Homaloidal nets and ideals of fat points I

TL;DR

This paper explores the algebraic and homological properties of base ideals associated with plane Cremona maps, focusing on the interplay between these ideals and fat point schemes, especially considering the role of the highest virtual multiplicity.

Contribution

It introduces conditions linking the algebraic properties of base ideals and fat point ideals, emphasizing the influence of the highest virtual multiplicity on Cremona maps.

Findings

Identifies conditions regulating the association between base ideals and fat point ideals.

Describes classes of Cremona maps based on the highest virtual multiplicity.

Analyzes when the base ideal is non-saturated and characterizes its saturation structure.

Abstract

One considers plane Cremona maps with proper base points and the {\em base ideal} generated by the linear system of forms defining the map. The object of this work is the interweave between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. One reveals conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here one emphasizes also the role of one single highest multiplicity. In this vein one describes classes of Cremona maps for large and small value of the highest virtual multiplicity. One also deals with the delicate property as to when the base ideal is non-saturated and the structure of its saturation.