An algebraic approach to finite projective planes

TL;DR

This paper explores algebraic structures related to finite projective planes, providing detailed descriptions of their algebraic invariants and conditions for certain algebraic properties to hold.

Contribution

It offers a comprehensive analysis of the Stanley-Reisner and inverse system algebras associated with finite linear spaces, including projective planes, and classifies characteristics for Lefschetz properties.

Findings

Full description of graded Betti numbers for the algebras

Classification of characteristics with Weak or Strong Lefschetz Property

Extension of results from projective planes to general linear spaces

Abstract

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring $R/I_\Lambda$ and the inverse system algebra $R/I_\Delta$. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the Weak or Strong Lefschetz Property.