Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in P 2 . Given a valid Hilbert function H of a zero-dimensional scheme in P 2 , we show how to construct a set of fat points Z ⊆ P 2 of double and reduced points such that H Z , the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of a set a positive number of double …