Ultraquadrics associated to affine and projective automorphims

Abstract

In this extended abstract, we study the properties of ultraquadrics associated with automorphisms of the field K(\alpha )(t1, . . . , tn), defined by linear rational (with common denominator) or by polynomial (with polynomial inverse) coordinates. We conclude that ultraquadrics related to polynomial automorphisms can be characterized as varieties K−isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. This information helps us to compute a parametrization of some ultraquadrics.