On the Hilbert Function of Partially General Unions of Double Points

We study the Hilbert function of a union of a fixed set of double points and a prescribed number of general double points. Our main results are for double points of Veronese varieties and of Segre–Veronese varieties. Among the Segre–Veronese varieties, we study the ones with all factors of dimension one and the ones with two factors, one of them of dimension one. We give many examples with exceptional or controlled behavior for a small number of double points.