Abstract
We describe the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic monomial transformations, and geometric descriptions in terms of fans.
Highlights
The best known birational map of Pr is maybe the standard Cremona transformation Sr : Pr Pr defined by Sr = (X0−1 : · · · : Xr−1)
For r = 2 there is a geometric description in the classic references on this subject; a Max Noether’s famous theorem shows that every birational map of P2 is a composition of automorphisms and S2, and that every Cremona transformation is a composition of quadratic ones
Our approach is based in a toric point of view and the property that Sr stabilizes the open set X0 · · · Xr = 0
Summary
The best known birational map of Pr is maybe the standard Cremona transformation Sr : Pr Pr defined by Sr = (X0−1 : · · · : Xr−1). For r = 2 there is a geometric description in the classic references on this subject; a Max Noether’s famous theorem shows that every birational map of P2 is a composition of automorphisms and S2, and that every Cremona transformation is a composition of quadratic ones. In this note we consider birational maps generalizing Sr. Our approach is based in a toric point of view and the property that Sr stabilizes the open set X0 · · · Xr = 0. We consider birational maps of Pr with this property. In the second paragraph of (Russo and Simis 2001) the birationality of these maps is characterized in terms of certain syzygies as an application of a more general criterion; see (Simis and Villarreal 2002): compare their Proposition 1.1 with our Proposition 3.1
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