On the uncommonness of minimal rank-2 systems of linear equations

TL;DR

This paper proves that generic pairs of linear equations on an even number of variables are rare, confirming a conjecture, and shows that large systems containing such minimal subsystems are also uncommon.

Contribution

It verifies a conjecture about the rarity of certain minimal rank-2 linear systems and extends the result to larger systems containing these subsystems.

Findings

Generic pairs of linear equations are uncommon.

Large systems containing minimal rank-2 subsystems are also uncommon.

Confirmed a conjecture by Kamčev, Morrison, and the second author.

Abstract

We prove that suitably generic pairs of linear equations on an even number of variables are uncommon. This verifies a conjecture of Kam\v{c}ev, Morrison and the second author. Moreover, we prove that any large system containing such a $(2\times k)$-system as a minimal subsystem is uncommon.