Minimal defining equations for PR varieties

Identify a minimal set of equations defining each Pawlowski–Rhoades (PR) variety \(\overline{C}_w\) associated with a Fubini word \(w\).

Background

The paper develops progressively smaller collections of equations defining the PR varieties $\overline{C}_w$ inside the space $X_{n,k}$ of spanning line configurations. It first replaces all vanishing minors by vanishing flag minors and then introduces ranked essential sets that yield a minimal set of rank conditions in the sense that no smaller set of those rank conditions suffices.

The authors note that analogous improvements to the Alpha Test for medium and espresso Fubini–Bruhat orders may reduce the number of equations further, paralleling work on minimal defining equations for Schubert varieties. The unresolved problem is to determine a genuinely minimal defining set of equations for a PR variety.