Reduction theory of point clusters in projective space

TL;DR

This paper extends the reduction theory of binary forms to positive zero-cycles in higher-dimensional projective spaces, aiming to facilitate applications to broader classes of projective varieties.

Contribution

It generalizes existing reduction results from binary forms to point clusters in P^n, providing a framework for applications to various projective varieties.

Findings

Generalization of reduction theory to higher dimensions

Application framework for projective varieties

Discussion of smooth plane curves case

Abstract

In this paper, we generalise results obtained earlier by John Cremona and the author on the reduction theory of binary forms, which describe positive zero-cycles in P^1, to positive zero-cycles (or point clusters) in projective spaces of arbitrary dimension. This should have applications to more general projective varieties in P^n, by associating a suitable positive zero-cycle to them in an PGL(n+1)-invariant way. We discuss this in the case of (smooth) plane curves.