Steiner Configurations Ideals: Containment and Colouring

Abstract

Given a homogeneous ideal I⊆k[x0,…,xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m,r∈N, I(m)⊆Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pkn. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System.

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

“…The classifiers discussed above seem perfectly fit for this task (like Table 10 contains a summary for Figure 1 as the exemplified case). 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27, 28} U2, D2: {1, 2, 3,4,6,7,9,10,11,12,13,14,15,16,18,19,20,21,22,24,25,26,27 Regarding the pairing (U2, D2) appearing for C 28 − T d (Figure 27), this pairing does not appear for C 28 − D 2 (Figure 25 vs. Figure 26) suggesting that its ...…”

“…In some cases, operating with connected undirected unweighted graphs [9] provides all necessary information, whereas in other cases more specificity is needed [10]. On the other hand, coloring graphs (of the vertices [11], edges [12] and planes [13]) can provide useful visual information.…”