A theorem about Cremona maps and symbolic Rees algebras

TL;DR

This paper investigates the structure of the symbolic Rees algebra associated with Cremona maps, providing conditions for it to have an expected form, supported by examples and a birationality criterion.

Contribution

It introduces sufficient conditions for the symbolic Rees algebra of Cremona map base ideals to have a standard form, unifying previous results and emphasizing Cohen--Macaulay cases.

Findings

Conditions ensuring the expected form of the symbolic Rees algebra

Application of a birationality criterion and inversion factor analysis

Extensive examples of Cremona maps tested against the conditions

Abstract

This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers all previous results on the subject so far. The proof relies heavily on a criterion of birationality and the use of the so-called inversion factor of a Cremona map. One adds a pretty long selection of examples of plane and space Cremona maps tested against the conditions of the theorem, with special emphasis on Cohen--Macaulay base ideals.