Steiner systems and configurations of points. (English) Zbl 1486.51005
Combinatorial design theory is a field of combinatorics connected to several other areas of mathematics including number theory and finite geometries. Steiner systems are important contents in design theory. Usually, the geometric study of Steiner systems focus on their automorphism groups. In this paper, the authors study special configurations of reduced points in \(\mathbb{P}^n\) constructed on Steiner systems, combining combinatorial algebraic geometry and commutative algebra.
Given any Steiner system \(S(t, n, v)\), the authors associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points in \(\mathbb{P}^n\) and its complement. They focus on the complement of a Steiner configuration of points because it is a proper hyperplanes section of a monomial ideal that is the Stanley-Reisner ideal of a matroid. They study homological invariants of the complement of a Steiner configuration of points, such as Hilbert function and Betti numbers. They 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. However, the monomial ideal associated to a Steiner configuration of points does not lead to a Cohen-Macaulay algebra and they cannot apply the same methods as for its complement. In addition, they also compute the parameters of linear codes associated to any Steiner configuration of points and its complement.
Given any Steiner system \(S(t, n, v)\), the authors associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points in \(\mathbb{P}^n\) and its complement. They focus on the complement of a Steiner configuration of points because it is a proper hyperplanes section of a monomial ideal that is the Stanley-Reisner ideal of a matroid. They study homological invariants of the complement of a Steiner configuration of points, such as Hilbert function and Betti numbers. They 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. However, the monomial ideal associated to a Steiner configuration of points does not lead to a Cohen-Macaulay algebra and they cannot apply the same methods as for its complement. In addition, they also compute the parameters of linear codes associated to any Steiner configuration of points and its complement.
Reviewer: Zihong Tian (Shijiazhuang)
MSC:
| 51E10 | Steiner systems in finite geometry |
| 51E22 | Linear codes and caps in Galois spaces |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 14G50 | Applications to coding theory and cryptography of arithmetic geometry |
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
| 94B05 | Linear codes (general theory) |
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