Maximal families of superabundant point systems in \(\mathbb{P}^ 2\)
The author addresses the problem of classifying point systems (or \(N\)- uples of points) in the projective plane. More precisely, if \(Z\subset\mathbb{P}^ 2\) is a finite subscheme of degree \(N\) and \(d\) is a positive integer, consider the linear system \(I_ Z(d)\) of plane curves of degree \(d\) containing \(Z\). The subscheme \(Z\) is called superabundant (for the linear system of plane curves of degree \(d)\) if the dimension of \(I_ Z(d)\) is strictly greater than the expected dimension \({d+2\choose 2}-1-N\). The set of superabundant subschemes \(Z\) is a subscheme \(W_ N[d]\) of the Hilbert scheme \(\text{Hilb}^ N\mathbb{P}^ 2\). The author's main result is to give the dimension of \(W_ N[d]\) and determine when it is irreducible, in terms of \(d\) and \(N\).
scientific article; zbMATH DE number 59944
scientific article