7 research outputs found
Cremona maps defined by monomials
Cremona maps defined by monomials of degree 2 are thoroughly analyzed and
classified via integer arithmetic and graph combinatorics. In particular, the
structure of the inverse map to such a monomial Cremona map is made very
explicit as is the degree of its monomial defining coordinates. As a special
case, one proves that any monomial Cremona map of degree 2 has inverse of
degree 2 if and only if it is an involution up to permutation in the source and
in the target. This statement is subsumed in a recent result of L. Pirio and F.
Russo, but the proof is entirely different and holds in all characteristics.
One unveils a close relationship binding together the normality of a monomial
ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter
suggests that facets of monomial Cremona theory may be NP-hard
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 {\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 -fat
ideal and a seemingly novel ideal defined in terms of valuations.Comment: New version only 36 pages, one typo correcte