### On the 3-3 Birational Transformation in Three Dimensions

Proceedings of the London Mathematical Society, 2nd Ser., Vol. 9 (1911), 51-66

Introduction

In a paper read before the London Mathematical Society in 1869, Cayley was the first to call attention to birational transformations in three dimensions, as an immediate extension of Cremona's two-dimensional work. In particular he investigated the 3-3 transformation, which is most compactly expressed by three bilinear relations between the two sets of homogeneous coordinates in the two spaces. He shewed that planes in one space correspond to cubic surfaces in the other, and since three planes meet in only one point, three of these cubic surfaces meet in only one variable point, and their remaining intersection is a fixed sextic curve with seven apparent double points. Cayley gave the formulae of transformation for two cases: (i) when the sextic does not degenerate, and (ii) when it becomes four non-intersecting straight lines and their two transversals. Soon afterwards Cremona and Noether both gave many examples proving that the sextic may degenerate in different ways in the two spaces.

The method of inversion, which is a special form of this transformation, had long been known; but since the time of Cremona the general theory has been greatly developed, and birational transformations are constantly employed in connection with such problems as singularities of surfaces. In particular, the 3-3 transformation, and some of its special cases, have been very useful, and it seems worth while to attempt a systematic treatment of all its varieties. Many of these have been discussed, but, as far as I know, with no attempt at completeness.

The following paper deals with the cases where the first principal sextic curve consists of six straight lines with eight simple intersections, or their equivalent in multiple intersections, taking account of all the possible arrangements of these intersections on the lines.