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The geproci property in positive characteristic

Jake Kettinger

TL;DR

The paper explores geproci sets in positive characteristic by leveraging spreads and maximal partial spreads over finite fields to construct abundant geproci configurations, including $(q+1,q^2+1)$-half grids and various non-half grids in $\mathbb{P}^3_{\mathbb{F}_q}$. It develops a framework connecting spreads to geproci properties, provides explicit constructions and computational verifications (e.g., Gorensteinness), and shows that positive characteristic yields numerous new geproci examples beyond the characteristic-zero taxonomy. It further extends the notion to infinitely-near points, producing a $(3,3)$-geproci non-half grid in characteristic $2$ via a quasi-elliptic fibration, and highlights phenomena like unexpected cones and Dynkin-diagram connections. Overall, the work demonstrates that positive characteristic offers robust, diverse tools for generating and analyzing geproci configurations, expanding the landscape beyond prior characteristic-zero results and linking finite-field geometry with projection-based intersection properties.

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.

The geproci property in positive characteristic

TL;DR

The paper explores geproci sets in positive characteristic by leveraging spreads and maximal partial spreads over finite fields to construct abundant geproci configurations, including -half grids and various non-half grids in . It develops a framework connecting spreads to geproci properties, provides explicit constructions and computational verifications (e.g., Gorensteinness), and shows that positive characteristic yields numerous new geproci examples beyond the characteristic-zero taxonomy. It further extends the notion to infinitely-near points, producing a -geproci non-half grid in characteristic via a quasi-elliptic fibration, and highlights phenomena like unexpected cones and Dynkin-diagram connections. Overall, the work demonstrates that positive characteristic offers robust, diverse tools for generating and analyzing geproci configurations, expanding the landscape beyond prior characteristic-zero results and linking finite-field geometry with projection-based intersection properties.

Abstract

The geproci property is a recent development in the world of geometry. We call a set of points an -geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point to a plane is a complete intersection of curves of degrees . 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.
Paper Structure (6 sections, 6 theorems, 28 equations, 2 figures)

This paper contains 6 sections, 6 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Geproci Property over Finite Fields
  3. Spreads
  4. Maximal Partial Spreads
  5. Examples
  6. The Geproci Property with Infinitely-Near Points

Key Result

Theorem 1

Let $\mathbb{F}_q$ be the field of size $q$, where $q$ is some power of a prime. Then $Z=\mathbb{P}^3_{\mathbb{F}_q}\subseteq \mathbb{P}^3_{\overline{\mathbb{F}}_q}$ is a $(q+1,q^2+1)$-geproci half grid.

Figures (2)

  • Figure 1: $D_4$ consists of 12 points arranged in 16 sets of 3 collinear points, and is covered by four skew lines as shown.
  • Figure 2: A $D_4$ in any characteristic.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 1
  • Example 2
  • Theorem 1
  • proof
  • Lemma 1
  • ...and 21 more