Bounding Pinch Point Schemes of Projected Surfaces

TL;DR

This paper establishes a lower bound for the length of pinch schemes in general linear projections of smooth surfaces into projective 3-space, characterizing the case of equality as rational normal scrolls.

Contribution

It provides a new lower bound for pinch scheme lengths and characterizes when this bound is achieved, specifically identifying rational normal scrolls.

Findings

Lower bound for pinch scheme length in projections to P^3

Equality case characterized by rational normal scrolls

Applicable to smooth surfaces with finitely ramified birational maps

Abstract

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.