A self-contained construction of Silverman's height functions attached to closed subschemes, generalizing the classical heights attached to Cartier divisors. Over a field K carrying a proper set M_K of absolute values, a closed subscheme Y meeting no associated point of X is presented as an intersection of effective Cartier divisors, each equipped with two globally generated invertible sheaves and regular generating sections; the local height lambda_Y(x,v) = min_i log max_j min_k |(s_j / s_D t_k)(x)|_v measures v-adically how close the point x lies to Y, growing without bound as x approaches Y. On a projective scheme two presentations of the same Y agree up to an M_K-constant, so lambda_Y depends only on Y; on a quasi-projective scheme the larger presentation ambiguity is controlled by a boundary function, the local height of the complement in a good projectivization. Local heights turn scheme intersection into minimum, ideal product into sum, inclusion into inequality, and commute with pullback along morphisms, each up to these error terms. Applied to the diagonal in X x X the construction yields the arithmetic distance function delta_X, symmetric and satisfying ultrametric-style triangle inequalities; averaging local heights over all absolute values of a field of definition, weighted by local degrees and normalized by the field degree, yields the global height h_Y, independent of that field and of the presentation up to a bounded function. Unlike the standard references, the ambient schemes are not assumed reduced or irreducible.
Expository lecture notes, with complete proofs, on height functions attached to closed subschemes of projective and quasi-projective schemes -- a generalization, due to Silverman (1987), of the class…