# Reduction theory of point clusters in projective space

### Michael Stoll

Universität Bayreuth, Germany

## 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.

## Cite this article

Michael Stoll, Reduction theory of point clusters in projective space. Groups Geom. Dyn. 5 (2011), no. 2, pp. 553–565

DOI 10.4171/GGD/139