# Groups, Geometry, and Dynamics

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**Volume 5, Issue 2, 2011, pp. 553–565**

**DOI: 10.4171/GGD/139**

Published online: 2011-03-06

Reduction theory of point clusters in projective space

Michael Stoll^{[1]}(1) Universität Bayreuth, Germany

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.

*Keywords: *Reduction theory, point clusters

Stoll Michael: Reduction theory of point clusters in projective space. *Groups Geom. Dyn.* 5 (2011), 553-565. doi: 10.4171/GGD/139