Cremona maps defined by monomials

TL;DR

This paper thoroughly analyzes and classifies monomial Cremona maps of degree 2, providing explicit descriptions of their inverses and degrees, and explores their algebraic and combinatorial properties.

Contribution

It offers a new proof for the characterization of degree 2 inverse maps, relates Cremona maps to monomial ideals and polyhedral cones, and discusses computational complexity implications.

Findings

Inverse of degree 2 monomial Cremona maps is degree 2 iff they are involutions up to permutation.

Explicit structure of inverse maps and their degrees is provided.

Connections between monomial Cremona maps, monomial ideals, and Hilbert bases are established.

Abstract

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard.