Height of varieties over finitely generated fields
Jose Ignacio Burgos Gil, Patrice Philippon, Martin Sombra
TL;DR
This paper develops a method to express the height of varieties over finitely generated fields as an integral over local heights, enabling new computations of arithmetic intersection numbers for non-toric varieties.
Contribution
It extends previous work on toric varieties to non-toric cases by providing a formula for heights as integrals, facilitating arithmetic intersection calculations.
Findings
Height expressed as an integral over local heights.
Extended combinatorial formulas to non-toric varieties.
Enabled computation of arithmetic intersection numbers.
Abstract
We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of non toric arithmetic varieties over the rational numbers.
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