Volume and Hilbert functions of R-divisors

Abstract

Let X be a proper, normal algebraic variety of dimension n over a field K and D an R-divisor on X. The Hilbert function of D is the function H(X,D) : m 7→ h(mD) := dimK H(X,OX(bmDc)); defined for all m ∈ R. If D is an ample Cartier divisor then H(X,D) agrees with the usual Hilbert polynomial whenever m 1 is an integer, but in general H(X,D) is not a polynomial, not even if D is a Z-divisor and m ∈ Z. The simplest numerical invariant associated to the Hilbert function is the volume of D, defined as