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arXiv · 1109.2815

Plane Cremona maps: saturation and regularity of the base ideal

S. H. Hassanzadeh·A. Simis
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Abstract

One studies plane Cremona maps by focusing on the ideal theoretic and homological properties of its homogeneous base ideal ("indeterminacy locus"). The {\em 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 $\boldsymbol\mu$-fat ideal and a seemingly novel ideal defined in terms of valuations.

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S. H. Hassanzadeh, A. Simis. 2011-09-13. Plane Cremona maps: saturation and regularity of the base ideal. https://arxiv.org/abs/1109.2815

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