On Frobenius incidence varieties of linear subspaces over finite fields

TL;DR

This paper introduces Frobenius incidence varieties over finite fields, exploring their properties like supersingularity and Betti numbers, and constructs dense sphere packings from these varieties, linking algebraic geometry with lattice theory.

Contribution

It defines Frobenius incidence varieties and investigates their geometric and arithmetic properties, including their application to dense sphere packings in characteristic 2.

Findings

Frobenius incidence varieties are supersingular with specific Betti numbers.

A positive-definite lattice of rank 84 is constructed from these varieties.

A dense sphere packing is achieved from a 4-dimensional Frobenius incidence variety in characteristic 2.

Abstract

We define Frobenius incidence varieties by means of the incidence relation of Frobenius images of linear subspaces in a fixed vector space over a finite field, and investigate their properties such as supersingularity, Betti numbers and unirationality. These varieties are variants of the Deligne-Lusztig varieties. We then study the lattices associated with algebraic cycles on them. We obtain a positive-definite lattice of rank 84 that yields a dense sphere packing from a 4-dimensional Frobenius incidence variety in characteristic 2.