Steiner systems and configurations of points

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.

Steiner Configurations Ideals: Containment and Colouring

“…We also point out that the two above preprints [21,22] do not compute the Waldschmidt constant exactly for any class of ideals, they study lower bounds for the Waldschmidt constant. Furthermore, as in [1] the authors found the exact value of the Waldschmidt constant for the Complement of a Steiner configurations of points, then Chudnovsky and Demailly's Conjectures easily follow for our class of ideals (see Section 3).…”

“…We will focus on the Containment problem, and we will show that the Stable Harbourne Conjecture and the Stable Harbourne-Huneke Conjecture hold for the defining ideal of a Complement of a Steiner configuration of points in P n k := P n . As pointed out in Remarks 2.5 and 2.6 in [1] in the language of Algebraic Geometry/Commutative Algebra, Steiner configurations of points and their Complement are special subsets of star configurations.…”

“…In this paper, we continue the study of Steiner configurations of points and their invariants, such as Hilbert Function, Betti numbers, Waldschmidt constant, regularity and resurgence found in [1]. We will focus on the Containment problem, and we will show that the Stable Harbourne Conjecture and the Stable Harbourne-Huneke Conjecture hold for the defining ideal of a Complement of a Steiner configuration of points in P n k := P n .…”

“…The ideal I X defining the Complement of a Steiner system is not a monomial ideal. However, the authors proved that the symbolic powers of I X and J share the same homological invariants (see Proposition 3.6 in [1]). This was possible because J is the Stanley-Reisner ideal of a matroid, so its symbolic powers define an arithmetically Cohen-Macaulay (ACM) scheme which gives, after proper hyperplane sections, the scheme of fat points supported on X.…”

“…In this paper, we study the containment properties of the ideal defining a Complement of a Steiner configuration of points in P n . Section 2 is devoted to recall notation, definitions and known results from the work in [1] that we will use in the next sections. The main result of Section 3 is Theorem 4 where we prove that an ideal defining the Complement of a Steiner Configuration of points in P n satisfies both the Stable Harbourne Conjecture and the Stable Harbourne-Huneke Conjecture.…”