The geproci property in positive characteristic

TL;DR

This paper explores the geproci property in algebraic geometry over fields of positive characteristic, introducing new methods to generate diverse geproci point sets beyond previously known examples.

Contribution

It develops novel geometric techniques in positive characteristic to construct and analyze geproci point sets, expanding the known classes of such configurations.

Findings

New methods for producing geproci half grids

Construction of non-half grid geproci sets in positive characteristic

Enhanced understanding of geproci configurations in different characteristics

Abstract

The geproci property is a recent development in the world of geometry. We call a set of points $Z\subseteq\mathbb{P}_k^3$ 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 $a\leq b$. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were found starting in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids. Before the work in this paper -- based partly on the author's thesis -- only a few examples of geproci nontrivial non-grid non-half grids were known and there was no known way to generate more. Here, we use geometry in the positive characteristic setting to give new methods of producing geproci half grids and non-half grids.