Hilbert Polynomials of Non-standard Bigraded Algebras

If R is standard bigraded, i.e. R is generated by elements of bidegrees (1, 0) and (0, 1), then HR(u, v) is equal to a polynomial in u, v for u, v large enough [Ba], [KMV], [W]. This fact does not hold if R is not standard bigraded. In this paper we will study the case R is generated by elements of bidegrees (1, 0), (d1, 1), . . . , (dr, 1), where d1, . . . , dr are non-negative integers. This case was considered first by P. Roberts in [Ro1] where it is shown that there exist integers c and v0 such that HR(u, v) is equal to a polynomial PR(u, v) for u ≥ cv and v ≥ v0. He calls PR(u, v) the Hilbert polynomial of the bigraded algebra R. It is worth to notice that Hilbert polynomials of bigraded algebras of the above type appear in Gabber’s proof of Serre non-negativity conjecture (see e.g. [Ro2]) and that the positivity of certain coefficient of such a Hilbert polynomial is strongly related to Serre’s positivity conjecture on intersection multiplicities [Ro3]. Using a different method we are able to show that for