Configurations of points in projective space and their projections

TL;DR

This paper systematically constructs and analyzes nongrid, nondegenerate $(a,b)$-geproci point sets in projective space, revealing new examples, their properties, and connections to classical geometric configurations and unexpected cones.

Contribution

It introduces a systematic method for constructing nongrid nondegenerate $(a,b)$-geproci sets for all $4 \,\leq\, a \leq b$, and explores their properties and relations to classical configurations.

Findings

Constructed new nongrid nondegenerate $(a,b)$-geproci sets for all $a, b \geq 4$

Identified the unique $(3,4)$-geproci set from the $D_4$ root system

Explored the relation between geproci sets, unexpected cones, and $d$-Weddle schemes.

Abstract

We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees $a$ and $b$. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had $ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci set coming from the $D_4$ root system, and we describe the $D_4$ configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of $d$-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.