The asymptotic growth of graded linear series on arbitrary projective schemes

Recently, Okounkov, Lazarsfeld and Mustata, and Kaveh and Khovanskii have shown that the growth of a graded linear series on a projective variety over an algebraically closed field is asymptotic to a polynomial. We give a complete description of the possible asymptotic growth of graded linear series on projective schemes over a perfect field. If the scheme is reduced, then the growth is polynomial like, but the growth can be very complex on nonreduced schemes. We also give an example of a graded family of m-primary ideals {I_n} in a nonreduced d-dimensional local ring R, such that the length of R/I_n divided by n^d does not have a limit, even when restricted to any arithmetic sequence.