### abstract

- A k-configuration is a set of points X in P2 that satisfies a number of geometric conditions. Associated to a k-configuration is a sequence (d1, …, ds) of positive integers, called its type, which encodes many of its homological invariants. We distinguish k-configurations by counting the number of lines that contain ds points of X. In particular, we show that for all integers m ≫ 0, the number of such lines is precisely the value of ΔHmX (mds − 1). Here, ΔHmX (−) is the first difference of the Hilbert function of the fat points of multiplicity m supported on X.