Castelnuovoʹs Lemma and h-vectors of Cohen-Macaulay homogeneous domains Original Research Article

This paper gives a new postulation of the Hubert function of a Cohen-Macaulay homogeneous domain. If A is a Cohen-Macaulay homogeneous algebra over a field k, there are positive integers h0, h1, …, hs satisfying ∑i ≥ 0dimkAiλi = (h0 + h1λ + … + hsλs)/(1 − λ)d, where d is the Krull dimension of A. We call the vector (h0, h1, …, hs) the h-vector of A. Let A be a Cohen-Macaulay homogeneous domain over C with the h-vector (h0, h1, …, hs). It is well known that hi ≥ h1, for all 2 ≤ i ≤ s − 1. We will show that if the equality holds for some 2 ≤ i ≤ s − 2 then h1 = h2 = … = hs − 1 and hs ≤ h1 (when hs ≥ 2, the condition hs − 1 = h1 also implies the same assertion). To prove this result, we will modify Castelnuovoʹs argument in his study on curves of maximal genus.