  In this paper we compute the Hilbert functions of irreducible (or smooth) and
reduced arithmetically Gorenstein schemes that are twisted anti-canonical
divisors on arithmetically Cohen-Macaulay schemes. We also prove some folklore
results characterizing the Hilbert functions of irreducible standard
determinantal schemes, and we use them to produce a new class of functions that
occur as Hilbert functions of irreducible (or smooth) and reduced
arithmetically Gorenstein schemes in any codimension.
