Search

1. Statement of the problem. It has been shown by Enriquest that if an algebraic surface contains a pencil of elliptic curves and also two curves cutting each curve of the pencil in two groups of points which are not equivalent and which do not possess equivalent multiples, the surface is invariant under a discontinuous infinite series of birational transformations. In a more recent memoir: the same author states that, with the exception of the case in which

It was shown by SCHWARZ t that no algehraic curve of genus greater than unity can remain invariatlt under a continuous group of birational transforma-tions. Later HURWITZ t showed that no such curve could belong to any birational group of infinite order. The corresponding theory for surfaces is by no means complete. WhileA those belonging to continuous groups have been determined, only a few isolated examples are known of surfaces having an infinite discontinuous group. § All the groups which have been discussed are illustrations of two principles, the first of which refers to quartic surfaces and will be considered in two parts, the latter including the former as a particular case; the second principle is appli-cable to a much wider category. I propose to discuss these principles and apply the second to the determination of an extended family of new surfaces having an infinite group.

n/a

Cubic varieties of 003 poillts in space of four dimensions have been extensively studied by CASTELNUOVO t and by SEGRE, t showing the close connection between the section of the enveloping cone by ordinary space (apparent contour) and the well knowll congruences of the second and third orders, with their focal surfaces. The treatment is exclusively synthetic and the methods there employed do not lend themselves readily to the discussion of particular cases. In the following paper I wish to call attention to two particular projections of the variety having nine double points. This variety is given a three page discussion in SEGRE'S first paper, containing most of the results of Articles 1, 2 3, 4, 6 of the present paper. The result of Art. 5 is stated by CASTELNUOVO in a footnote without proof, but no use is there made of it. The results in the remaining articles are new. I have made extensive use of all the papers cited, but as I shall proceed analytically, the development will be along diflerellt lines, and no knowledge of their contents will be assumed. @ Let r 1Z2z3 + \z4z5z6 ° define a cubic variety in space of four dimensions 4, where Ewt O. § Every plane defined by wi ° ( i 1, 2, 3 ) and wk-O (k 4, 5, 6 ) will lie entirely on r. These nine planes may be designated by av. Any two of these having neither subscript in common will intersect in just one point which is a double point on r. Since two values of i and of k have been used to define each point, it ulay be denoted by A with the remaining pair as subscripts. It follows immediately that r contains nine double ?oints; through every double point pass four planes and ill every plane lie four double points.