Distinguishing \(\Bbbk\)-configurations (Q1670786)

In the late 1980's, \textit{L. G. Roberts} and \textit{M. Roitman} [J. Pure Appl. Algebra 56, No. 1, 85--104 (1989; Zbl 0673.13015)] introduced special configurations of points in \(\mathbb{P}^2\) which they named \(\Bbbk\)-configurations. A \(\Bbbk\)-configuration is a set of points \(\mathbb{X}\) in \(\mathbb{P}^2\) that satisfies a number of geometric conditions. Associated to a \(\Bbbk\)-configuration is a sequence \((d_1,\cdots, d_s)\) of positive integers, called its type, which encodes many of its homological invariants. The definition of \(\Bbbk\)-configuration was first extended to \(\mathbb{P}^3\) by \textit{T. Harima} [J. Pure Appl. Algebra 103, No. 3, 313--324 (1995; Zbl 0847.13003)], and later to all \(\mathbb{P}^n\) by Geramita, Harima, and Shin. As shown by Roberts and Roitman, all \(\Bbbk\)-configurations of type \((d_1,\cdots, d_s)\) have the same Hilbert function. This result was later generalized by \textit{A. V. Geramita} et al. [Adv. Math. 152, No. 1, 78--119 (2000; Zbl 0965.13011)]. The authors of the present paper determine how one can distinguish the \( \Bbbk\)-configurations from an algebraic point-of-view. They show that the number of lines that contain \(d_s\) points of \(\mathbb{X}\) is encoded in the Hilbert function of fat points supported on \(\mathbb{X}\). That is, one can obtain an exact value if one instead considers the Hilbert function of the set of fat points supported on the \(\Bbbk\)-configuration.