Algebraic affine rotation surfaces of parabolic type

On the Affine Image of a Rational Surface of Revolution

“…stands for lower order terms. Since the form of highest order of an affine surface of rotation has a very specific structure (see Theorem 6 in [8], Theorem 6 in [10], Theorem 6 in [9]), we deduce that S 1 is not an affine surface of rotation. In order to compute cubic equations defining the variety V, we consider Equation (15) for the points corresponding to (t i , s i ) with t i , s i ranging from −3 to 3.…”

“…Because S 2 is rational and therefore irreducible, we get that S 2 must be a paraboloid of revolution, so S 1 = f (S 2 ) must also be a paraboloid. But this is a contradiction, because from Corollary 5 in [10] the only quadrics that are affine surfaces of rotation of parabolic type are either cones (which are developable surfaces), or hyperboloids. Therefore, we have proved the following result.…”

“…These surfaces can be of three different subtypes, elliptic, hyperbolic and parabolic, the first of them being the classical surfaces of revolution. Theoretical properties of algebraic affine surfaces of rotation are treated in some recent papers: the elliptic case is studied in [8], the hyperbolic case is addressed in [9], and the parabolic in [10]. Furthermore, an algorithm for recognizing algebraic affine surfaces of revolution is provided in [11].…”