Steiner systems and configurations of points

TL;DR

This paper explores the algebraic and geometric properties of Steiner systems by associating ideals to their configurations of points, analyzing their invariants, powers, and related coding parameters.

Contribution

It introduces a novel connection between Steiner systems and algebraic geometry, studying homological invariants and coding parameters of associated point configurations.

Findings

Computed Hilbert functions and Betti numbers for Steiner configurations

Analyzed symbolic and regular powers, including Waldschmidt constants and resurgence bounds

Determined parameters of linear codes from Steiner configurations

Abstract

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.