Reduction theory of point clusters in projective space

Abstract

We generalise earlier results of John Cremona and the author on the reduction theory of binary forms, whose zeros give point clusters in ${\mathbb{P}}^1$, to point clusters in projective spaces ${\mathbb{P}}^n$ of arbitrary dimension. In particular, we show how to find a reduced representative in the SL($n+1,\mathbb{Z}$)-orbit of a given cluster. As an application, we show how one can find a unimodular transformation that produces a small equation for a given smooth plane curve.