called a [*pencil submersion*]{} and the restriction of $A/B$ to $C_0$ is a ramified map of degree $d$ (any local submersion that leaves such a trace in $C_0$ is in fact a pencil submersion). We see that to be equivalent to a neighborhood of $C_0$, a neighborhood of $C$ has to carry also many submersions to $\Pp^1$, the surprising feature in [@FA] is that only three submersions are needed. The converse is not true as we can see in the following example.

\[fake-example\] Consider the rational curve in $\mathbb P^2$ defined in affine coordinates by the equation $y^2=x^2(x+1)$; it is a smooth rational curve except for the node at the point $(0,0)$. We blow up first at a point in the curve different from $(0,0)$, and then we blow up at $(0,0)$. The strict transform is a smooth rational curve $C$ of self intersection number equal to 4 with many local submersions (which come from submersions constructed in the plane as above), but its neighborhood can not be embedded in the plane: given a submersion constructed using $l=1$ as above (before blow up’s), we notice that it induces a ramified map from $C$ to $\Pp^1$ of degree 3; but for 