Bounding Pinch Point Schemes of Projected Surfaces

Let $X$ be a smooth surface and let $\phi :X\to {\mathbb{P}}^N$, with $N\ge 4$, be a finitely ramified map which is birational onto its image $Y=\phi (X)$, 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 ${\mathbb{P}}^3\mathrm{.}$ We then prove that the lower bound is realized if and only if $Y$ is a rational normal scroll.