Resurgence and Waldschmidt constant of the ideal of a fat almost collinear subscheme in ℙ²

Let Zn = p₀ + p₁ + ··· + pn be a configuration of points in ℙ², where all points pi except p₀ lie on a line, and let I(Zn) be its corresponding homogeneous ideal in 𝕂 [ℙ²]. The resurgence and the Waldschmidt constant of I(Zn) in [5] have been computed. In this note, we compute these two invariants for the defining ideal of a fat point subscheme Zn,c = cp₀ + p₁ +··· + pn, i.e. the point p₀ is considered with multiplicity c. Our strategy is similar to [5].