surface;

-   if $z_0\!\in\!\Si_{u_0}$ is a smooth point, there are neighborhoods $U_{z_0}$ of $u_0$ in $\ov\M$ and $\ti{U}_{z_0}$ of $z_0$ in $\fU$ and a stratified isomorphism of fiber bundles $$\ti\phi_{z_0}\!:\ti{U}_{z_0}\lra U_{z_0}\!\times\!(\Si_{u_0}\!\cap\!\ti{U}_{z_0})$$ over $U_{z_0}$ such that the restriction of $\ti\phi_{z_0}$ to each fiber of $\pi$ is holomorphic and the restriction of $\ti\phi_{z_0}$ to $\Si_{u_0}\!\cap\!\ti{U}_{z_0}$ is the identity;

-   if $z_0\!\in\!\Si_{u_0}$ is a node, there are neighborhoods $U_{z_0}$ of $u_0$ in $\ov\M$ and $\ti{U}_{z_0}$ of $z_0$ in $\fU$, a stratified space $U_{z_0}'$, and stratified embeddings $$\phi_{z_0}\!: U_{z_0}\lra U_{z_0}'\times\C \qquad\hbox{and}\qquad
    \ti\phi_{z_0}\!:\ti{U}_{z_0}\lra U_{z_0}'\times\!\fU_{\std}$$ such that the diagram $$\xymatrix{ \ti{U}_{z_0} \ar[d]^{\pi} \ar[r]^-{\ti\phi_{z_0}}& 
    U_{z_0}'\!\times\!\fU_{\std} \ar[d]^{\id\times\pi_{\std}}\\
    U_{z_0} \ar[r]^-{\phi_{z_0}}& U_{z_0}'\!\times\!\C }$$ commutes and the restriction of $\ti\phi_{z_0}$ to each fiber of $\pi$ is holomorphic.

\[flatfam\_df2\] If $S$ is a finite set, a stratified map $\pi\!:\fU\!\lra\!\ov\M$ with stratified sections $z_j\!:\ov\M\!\lra\!\fU$, $j\!\in\!S$, is a  if

-   $\pi\!:\fU\!\lra\!\ov\M$ is a flat stratified family of Riemann surfaces;

-   $z_j(u)\!\in\!\Si_u$ is a smooth point for every $u\!\in\!\ov\M$ and $j\!\in\!S$;

-   $z_{j_1}(u)\!\neq\!z_{j_2}(z)$ for every $u\!\in\!\ov\M$, $j_1,j_2\!\in\!S$ with $j_1\!\neq\!j_2$.

\[flatfam\_df3\] If $\pi\!:\fU\!\lra\!\ov\M$ is a flat stratified family of $S$-marked Riemann surfaces and $Y$ is a smooth manifold, a continuous map $F\!:\fU\!\lra\!Y$ is a if

-   for every $u\!\in\!\ov\M$, the restriction of $F$ to $\Si_u\!\equiv\!\pi^{-1}(u)$ is 