Finite sets of points in $\mathbb{P}^4$ with special projection properties

TL;DR

This paper introduces and studies special point sets in projective 4-space with unique projection properties, revealing existence conditions and their configurations, and posing open questions for further research.

Contribution

It defines the notion of $(b,d)$-geprofi sets, establishes their existence criteria, and explores their properties and configurations in $ ext{P}^4$.

Findings

Nontrivial $(b,d)$-geprofi sets exist iff $b ext{ } ext{geq} ext{ } 4$ and $d ext{ } ext{geq} ext{ } 2.

Such sets can exist in linear general position for infinitely many $(b,d)$ pairs.

The paper raises open questions about the properties and classifications of these sets.

Abstract

In this note we introduce the notion of $(b,d)$-geprofi sets and study their basic properties. These are sets of $bd$ points in $\mathbb{P}^4$ whose projection from a general point to a hyperplane is a full intersection, i.e., the intersection of a curve of degree $b$ and a surface of degree $d$. We show that such nontrivial sets exist if and only if $b\geq 4$ and $d\geq 2$. Somewhat surprisingly, for infinitely many values of $b$ and $d$ there exist such sets in linear general position. The note contains open questions and problems.