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.

中文翻译：

---

Steiner 系统和点的配置

本文的目的是在设计理论和代数几何/交换代数之间建立联系。特别是，给定任何 Steiner 系统 $S(t,n,v)$，我们将两个理想关联在一个合适的多项式环中，定义点的 Steiner 配置及其补。我们专注于后者，研究其同调不变量，例如希尔伯特函数和贝蒂数。我们还研究了与定义点的 Steiner 配置的补集的理想相关的符号和规则幂，找到它的 Waldschmidt 常数、规律性、其回潮和渐近回潮的界限。我们还计算与点的任何 Steiner 配置及其补码相关联的线性代码的参数。