Plane Cremona maps: Saturation and regularity of the base ideal

One studies plane Cremona maps by focusing on the ideal theoretic and homological properties of its homogeneous base ideal (“indeterminacy locus”). The leitmotiv driving a good deal of the work is the relation between the base ideal and its saturation. As a preliminary one deals with the homological features of arbitrary codimension 2 homogeneous ideals in a polynomial ring in three variables over a field which are generated by three forms of the same degree. The results become sharp when the saturation is not generated in low degrees, a condition to be given a precise meaning. An implicit goal, illustrated in low degrees, is a homological classification of plane Cremona maps according to the respective homaloidal types. An additional piece of this work relates the base ideal of a rational map to a few additional homogeneous “companion” ideals, such as the integral closure, the μ-fat ideal and a seemingly novel ideal defined in terms of valuations.