of a number of local submersions is enough to guarantee an embedding into the projective plane $\Pp^2$. It is in fact a necessary condition.

In order to discuss this, let us suppose that a curve $C$ contained in some surface $S$ can be embedded in $\mathbb P^2$ as a curve $C_0$ of degree $d \geq 2$ (we have of course to start with $C\cdot C= d^2$ in $S$). It is easy to find infinitely many submersions in a neighborhood of $C_0$. For example, we take two curves $\{A=0\}$ and $\{B=0\}$ of the same degree $l\in {\mathbb N}$ which cross each other in $l^2$ distinct points not in $C_0$. It can be seen that the map $A/B$, which is well defined outside $\{A=0\}\cap \{B=0\}$, has no multiple fibers so that it has only a finite number of critical points; if $C_0$ avoids all these points then $A/B$ is a submersion in some neighborhood of $C_0$ and the restriction of $A/B$ to $C_0$ is a ramified map from $C_0$ to $\Pp^1$ of degree $l.d$. We will be particularly interested in the case $l=1$, that is, $A=0$ and $B=0$ are lines whose common point is not in $C_0$; the submersion $A/B$ will be 