Reduction theory of point clusters in projective space (Q664246)

The author generalizes the results of [\textit{M. Stoll} and \textit{J. E. Cremona}, J. Reine Angew. Math. 565, 79--99 (2003; Zbl 1153.11317)] on the \textit{reduction theory} of binary forms, which describes positive zero-cycles in \(\mathbb P^1\), to \textit{positive zero cycles (or point clusters) } in projective spaces of arbitrary dimension \(\mathbb P^n\). It has also application to more general projectives varieties in \(\mathbb P^n\), by associating a suitable zero-cycle to them in an \(\text{PGL}(n+1)\)-invariant way. The author considers projectives varieties over \(\mathbb Q\) in some \(\mathbb P^n\), with fixed discrete invariants. On this set, there is an action of \(\text{SL}(n+1,\mathbb Z)\) by linear substitution of the coordinates. The author selects a specific representative of each \(\text{SL}(n+1,\mathbb Z)\)-orbit of a given cluster, which is called \textit{reduced} in a way as canonical as possible. This representative will then allow a description as the zero set of polynomials with fairly small integer coefficients. As an application, the author shows how one can find a unimodular transformation that produces a small equation for a given smooth plane curve. In [loc. cit.], the key role is played by a map \(z\) from binary forms of degree \(d\) into the symmetric space of \(\text{SL}(2,\mathbb R)\) (which is the hyperbolic plane \(\mathcal H\) in this case) that is equivariant with respect to the action of \(\text{SL}(2,\mathbb R)\). The author then defines a form \(F\) to be \textit{reduced} if \(z(F)\) is in the standard fundamental domain for \(\text{SL}(2,\mathbb Z)\) in \(\mathcal H\). In order to make the map \(z\) as canonical as possible, he uses a larger group than \(\text{SL}(2,\mathbb Z)\), namely \(\text{SL}(2,\mathbb C)\). He then looks for a map \(z\) from binary forms with complex coefficients in the symmetric space \(\mathcal H_{\mathbb C}\) for \(\text{SL}(2,\mathbb C)\) that is \(\text{SL}(2,\mathbb C)\)-equivariant and commutes with complex conjugation. This map restricted to real forms will have image contained in \(\mathcal H\) and satisfies the initial requirements. In the more general situation of the article, the author works with the space \(\mathcal H_{n,\mathbb R}\) of positive definite quadratic forms in \(n+1\) variables, modulo scaling, and the space \(\mathcal H_{n,\mathbb C}\) of positive definite Hermitian forms in \(n+1\) variables, modulo scaling by positive real factors. There is a natural action of complex conjugation on \(\mathcal H_{n,\mathbb C}\); the subset fixed by it can be identified with \(\mathcal H_{n,\mathbb R}\). The author defines a distance function on \(\mathcal H_{n,\mathbb C}\) depending on a collection of points in \(\mathbb P^n(\mathbb C)\). Under a suitable condition on the point-cluster or zero-cycle \(\mathbb Z\), this distance function has a unique critical point, which provides a global minimum. This point is assigned to \(Z\) as its covariant \(z(Z)\), which solves the problem.