \pi^{-1} C ^{sm}$, and $\pi: \cC \to C$ is an isomorphism over a dense open subset of $C$.
\item Consider a node $\bar x \to C$, where the strictly henselian local ring  $\cO_{C,\bar x}$ is the strict henselization of $k[x,y]/(xy)$. Then 
$$\cC \times_C\Spec \cO_{C,\bar x}\ \simeq\
 \left[ \Spec \cO_{C,\bar x}[z,w]/(zw,z^m-x,w^m-y) \ \ \bigg/ \ \ \mu_m\right],$$
 where $\zeta \in \mu_m$ acts by $(z,w) \mapsto (\zeta z, \zeta^{-1}w)$.
\end{enumerate}
\end{definition}
An action such as (2) above is called {\em balanced} - it is crucial to our discussion of log structures below. Note that $\cC$ may have a stack structure at isolated smooth points as well - such points will behave like $[\AA^1/\mu_a]$, where $\mu_a$ acts by multiplication.

Over a general base $S$ twisted curves are detected by their geometric fibers: a {\em twisted curve} $\cC \to S$ is a flat, tame Deligne--Mumford stack locally of finite presentation, all of whose  geometric fibers are twisted curves as in the definition above.

The {\em genus} of $\cC$ is simply the genus of $C$. One typically needs to consider $n$-pointed twisted curves, where the markings are described in families as follows:

\begin{definition}
An {\em $n$-pointed twisted curve} $\cC /S$ marked by disjoint closed substacks $\{\Sigma_i\}_{i=1}^n$ in $\cC$ is assumed to satisfy the following:
\begin{enumerate}
\item the $\Sigma_i$ are contained in the smooth locus $\cC^{sm}$,
\item each  $\Sigma_i$ is a tame \'etale gerbe over $S$, and 
\item $\cC_{gen} := \cC^{sm} \smallsetminus \cup_i\Sigma_i \ \ \longrightarrow\ \  C $ is an open embedding.
\end{enumerate}
\end{definition}
\begin{remark}
When $S=\Spec k$ where $k=\bar k$, then $\Sigma_i = B\mu_{a_i}$, and moreover $a_i$ is locally constant in families.
\end{remark}

\begin{remark}
When $(\cC/S, \{\Sigma_i\})$ is an $n$-pointed twisted curve, then the coarse moduli space of $\Sigma_i$ is isomorphic to $S$. This means that the composite morphism $\Sigma_i\to \cC \to C$ factors through a section $p_i: S \to C$. It follows that $(C,\{p_i\})$ is an $n$-pointed curve in the usual sense. This gives a functor 
$$(\cC/S, \{\Sigma_i\}) \mapsto (C,\{p_i\})$$
\end{remark}

One can ask oneself: what does one need in order to recover a twisted $n$-pointed curve  $(\cC/S, \{\Sigma_i\})$ from a usual $n$-pointed curve $(C,\{p_i\})$? In other words, can we enrich the functor above to something like
$$(\cC/S, \{\Sigma_i\}) \mapsto (C,\{p_i\})+\text{?}$$
which is nice and explicit and actually an equivalence of categories?

The stack structure at the marking definitely needs the data of the integers $a_i$, but in fact this is all that is necessary for the markings: near $p_i$, the curve $\cC$ is canonically isomorphic to the root stack $C(\sqrt[a_i]{p_i})$. If $x$ is a local generator of the ideal of $p_i$, then Zariski locally we have
$$\cC\ \  \simeq \ \ \left[ \Spec \cO_C[z]/(z^{a_i} -x)\ \  \bigg/\ \  \mu_{a_i}\right].$$

The story is a bit more interesting at a node. It has to be - a twisted curve $\cC$ with a node of index $m> 1$ has ``ghost" automorphisms in $\mu_m$ which are not detectible on the coarse 
curve $C$: using the local coordinates given in Definition \ref{Def:twisted-curve} (2), the $\mu_m$ action 
$$(z,w) \mapsto (\zeta z, w)$$
on $\Spec \cO_{C,\bar x}[z,w]/(zw,z^m-x,w^m-y)$ commutes with the action defining the quotient stack. It  therefore descends to a nontrivial action on $\cC$ which becomes trivial on the coarse moduli space $C$. 

\subsection{Log twisted curves} 
Let $X$ be a Deligne--Mumford stack. Recall from Definition \ref{Def:fine} that a fine log structure $M$ on $X$ is said to be {\em locally free} if for every geometric point $\bar x \to X$ we have that the characteristic sheaf $\overline M_{\bar x}$ is isomorphic to $\NN^r$ for some $r$.

In this situation we say that a morphism of sheaves of monoids $M \to M'$ is {\em simple} if for   every geometric point $\bar x \to X$  we can identify the map as the diagonal map 
$$\xymatrix{ \overline M_{\bar x} \ar[rr]\ar^\simeq[d]&& \overline M'_{\bar x}\ar^\simeq[d]\\
\NN^r\ar[rr]^{(m_1,\ldots,m_r)}&& \NN^r}$$
where all $m_i$ are prime to the characteristic of the field.

\begin{definition}
An {\em $n$-pointed log twisted curve over $S$} is the data
$$(C/S, \{\sigma_i,a_i\},\ell:M_S \to M_S')$$ where
\begin{itemize}
\item $(C,\{\sigma_i\})/S$ is an $n$-pointed nodal curve.
\item $M_S$ is the canonical log structure coming from the family $(C,M_C) \to (S,M_S)$ (see section \ref{Satriano}).
\item $a_i: S \to \ZZ_{>0}$ are locally constant, with $a_i(s)$ invertible in the residue field $k(s)$. 
\item $\ell: M_S\to M_S'$ is a simple morphism.
\end{itemize}
\end{definition}

And we have the following:
\begin{theorem}[\protect{\cite[Theorem 1.8]{Olsson-curves}}]
The fibered category of $n$-pointed twisted curves is naturally equivalent to the stack of $n$-pointed log twisted curves.
\end{theorem}

The picture is as follows: we have already noted that we can replace a marking $p_i$ by a stacky marking just using the data $a_i$.  Now the $j$-th node which looks like $\Spec \cO_S[x,y]/(xy-t)$ needs to be replaced by $\left[\Spec \cO_S[z,w]/(zw-t^{1/m_j})\ \big/\ \mu_{m_j}\right]$, and the data is encoding by deviding the $j$-th generator of $\NN^r$ by $m_j$.\footnote{put here Martin's picture!!}


\begin{remark}
We can decompose the stack according to $a_i$:
$$\xymatrix{\mathop{\coprod}\limits_{\underline a}^{\vphantom{a}} \cM^{tw}_{g,n,\underline a} \ \ \ \ \ = 
%\ar@{=}[r]
& \cM^{tw}_{g,n}\ar[d]&= \ \ \ \text{ stack of $n$-pointed twisted curves}\\
&\cS_{g,n}&= \ \ \ \text{ stack of $n$-pointed nodal curves}\ \ 
} $$
and it can be deduced form the theorem that $\cM^{tw}_{g,n,\underline a} $ is obtained from $\cS_{g,n}$ using a root construction applied to the boundary divisor of $\cS_{g,n}$, see \cite[Remark 1.10]{Olsson-curves}. 

In fact we have to apply all possible roots, accounting for all possible twisting of nodes, and glue together, so $\cM^{tw}_{g,n,\underline a} $ is highly non-separated.
\end{remark}

Below we sketch the main ideas in proving this. We stress that the assumption that our twisted curves are {\em balanced} is crucial - the case of unbalanced curves has not been investigated.

\subsection{From twisted curves to log twisted curves}



Fix a twisted curve $f:\cC \to S$.

We can follow F. Kato \cite{FKato}, giving log structures on nodal curves: consider all possible triples $$(M_S,M_{\cC}, f^\flat: f^{-1}M_S \to M_\cC)$$ such that 
\begin{enumerate}
\item $(\cC,M_\cC) \to (S,M_S)$ is log smooth;
\item $M_\cC,M_S$ are locally free; and
\item for all geometric points $\bar x \to \cC$ mapping to nodes we have
$$\xymatrix{\overline M_{\cC,\bar x}\ar[r]^\sim & \NN^{r-1}\oplus \NN^2 \\
\overline M_{S,f(\bar x)}\ar[r]^\sim\ar[u] & \NN^{r-1}\oplus \NN\ar[u]_{id\oplus \Delta}
}$$
\end{enumerate}

If $S = \Spec k$ we get a natural map $$\NN^\text{number of nodes} \to \overline M_S. $$ We say that $(M_\cC,M_S,f^\flat)$ is {\em special} if for every geometric point $\bar s \to S$ this map is an isomorphism. 

A result \cite[Theorem 3.6]{Olsson-curves} analogous to F. Kato's \cite[Theorem 2.1]{FKato} says that there is a unique special triple $(M_\cC,M'_S,f^\flat)$ associated to $f: \cC \to S$. Analyzing the coarse moduli space we obtain a unique diagram

$$\xymatrix{
(\cC, M_\cC)\ar[r]\ar[d] & (C, M_C)\ar[d]\\
(S, M'_S) \ar[r]^{(id,\ell)} & (S, M_S)
}$$
where $\ell: M_S \to M'_S$ is simple. In particular we obtain a log twisted curve
$$(C/S, \{\sigma_i,a_i\}_{i=1}^n, \ell: M_S \to M'_S)$$
\subsection{From log twisted curves to twisted curves}
Now we fix a log twisted curve $(C/S, \{\sigma_i,a_i\}_{i=1}^n, \ell: M_S \to M'_S)$.
In particular we have the log smooth curve 
$$(C, M_C) \to (S, M_S)$$ which is the coarse moduli space of a putative twisted curve. 
We want to describe $\cC/C$ as the stack parametrizing natural objects over $T \to C$. Here it is! If we denote the relevant maps as follows
$$\xymatrix{T\ar[r]^s\ar[dr]_h& C \ar[d]\\& S,}$$ then $\cC$ is the groupoid of diagrams

$$\xymatrix{h^*M_S\ar[r]^\ell\ar[d] & H^* M'_S\ar[d]\\
    s^*M_C\ar[r]^k&M'_C,
}$$
where
\begin{enumerate}
\item $k$ is simple and for every geometric point $\bar t \to T$ the map 
$\overline M'_{S,\bar t} \to \overline M'_{C,\bar t}$ is either an isomorphism (at a general point), or of the form
$$\xymatrix{\NN^r\ \ \ar@{^(->}[rr]^{(id,0)}&&\ \  \NN^r\oplus \NN}$$ (at a marked point), or 
$$\xymatrix{\NN^{r-1} \oplus \NN\ \ \ar@{^(->}[rr]^{(id,\Delta)}&&\ \  \NN^{r-1}\oplus \NN^2}$$ (at a node).
\item for all $i$ and geometric point  $\bar t \to T$ with $s(\bar t)\subset \sigma_i(S)\subset C$ the group 
$$\text{Coker} ( \overline{M}^{gp}_{C,\bar t} \to \overline{M'}^{gp}_{C,\bar t})$$ is cyclic of order $a_i$. 
\end{enumerate}