Fattening of ACM arrangements of codimension 2 subspaces in ℙN

Abstract

In the present note we study configurations of codimension 2 flats in projective spaces and classify those with the smallest rate of growth of the initial sequence. Our work extends those of Bocci, Chiantini in P 2 and Janssen in P 3 to projective spaces of arbitrary dimension.

An invitation to the positivity and geometry of algebraic cycles

“…They showed that the constrain is serious and the points in Z either form what is now known as a star configuration (see [10]) or they are all collinear. This result has been generalised for flats of codimension 2 by Janssen [18] and Haghighi, Zaman Fashami and Szemberg [8] under the additional assumption that the union of studied flats is an arithmetically Cohen-Macaulay variety. Already in P 3 it is natural to study the same problem for connected curves.…”

“…In particular, Problem 1.5 is stated without assuming that the considered curve is an ACM-subscheme. Note that this assumption was inevitable in the approach taken on by Janssen [18] and in the generalisations proved in [8]. The reason is the application of the Hilbert-Burch theorem, which gives a useful description of the defining ideal of an arithmetically Cohen-Macaulay subvariety of codimension 2 in a projective space (or more generally: in a smooth projective variety).…”