On Codimension Two Flats in Fermat-Type Arrangements

Abstract

In the present note we study certain arrangements of codimension 2 flats in projective spaces, we call them Fermat arrangements. We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.

Fermat-type configurations of lines in P3 and the containment problem

“…In [13] we show that in codimension 2 for I = I(N, n) = I(RF n N (N − 2)), the containment I (3) ⊆ I 2 still fails. On the other hand, there are several interesting algebraic properties of ideals I(F n N (k)), which we study in the forthcoming article [12].…”

Configurations of Linear Spaces of Codimension Two and the Containment Problem

“…(3) N,n ⊆ I 2 N,n has been found by Grzegorz Malara and Justinya Szpond and can bee seen in their upcoming paper [MS2].…”

Singular loci of reflection arrangements and the containment problem

“…Notably, most known counterexamples come from singular points of line arrangements: one family of counterexamples known in the literature under the name of Fermat configurations of points [8,14], corresponds in hindsight to the singular loci of the monomial groups G(m, m, 3), while two other sporadic counterexamples known as the Klein and the Wiman configurations [2] correspond to the singular loci of the groups G 24 and G 27 in the Shephard-Todd classification. The former family has been recently generalized to Fermat-like configurations of lines in P 3 in [20,21], which correspond to the singular loci of rank four monomial groups G(m, m, 4). For each of the ideals J defining one of these special configurations the non-containment J (3) ⊆ J 2 has been proven in the cited source.…”

“…The above-mentioned examples show the sharpness of the results in [9,17,19] for the pair m = 3, r = 2, leaving open this problem for all other pairs as well as Harbourne's conjecture for r > 2. Moreover, while the papers [20,21] give a negative answer to Harbourne's question in projective spaces of dimension n > 2 along the lines of the Fermat examples in the plane, they leave open the possibility of higher dimensional counterexamples of sporadic flavor which would parallel the Klein and Wiman examples. Indeed, in this paper we find several new sporadic examples of hyperplane arrangements A one each in P 3 , P 4 and P 5 for which J(A) (3) ⊆ J(A) 2 .…”