Computing heights via limits of Hodge structures

TL;DR

This paper introduces a practical method for computing Beilinson--Bloch heights of cycles on varieties over number fields by analyzing limit mixed Hodge structures, demonstrated on specific nodal varieties.

Contribution

It develops a new computational approach linking limit mixed Hodge structures to Beilinson--Bloch heights, enabling explicit calculations for certain varieties.

Findings

Successfully computed heights for a nodal quartic curve.

Successfully computed heights for a nodal cubic threefold.

Identified primes involved in the height congruences.

Abstract

We consider the problem of explicitly computing Beilinson--Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson--Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson--Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases, we explain the nature of the primes occurring in the congruence.