Combinatorics of Cremona monomial maps (Q2894537)

The \(n\)-dimensional Cremona group \(\mathrm{Cr}_n(k)\) is the group of birational automorphisms of the projective space \(\mathbb{P}^n_k\) defined over a field \(k\). The structure of \(\mathrm{Cr}_n(k)\) is well known for \(n=1, 2\) but little is known for \(n \geq 3\). In this paper the authors study the subset of \(\mathrm{Cr}_n(k)\) which consists of birational maps of \(\mathbb{P}^n\) defined by monomials of fixed degree. The main result of the paper is that the inverse of a birational map of \(\mathbb{P}^n\) defined by monomials of fixed degree is also defined by monomials of fixed degree and hence such maps form a subgroup of \(\mathrm{Cr}_n(k)\). Their proof is purely combinatorial. Let \(f : \mathbb{P}^n \dasharrow \mathbb{P}^n\) be a birational map given by \(f=(x^{v_0}, \ldots, x^{v_n})\), where \(x_0, \ldots, x_n\) are coordinates of \(\mathbb{P}^n\) and for any \(a=(a_0,\ldots, a_n) \in \mathbb{N}^{n+1}\), \(x^a=x_0^{a_0}\cdots x_n^{a_n}\). To any such map corresponds an \((n+1) \times (n+1)\) matrix \(A\) whose columns are \(v_0, \ldots , v_n\). By using basic linear algebra methods the authors show that there exist matrices \(B\) and \(\Gamma\) such that \(AB=\Gamma + I_{n+1}\), where \(\Gamma\) is a matrix with repeated column. Then the columns of \(B\) define the inverse birational monomial map of \(f\). Finally the authors show that for \(n=2\), the subgroup of \(\mathrm{Cr}_2(k)\) of monomial maps is generated by the standard quadratic transformation and \(f : \mathbb{P}^2 \dasharrow \mathbb{P}^2\) given by \(f(x_1,x_2,x_3)=(x_1^2,x_1x_2,x_2,x_3)\).