  Let $X$ be a smooth surface and let $\varphi:X\to\mathbb{P}^N$, with $N\geq
4$, be a finitely ramified map which is birational onto its image $Y =
\varphi(X)$, with $Y$ non-degenerate in $\mathbb{P}^N$. In this paper, we
produce a lower bound for the length of the pinch scheme of a general linear
projection of $Y$ to $\mathbb{P}^3.$ We then prove that the lower bound is
realized if and only if $Y$ is a rational normal scroll.
