Bounds on the degrees of birational maps with arithmetically Cohen-Macaulay graphs

TL;DR

This paper investigates bounds on the degrees of birational maps with Cohen-Macaulay graphs, establishing an upper limit of n^2 for Cremona maps in n-dimensional projective space, based on algebraic properties of their Rees algebras.

Contribution

It provides a new upper bound on the degree of birational maps with Cohen-Macaulay graphs in arbitrary dimension, extending known results from the plane case.

Findings

In the plane case, a complete classification of Cremona maps with Cohen-Macaulay graphs is achieved.

For any dimension n, a Cremona map with Cohen-Macaulay graph has degree at most n^2.

The Cohen-Macaulay property of the Rees algebra imposes strong restrictions on the map's degree.

Abstract

A rational map whose source and image are projectively embedded varieties has an {\em Arithmetically Cohen-Macaulay graph} if the Rees algebra of one (hence any) of its base ideals is a Cohen-Macaulay ring. If the map is birational onto the image one considers how this property forces an upper bound on the degree of a representative of the map. In the plane case a complete description is given of the Cremona maps with Cohen-Macaulay graph, while in arbitrary dimension $n$ it is shown that a Cremona map with Cohen-Macaulay graph has degree at most $n^2$.