Containment counterexamples for ideals of various configurations of points in PN

Singular loci of reflection arrangements and the containment problem

“…We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in [2,8,14,20] under a unified approach.…”

“…Our interest in singular loci of hyperplane arrangements has been sparked by the peculiar behavior of some ideals in this class with regards to containments between ordinary and symbolic powers. It is known thanks to [9,17,19] that the containments J(A) (2r) ⊆ J(A) r are satisfied for every positive integer r. What is more interesting, however, is that several examples of ideals J(A) have arisen in the literature as witnesses to the optimality of the above containment, in the sense that they have also been shown to satisfy J(A) (3) ⊆ J(A) 2 for certain groups G. In hindsight, the groups for which the stated noncontainment was known to hold before our work are the infinite family of monomial groups G(m, m, 3) [8,14] and two classical groups studied by Klein (G 24 ) and Wiman (G 27 ) [1,2].…”

Symbolic Powers of Derksen Ideals

“…Let J (A) be a radical ideal which defines the singular locus of the reflection arrangement A. It is known that J (A) (3) ⊆ J (A) 2 for certain groups G. Dumnicki, Szemberg, and Tutaj-Gasińska [DSTG13] and Harbourne and Seceleanu [HS15] showed it for the infinite family of monomial groups G(m, m, 3); and Klein and Wiman showed it for two classical groups In this manuscript we give a positive answer in several cases for the following question.…”

On the resurgence and asymptotic resurgence of homogeneous ideals

“…Subsequently, Harbourne conjectured in [1,Conjecture 8.4.3] that if I is a radical ideal in a regular ring R, then I (ht) ⊆ I t could be sharpened to I (ht−h+1) ⊆ I t for all t ∈ N, where h is the big height of I. Though there are large classes of ideals which satisfy Harbourne's conjecture [1,16], counter examples have been constructed in [11] (see [25] for more information on these counterexamples). Grifo in [15] questioned whether Harbourne's conjecture is true asymptotically i.e., I (ht−h+1) ⊆ I t for all t ≫ 0.…”

“…follows from [15] and [18] that the stable Harbourne conjecture is true when the resurgence or the asymptotic resurgence is strictly smaller than the bigheight h. A great source of examples for which the resurgence and asymptotic resurgence are known, comes either from the geometric side ( [4,5,10,24,25]) or from combinatorial side (see [30,28,19,29]).…”