Projective and affine symmetries and equivalences of rational and polynomial surfaces (Q1713124)

An algebraic curve of genus 0 can be rationally parametrized. In fact it can be properly rationally parametrized; i.e., by a birational map from the line to the curve. All these proper parametrizations of a curve are related by bilinear reparametrizations, so-called Möbius transformations. This fact can be used for detecting Euclidean equivalences, symmetries, and similarities of curves. Rational surfaces are considered in this paper; more precisely, surfaces admitting a birational base-point free parametrization. Such parametrizations are shown to be unique up to projective transformations. This investigation leads to the classification of all projective equivalences between two given such surfaces, and also to a classification of the generic cases of quadratic surfaces.