About \(K\)-configurations in \(\mathbb{P}^ 2\) (Q1208224)

Very special finite subsets of a projective plane over a field \(K\) are considered. These subsets, named \(K\)-configurations, have \(d_ 1>0\) of points on a line, then \(d_ 2>d_ 1\) points on a line missing the former ones, and so on. The Hilbert function of such a subset is easily computed so that it determines and is in turn determined by the integers \(d_ i\). A even more special type of finite sets of points, the \(n\)- regular \(K\)-configurations, are also considered: the Hilbert functions of some of them are shown to be those of the complete intersections of plane curves.