{c}
{title2=$L\longmapsto h_L\pmod{O(1)}$}

On a projective variety over a <number field>, a very ample <line bundle> gives a logarithmic <projective height> by an embedding. Tensor products correspond to sums by the Segre embedding, and a sufficiently ample twist expresses any line bundle as a difference of very ample ones. This defines its height modulo bounded functions. It is additive and functorial under morphism pullback. An ample bundle has the <Northcott theorem> finiteness property for bounded height and bounded point degree; a trivial bundle need not. The <product formula> makes these absolute heights independent of scalar coordinates and of the number field used to compute them.


 Weil height machine (source code)