Uncrumpled: A scheme for bounding the pinch points on a variety of projected surfaces

Abstract

Let X be a smooth surface and let phi be a finitely ramified map from X to P^N with N at least 4, which is birational onto its image Y = phi(X), and with Y non-degenerate in 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 P^3. We then prove that the lower bound is realized if and only if Y is a rational normal scroll.

We describe an alternative way of viewing this problem, along with a review of the work done on some adjacent problems. We also include a small collection of examples where we compute the number of pinch points contained in a general linear projection to P^3 as evidence for the main theorem. Finally, we conclude with a strengthening of the main result and we describe several future problems stemming from this result.