Cremona transformations with an invariant rational sextic

Abstract

I t is well known that a Cremona transformation T with ten or fewer jF-points will transform a rational sextic S into a rational sextic S' when the F-points of T are all located at the nodes of S. I have shown (cf. [ l ] , p. 248, (9); [2], p. 255, (5)) that, even though the number of transformations T of the type indicated is infinite, the transforms S' are all included in 2 • 31 -51 classes, the members of any one class being all protectively equivalent and a member of one class being projectively distinct from a member of another class. The sextic 5 itself is in one of these classes, T being then the identity. If S' is in the same class as 5, and if C is the collineation which carries S' into 5, then TC is a Cremona transformation of the same type as T which transforms S into itself. There is thus an infinite group of Cremona transformations which carry 5 into itself. If / is a parameter on 5, the effect of an element of such a group on the points of 5 is represented by