Sets of points which project to complete intersections

TL;DR

This paper characterizes non-degenerate point sets in projective 3-space whose general projections are complete intersections in the plane, introducing $(m,n)$-grids and exploring the unexpected cone property with specific examples.

Contribution

It establishes a connection between point sets with projection properties and the unexpected cone property, including classification results and novel examples from root systems.

Findings

9 points form a (3,3)-grid if their projection is a complete intersection of two cubics

A set of 24 points from the $F_4$ root system has the projection property without being a grid

Subsets of 20, 16, and 12 points also exhibit the projection property

Abstract

The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the {\em unexpected cone property} ${\mathcal C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of ${\mathcal C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete intersection of two cubics if and only if the points form a $(3,3)$-grid. However, in an appendix we describe a set of $24$ points that are not a grid but nevertheless have the projection property. These points arise from the $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.