The numerical equivalence relation for height functions and ampleness and nefness criteria for divisors

In this paper, we study properties of Weil height functions associated with numerically trivial divisors. It helps us to define the fractional limit of h(E) with respect to h(D) on U, with D ample: Flim(D)(E, U) := liminf(P is an element of UhD(P)->infinity) h(E)(P)/h(D)(P). The value of Flim(D)(E, U) contains numerical information about a divisor E, enough to determine whether E is ample, numerically effective (nef) or pseudo-effective.