J. A. TODD .sx SOME IMPORTANT DOUBLY INFINITE LINE SYSTEMS IN HIGHER SPACE .sx By J. A. TODD .sx Received 5 January , 1931 .sx - Read 15 January , 1931 .sx Introduction .sx The principal object of the present paper is the study of the properties of certain elementary doubly infinite line systems in higher space .sx The method employed is to represent the lines of by their Grassmann coordinates , and to take these as point coordinates in higher space , so that to the lines of correspond the points of a certain locus in this space .sx To the lines of the system under consideration will then correspond the points of a surface on this locus .sx We examine the relations between the line system and this representative surface , and in particular the relations between the representative surfaces of two line systems , one of which is a projection of the other .sx This leads to the consideration of normal and supernormal loci , form which all the others may be derived .sx The systems considered are those whose representative surfaces are the non-ruled surfaces whose prime sections are rational or elliptic curves ; it is shown that the line systems generate threefold loci which are closely related to each other and which can be derived as projections of known loci .sx In 1 the necessary properties of the locus representing the lines of are briefly developed .sx The general properties of doubly infinite line systems are considered in 2 , in 3 the non-ruled surfaces whose prime sections are rational or elliptic curves are enumerated , and also the threefolds with elliptic curve sections which are not the locus of a single infinity of planes .sx The line systems which correspond to these surfaces are examined in the next three sections .sx The later sections deal with the particular case in which the line system is supposed to lie in 4 , and the relations of the representative loci to the which represents all the lines of 4 are considered .sx The paper concludes with an investigation of the representative locus for the lines lying on the general quadric primal in 4 .sx 1 The representation of lines of by points .sx 1.1 Line coordinates .sx It is well known that , if and are two points of , the ratios of the quantities uniquely determine the line and are uniquely determined by it .sx They are , in fact , the extension to n dimensional space of the Plcker line coordinates in ordinary space .sx They are connected by a number of linearly independent quadratic relations of the form ( 1.11 ) , where , , , represent any four of the suffixes 0, .sx ..,n. We may regard the as homogeneous coordinates in a space , where ; the points of whose coordinates satisfy ( 1.11 ) are then the points of a certain locus Gn which is in unexceptional correspondence with the lines of .sx 1.2 Elementary properties of the locus Gn .sx The general properties of the locus have been considered by Severi .sx The points of Gn which correspond to lines through a point A of the lie in a , so that Gn contains .sx Any two of these , a , a say , have a common point which maps the line joining the corresponding points A , A of the .sx The points of Gn which correspond to lines lying in a of the lie on a ; in particular the map of the lines of a plane of is a plane on Gn .sx If two lines of the intersect , the corresponding points of both lie in the which corresponds to the point of intersection , so that the line joining these points lies entirely on .sx Conversely , the points of any line lying on Gn map the lines of a plane pencil in the , and the line is the intersection of a definite and a definite plane .sx The lines of Gn which pass through a point L map all the plane pencils of which the corresponding line l is a member , so that the points of these lines represent all the lines of the which meet .sx The lines on lie in the tangent space to Gn at L and generate a cone whose vertex is L. This cone contains , corresponding to the points of l , and planes , corresponding to planes through l , and any one of the meets any of the planes in a line through L. The prime section of the cone is thus a locus generated by and lines , any two spaces of different systems intersecting ; it is accordingly the rational normal of , defined by the joining lines of corresponding points of two related which do not intersect .sx This accords with the fact that the dimension of the tangent space to Gn is , which is the freedom of the lines of .sx The points of a prime section of Gn correspond to the lines of whose coordinates satisfy a linear relation , that is , to the lines of a linear complex .sx In particular , the lines which meet a secundum , or , of the form such a system and are mapped on a prime section of Gn ; this is a special section and contains the tangent to Gn at all the points which map lines lying in the given secundum ; that is , the prime touches Gn at all points of a .sx It follows that a ruled surface of order m is mapped on Gn by a curve of order m ; the genus of the curve will be equal to that of the surface since the points of the former are in ( 1,1 ) correspondence with the generators of the latter .sx In particular , a rational scroll of order is represented by a rational curve of this order ; and we can show that if the surface lies in its normal space , namely , the same is true of the representing curve .sx We shall lose no generality in supposing k to be zero , since we could otherwise consider the corresponding to the in which the surface lies .sx Now we have seen that the tangent space at a point of Gn cuts Gn in a cone projecting the normal of ; the section of this by a general is a normal rational , and this maps the generators of a ruled surface of order special in the fact that it possesses a linear directrix .sx The general surface , not possessing this peculiarity , will a fortiori be mapped by a normal curve .sx It follows that a regulus is mapped by the points of a conic , a cubic surface in 4 by a twisted cubic , and so on .sx It is important to notice that , if a lies on Gn , representing the lines of some of the , the space which contains Gr cannot have any further intersection with Gn .sx To prove this we observe , first , that the theorem will follow by induction if we can prove if for the case in which , for we can consider a succession of loci , each of which contains all the ones succeeding it in the series , and apply the theorem to each consecutive pair .sx Now , if the theorem is not true for the , every linear complex which contains all the lines of some of the necessarily contains other fixed lines .sx In particular this will be true of the special linear complexes which consist of the lines meeting linearly independent of the .sx These have thus common lines not lying in the .sx But this is impossible , for , since the are independent , they have no common point , and any transversal line of them all must lie in the containing .sx This proves the result stated , when , and the general theorem follows .sx 1.3. Relations between projected line systems .sx It is clear from what has been said above that , if any line system of the is mapped on Gn by the points of a locus S , and if we consider the as being immersed in some space of higher dimension , then will be represented upon by a locus identical with S projectively , which lies on a particular of , namely that corresponding to the particular in which lies .sx It is thus appropriate to consider the representation of a system of lines by points quite apart from the consideration of any particular Grassmannian locus , and to consider merely the correspondence between the points of the representative locus S and the lines of the system .sx The geometrical properties of S , such as the nature of the curves , surfaces , .sx .. , which it contains , and the relations of these loci to one another , can all be translated directly into properties of subsystems of lines belonging to .sx Such properties are independent of any Gn on which may be supposed to lie .sx There are , however , properties of whose interpretation involves reference to the appropriate Gn ; such a property is that of two lines of intersecting , which is represented by the fact that a certain chord of S lies on Gn .sx We thus are able to distinguish among the properties of the system those which are essentially independent of the space in which it lies , and correspond to properties of the representative locus S , and those which essentially belong to the fact that the system lies in a definite , whose interpretation involves reference to Gn .sx We now suppose that the dimensions of does not exceed , being the dimension of the containing space .sx Then through a general point of the space no lines of pass .sx If from such a point we project the system on to a , we obtain , in this space , a system of lines which will be mapped on by some locus .sx Now , since the projection is from a point which is not on any line of , it follows that those properties of which simply concern its submanifolds are equally properties of , and that the only differences between and are those whose interpretation would involve reference to the Grassmannian loci .sx It follows that the relation between the loci and is such that in general a curve on the one corresponds to a curve on the other having precisely the same order and genus , that every point of gives rise to a unique point of , but that possibly , for particular points , a point of may arise from two or more points of ; this would happen , for example , if the vertex of projection lay in a plane with two of the lines of .sx It follows that the only difference between and projectively is that possibly may lie in a lower space than and be capable of derivation from by projection from some space not meeting .sx It follows that the representative locus of the projection of a system from a point not lying on any of its lines is either projectively identical with the representative locus of or can be obtained from this locus by a general projection .sx That both these cases can arise follows by the following examples .sx Consider , first , the lines of the rational normal of 5 , which join corresponding points of two projectively related planes .sx On the locus , these lines correspond to the points of a surface , and since the lines are clearly in rational correspondence with the points of one of the planes of , it follows that this surface is rational .sx The lines of which meet a line in this plane form a regulus , and thus correspond to the points of a conic on , whence it follows that the surface is a Veronese surface .sx Now consider the projection of the system into 3 from two arbitrary points .sx