Steiner Configurations ideals: containment and colouring

Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ 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 $\mathbb{P}^{n}_{k}$. 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.


Introduction
In this paper we continue the study of Steiner configurations of points and their invariants, such as Hilbert Function, Betti numbers, Waldschmidt constant, regularity, resurgence found in [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 [3] in the language of Algebraic Geometry/Commutative Algebra, Steiner configurations of points and their Complement are special subsets of star configurations.
First, we give an overview on the Containment problem to introduce the related conjectures. Then we devote Section 2 to recall notation, definitions and known results for a Steiner configuration of points and its Complement that we will use to prove the results of this paper. Let I be a homogeneous ideal in the standard graded polynomial ring R := k[x 0 , . . . , x n ], where k is a field. Given an integer m, we denote by I m the regular power of the ideal I. The m-th symbolic power of I is defined as where Ass(I) denotes the set of associated primes of I and R p is the localization of R at a prime ideal p.
If I is a radical ideal (this includes for instance squarefree monomial ideals and ideals of finite sets of points) then Symbolic powers of ideals play a significant role in the famous Zariski-Nagata Theorem (see [48,54]). If R is a polynomial ring over an algebraically closed field k, then I (m) consists precisely of those functions which vanish on the algebraic variety defined by I with multiplicity at least m. It is easy to show from the definition that I r ⊆ I (m) if and only if r ≥ m. The reverse inclusion I (m) ⊆ I r motivates the following question.
Question 1.1. (Containment problem) Given a homogeneous ideal I ⊆ k[x 0 , . . . , x n ], for which pairs m, r ∈ N, does I (m) ⊆ I r hold?
One of the initial works that introduce Question 1.1 is [51]. The problem is still open in general and in the last couple of decades it was extensively studied for several classes of ideals, in particular for ideals defining finite sets of points in projective and multiprojective spaces, see [8,9,17,13,24,25,26,35,36,37,46] just to cite some among all the known results. Containment problems are useful in giving lower bounds to nonzero homogeneous forms vanishing through a finite set of points with a fixed multiplicity.
It is of great interest to study the ideals of fat points. Given distinct points P 1 , . . . , P s ∈ P n and nonnegative integers m i (not all 0), let Z = m 1 p 1 + · · · + m s p s denote the scheme (called a fat point scheme) defined by the ideal I Z = ∩ s i=1 (I mi Pi ) ⊆ k[P n ], where I Pi is the ideal generated by all homogeneous polynomials vanishing at P i . Symbolic powers of I Z take the form I . We say that Z is reduced if I Z is a radical ideal.
The Containment problem also helps us to bound certain useful invariants like Waldschmidt constant, α(I), of an ideal I defined as where α(I) is the minimum integer d such that I d = (0), that is, it is the least degree of a minimal generator of I. This limit exists and was first defined by Waldschmidt [53] for ideals of finite sets of points in the context of complex analysis; specifically, in our language, the problem was to determine the minimal degree of a hypersurface that passed through a collection of points with prescribed multiplicities.
The following slight different version of Question 1.1 was introduced in [39]. Recall that the big height of an ideal I refers to the maximum of all the heights of its associated prime ideals. A counterexample to the above conjecture was initially found in [23].
A celebrated result of [24,41,46] is shown in the next theorem.
Theorem 1.4. Let R be a regular ring and I a radical ideal in R. Then for all n ∈ N, whenever h is the big height of I.
One could hope to sharpen the containment by reducing the symbolic power on the left hand side by a constant or increasing the ordinary power on the right hand side by a fixed constant.
In the study of finding the least degree of minimal generators of an ideal I, Chudnovsky made the following: Conjecture 1.7. (Chudnovsky's Conjecture). Suppose that k is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points X ⊆ P n k . Then, for all h > 1, A generalization of Chudnovsky's Conjecture is the following: Suppose that k is an algebraically closed field of characteristic 0. Let I be the defining ideal of a set of points X ⊆ P n k and let m ∈ N be any integer. Then, for all h > 1, Two recent preprints, [5,6], focus on the Containment problem and related conjectures. In the first one, the authors show that Chudnovsky's Conjecture holds for sufficiently many general points and to prove it they show that one of the containments conjectured by Harbourne and Huneke holds eventually, meaning for large powers (see Theorem 4.6 in [5]). They also show other related results, for example, that general sets of points have expected resurgence and thus satisfy the Stable Harbourne Conjecture.
In the second preprint, the authors show that Demailly's Conjecture (which is a generalization of Chudnovsky's) also holds for sufficiently many general points, for star configurations (in general, not just points) and for generic determinantal ideals.
In this paper we prove that the Stable Harbourne Conjecture and the Stable Harbourne-Huneke Conjecture hold for ideals defining the Complement of a Steiner Configuration of points in P n that are special subsets of star configurations and, hence, far from being general. We will give more details in Section 3.
We remark that the least degree of a minimal generator of the ideal defining the Complement of a Steiner Configuration of points in P n is strictly less than the least degree of a minimal generator of the ideal of a star configurations (see Theorem 2.6 and also Proposition 2.9 in [29]). So, it is worth investigating whether the Containment problem and its related conjectures hold for the Complement of a Steiner Configuration of points in P n .
In [3] the authors constructed a squarefree monomial ideal J associated to a set X of points in P n constructed from the Complement of a Steiner system. The ideal I X defining the Complement of a Steiner system is not a monomial ideal. But the authors proved that the symbolic powers of I X and J share the same homological invariants (see Proposition 3.6 in [3]). This was possible because J is the Stanley-Reisner ideal of a matroid, so its symbolic powers define an arithmetically Cohen-Macaulay (ACM for short) scheme which gives, after proper hyperplane sections, the scheme of fat points supported on X. But we point out that the regular powers of J are not necessarily ACM anymore and we cannot relate them to squarefree monomial ideals. Thus, the homological invariants of the regular powers of J are not immediately correlated to that of I X .
In [17] the authors proved that the Chudnovsky's Conjecture, the Harbourne's Conjecture and the Harbourne-Huneke containment conjectures hold for squarefree monomial ideals.
As previously remarked, since the ideal I X defining the Complement of a Steiner system is not a squarefree monomial ideal, we cannot recover the Stable Harbourne Conjecture and the Stable Harbourne -Huneke Conjecture using [17].
We also point out that the two above preprints [5,6] do not compute the Waldschmidt constant exactly for any class of ideals, they study lower bounds for the Waldschmidt constant. And, since in [3] 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).
For other results on this topic we can also see [27,47]. Another tool useful to measure the non-containment among symbolic and ordinary powers of ideals is the notion of resurgence ρ(I) of an ideal I, introduced in [9] that gives some notion of how small the ratio m/r can be and still be sure to have I (m) ⊆ I r . Definition 1.9. Let I be a nonzero, proper ideal in a commutative ring R, the resurgence of the ideal I is given by It always satisfies ρ(I) ≥ 1. The groundbreaking results of [24,41,46] show that ρ(I) ≤ h, where h is the big height of the radical ideal I. This motivates us to ask whether ρ(I) can strictly be less than its big height and which are some of the interesting consequences. Although there are few cases where the resurgence has been computed, in general, it is extremely difficult to estimate the exact value for ρ(I).
The reader can look at [22] for the first examples where the resurgence and the asymptotic resurgence are not equal. An asymptotic version of the resurgence was introduced in the paper [35]. Definition 1.10. For a nonzero, proper homogeneous ideal I ⊆ k[x 0 , . . . , x n ], the asymptotic resurgence ρ a (I) is defined as follows: It is clear from the definition that 1 ≤ ρ a (I) ≤ ρ(I). As pointed out in [40], DiPasquale, Francisco, Mermin, Schweig showed that ρ a (I) = sup{m/r : I (m) I r }, where I r is the integral closure of I r (see also [20] Corollary 4.14) .
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 [3] that we will use in the next sections. The main result of Section 3 is Theorem 3.5 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. In Lemma 3.2, we give a criterion for when the resurgence number can be computed in finite number of steps. This result improves the bounds found in Corollary 4.8 in [3]. We also point out that Lemma 3.2 is similar to results from [19,20]. As a consequence, in Corollary 3.6 we 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 (see [34] for the first definition). Moreover, using Theorem 2.6, Corollaries 2.7 and 2.8, we show that the ideal of a Complement of a Steiner Configuration of points satisfies Chudnovsky and Demailly's Conjectures (see Corollary 2.8, Corollary 3.7 and Corollary 3.8).
Finally, in Section 4, 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. The ideas come from the paper [28] where the authors start to study the natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction.
There exists an extensive literature on the subject of colourings both from Design Theory and Algebraic Geometry/Commutative Algebra point of view. Among all, we make use of [10,28,38,31] as some referring texts for preliminaries on hypergraph theory and associated primes and for an algebraic method to compute the chromatic number, respectively.
Most of the existing papers are devoted to the case of weak colourings (or vertex colourings), i.e. colourings where the colours are assigned to the elements in such a way that no hyperedge is monochromatic (i.e. no hyperedge has all its elements assigned the same colour). The reader can see [10] or Chapter 3 in [31] for other different types of colouring a hypergraph, such as strong vertex colouring, vertex equicolouring, good colouring of H.
In this paper we use the case of weak colouring to get result on the Containment problem since it is the one commonly used in Combinatorial Commutative Algebra. The main result of this section is Theorem 4.8 that more generally predicts the failure of the containement for squarefree monomial ideals based on the definition of coverability (see Definition 4.4). We apply these results in the case that H is a Steiner System.
We end the paper recalling some open questions posed in [3] and that are still under investigations and posing new ones as possible further research problems.

Notation, definitions and known results for Ideals of a Steiner configuration of points and its Complement
In this section we recall the main results from [3], where the authors studied the homological properties of ideals constructed from Steiner systems, especially in the zero dimensional case of P n .
The elements in V are called vertices or points and those of B are called blocks. In particular, a Steiner triple system of order v, ST S(v), is a collection of triples (3-subsets) of V , such that each unordered pair of elements is contained in precisely one block, and a Steiner quadruple system of order v, SQS(v), is a collection of quadruples (4-subsets) of V such that each triple is found in precisely one block.
The existence of a Steiner system strongly depends on the parameters (t, n, v). If a Steiner system We use [14] and [16] as main references for all the background on design theory.
We recall the most known example. Denoted by C (n,v) the set containing all the subsets of V with n elements the above definition applied to a Steiner system gives us two different sets of points. We call X H,C (v,n) \B the Complement of a Steiner configuration of points with respect to H (or C-Steiner X C if there is no ambiguity).
As pointed out in [3], Remarks 2.5 and 2.6 a Steiner configuration of points and its Complement are subschemes of a star configuration of v n points in P n (see [11,12,29,30] just to cite some reference on star configurations).
We also have We recall the most known construction of Steiner Configuration of points and its Complement.
In [3] the authors constructed a squarefree monomial ideal J associated to a set X H,C of points in P n constructed from the Complement of a Steiner system. The ideal I X H,C defining the Complement of a Steiner system is not a monomial ideal. But the authors proved that the symbolic powers of I X H,C and J share the same homological invariants (see Proposition 3.6 in [3]).
The following results give the least degree of a minimal generator and the regularity and the Waldschmidt constant of an ideal defining the Complement of a Steiner configuration of points, respectively. Theorem 2.6 ([3], Theorem 3.9). Let (V, B) be a Steiner system of type S(t, n, v). Set I X H,C := I X C the ideal defining the Complement of the Steiner configuration of points associated to S(t, n, v). Then X C ) + pv, where m = pn + q and 0 ≤ q < n and α(I (n)

Asymptotic resurgence and Stable Harbourne Conjecture
Containment problems have been of interest among commutative algebraists and algebraic geometers. In the last decade, several conjectures related to this problem have been posed creating an active area of current interests and outgoing investigations.
A celebrated result of [24,41,46] is that I (hn) ⊆ I n for all n ∈ N, whenever h is the big height of I. One could hope to sharpen the containment by reducing the symbolic power on the left hand side by a constant or increasing the ordinary power on the right hand side by a fixed constant. This motivates us to look at the Stable Harbourne Conjecture and the Stable Harbourne -Huneke Conjecture and study which class of ideals satisfies them. Here, we prove that the ideal defining a Complement of a Steiner configurations of points satisfies both conjectures. We need to recall some known results.
In [33], the Conjecture 1.5 is shown to hold (1) if there exists k > 0 such that I (hk−h) ⊆ I k ; (2) if I (hk−h+1) ⊆ I k for some k and I (r+h) ⊆ II (r) for all r ≥ k; (3) if the resurgence satisfies ρ(I) < h.
In particular, condition (2) gives a criterion for the Stable Harbourne Conjecture (SHC for short) to hold. Namely, for a radical ideal of big height h, if for all k ≥ 1, it is I (k+h) ⊆ II (k) and fix an integer C and m such that I (hm−C) ⊆ I m holds, then for all q ≥ m, we have Theorem 3.1 (Theorem 2.5, [33]). Let R be a regular ring containing a field, and let I be a radical ideal in R with big height h. If I (h(m−1)) ⊆ I m for some m ≥ 2, then I (h(k−1)) ⊆ I k for all k >> 0 (indeed for all k ≥ hm).
For the ease of the reader, we adapt the proof in our case. Proof. Using Briancon Skoda Theorem ( [52], Corollary 13.3.4), we have that I r+n ⊆ I r , where n + 1 is the number of variables in the polynomial ring and I denotes the integral closure of the ideal I. For s, r ∈ N such that I (s) I r then I (s) I r+n . Using [20], Lemma 4.12, we get s r+n < h(1 − 1/r 1 ) = ρ a (I), that is, s r < (1 + n/r)h(1 − 1/r 1 ).
If ρ(I) > ρ a (I), applying [19] Proposition 2.2, then there exist s 0 , r 0 , such that I (s0) I r0 and solving for r gives us the inequality r ≥ n s0/r0 and s is such that I (s) I r , we have s r < s0 r0 .
Hence, it suffices to look at r ≤ n s0/r0 Proof. Suppose we have a b = h, then I (hb) I b is a contradiction since I (hk) ⊆ I k for all k ∈ N (from [24,41,46]). Hence, ρ(I) < a b = h.
The next proposition shows that Conjecture 3.1 in [7] holds for the Complement of a Steiner Configuration of points. Proof. From Theorem 2.6, we have α(I (nr) ) = rv. From [3], Corollary 4.7, we have ω(I r ) = α(I r ) = r(v − n), where ω(I) is the maximum of the generating degrees of the ideal I. Since I (nr) ⊆ I r for all r ≥ 1, we have α(I (nr) ) ≥ rω(I) = rα(I) and α(I r ) = r(v − n), so α(I (nr) ) − rω(I) = rv − r(v − n) = rn. Since every minimal generator of I (nr) is contained inside I r and the difference between the degree of any nonzero homogeneous polynomial in I (nr) and that of generators of I r is at least rn, we have that I (nr) ⊆ M rn I r , the conclusion follows.
We prove the main result of this section: Proof. (1) Consider the Steiner Configuration of points S(t, n, v) in P n k and I := I C the ideal defining its Complement.
Hence I (nk−n) ⊆ I k for k 0.
As a consequence, we can 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 (see [34]). Corollary 3.6. Let I ⊆ k[x 0 , . . . , x n ] be the ideal defining the Complement of a Steiner Configuration of points in P n k . Then, ρ(I) < n.
We give an alternative proof of Chudnovky's Conjecture: Corollary 3.8. Let I ⊆ k[x 0 , . . . , x n ] be the ideal defining the Complement of a Steiner Configuration of points in P n k . Then Demailly's Conjecture holds for I.

Containment and colouring
In this section we focus on the relation between the colourability of a hypergraph H and the failure of the containment problem for the cover ideal associated to H. Then, we apply these results in the case that H is a Steiner System. There exists an extensive literature on the subject of colourings both from Design Theory and Algebraic Geometry/Commutative Algebra point of view. Among all, we make use of [10,28,38,31] as some of referring texts.
Most of the existing papers are devoted to the case of weak colourings (or vertex colourings), i.e. colourings where the colours are assigned to the elements in such a way that no hyperedge is monochromatic (i.e. no hyperedge has all its elements assigned the same colour). The reader can see [10] or Chapter 3 in [31] for other types of colouring a hypergraph, such as strong vertex colouring, vertex equicolouring, good colouring of H.
In this paper we use the case of weak colouring to get results on Containment problem.
We first recall some known definitions and results from [10] or [31], Chapter 2.
A hypergraph is a pair H = (V, E), where E = {x 1 , . . . , x n } is a finite nonempty set containing n elements called vertices and E = {e i } i∈I (I set of indices) is a family of subsets of X, called edges, or otherwise hyperedges, such that for all e ∈ E, e = ∅ and ∪ e∈E e = X. As for graphs, proper colourings generate partitions of the vertex set into a number of stable (independent) non-empty subsets called colour classes, with as many classes as the number of colours actually used.
Notice also that no colouring of H with two colours exists. Then χ(H) = 3. Remark 4.7. We refer the reader to Section 3.5 in [31] to see examples of different types of colourings that give different chromatic numbers for the same H. In particular, in Example 8 of [31], the strong vertex colouring of H as in (4.1) gives χ(H) = 7 (recall that a mapping c is a strong colouring of vertices of H if for all e ∈ E it is |c(e)| = |e|).
For a non empty hypergraph H, i.e. H ⊂ 2 V , we define the ideal called the cover ideal of H, where for a subset of V , σ := {i 1 , i 2 , . . . , i n } ⊆ V, the ideal denotes the prime ideal generated by the variables indexed by σ. We study some properties of the cover ideals of B. The following results show a relation between the coverability of a hypergraph H and that the Containment problem can fail. Proof. We put τ := τ (H) and w := x 1 · x 2 · · · x n . In order to prove the statement it is enough to show that w ∈ J (τ ) H . By contradiction, assume w ∈ J d H . Thus, there exist w 1 , . . . , w d ∈ J H such that w = w 1 · · · w d . We U j := {x u ∈ V | x u divides w j }, then U 1 , . . . , U d is a partition of V . Thus, for each b ∈ B we have w i ∈ p b , therefore U i ∩ e = 0 with i = 1, . . . , d. This contradicts that H is not d-coverable.
Recall that an m-colouring of (V, B) is called an m-bicolouring if the vertices of each b ∈ B are coloured with exactly two colours. A Steiner Triple Systems (V, B) admitting an m-bicolouring is m-bicolourable. Thus, in a bicolouring of a Steiner Triple System (V, B), every triple has two elements in one colour class and one in another class, so there are no monochromatic triples nor polychromatic triples (i.e. triples receiving three colours). For instance, for a deep investigation of colouring properties of Steiner Triple Systems the reader can see [15].
As a consequence, we get a failure of the containment for the cover ideals associated to Steiner Triple Systems (V, B) of type S(t, n, v).
Proof. It is enough to show that B in not 2-coverable. Assume by contradiction V = U 1 ∪ U 2 . By definition, for each {i, j, k} ∈ B we have {i, j, k} ∩ U 1 = ∅ and {i, j, k} ∩ U 2 = ∅. This implies that S(2, 3, v) is 2-bicolourable contradicting a well known fact about Steiner Triple Systems, see [50].
We end the paper showing the failure of the containment for the cover ideals associated to Steiner Systems. Proof. Note that, from Theorem 4.8, it is enough to show that B is not t-coverable. Assume by contradiction there is a partition of V in t colour classes, V = C 1 ∪ · · · ∪ C t such that B ∩ C j = ∅ for each j = 1, . . . , t and B a block of S. We denote by c j the number of elements in C j for j = 1, . . . , t. Note that c j ≤ a + 1. Indeed if i 1 , i 2 , . . . , i a+2 ∈ C j are different elements, then a block containing i 1 , i 2 , . . . , i a+2 cannot intersect t colour classes. This implies v ≤ (a + 1)t.
The next example shows that Theorem 4.8 does not characterize the failure of the containment.
B ⊆ J 2 B cannot be motivated from Theorem 4.8.

Conclusions
Several conjectures have been posed on the Containment problem, creating an active area of current interests and ongoing investigations. In this paper, we 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 . Moreover, given a hypergraph H, we also study the relation between its colourability and that the Containment problem can fail for the cover ideal associated to H. We wish to continue the study of Steiner configurations of points and their Complements, since they are special subsets of star configurations whose Hilbert Function is the same as sets of generic points while geometrically they are far of being generic.
We end this section recalling some open questions that are still under investigations and posing new ones.
We recall from [3], that from a combinatorial point of view, two Steiner systems having the same parameters could have very different properties and such differences effect the homological invariants. Using experiments with [1] and [32] we ask: Question 5.1. Let (V, B) be a Steiner system of type S(t, n, v), and X H,B the associated Steiner configuration of points. Assume that the hyperplanes in H are chosen generically. Do the Hilbert function and the graded Betti numbers of X H,B only depend on t, n, v? Question 5.2. Let (V, B) be a Steiner system of type S(t, n, v), and X H,B the associated Steiner configuration of points. Assume that the hyperplanes in H are chosen generically. Are the Hilbert function and the graded Betti numbers of X H,B generic with respect to the Hilbert function? (i.e. the same as a set of |B| generic points in P n ?) 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 suggest the following