A geometry paper by T L Wren
In the following paper certain properties of the configuration of n straight lines meeting a common transversal have been established with the aid of the two-three birational space transformation. This transformation, first investigated by Cayley, has been used by Noether in mapping certain types of surface point for point upon a plane.
The present paper consists of four parts. In Part I an account is given of the general properties of the transformation used in the subsequent work. All the results are derived from first principles without any reference to the properties of birational transformation in general.
Part II is devoted to the establishment of certain existence theorems for surfaces of order n with (n - 1)-ple or (n - 2)-ple straight line, and for the rational space curve of order n with an (n - 1)-point secant. The results of sections 6-9 may be summarised as follows, the word "line" being understood to mean "straight line" throughout the paper.
(i) A surface of order n with (n - 2)-ple line is uniquely determined by 2n - 1 lines meeting the multiple line.
(ii) A surface of order n with (n - 1)-ple line is uniquely determined by p lines meeting the multiple line and 3n - 2p points.
(iii) A curve of order n with an (n - 1)-point secant is determined uniquely by p points and 3n + 1 - 2p lines meeting the multiple secant.
The proof in each case is by induction, the transformation being used to reduce the order of the surface or curve while preserving its type. The results just related are thus deduced from the corresponding theorems for the quadric surface and straight line. In none of these cases does the proof involve consideration of elements of either space corresponding to fundamental elements of the other. In section 10 are discussed some relations between ruled surfaces and rational curves of the type already considered.
The method of sections (6-9) is really equivalent to a succession of applications of the two-three transformation, the principal fundamental line remaining the same throughout. The transformation resulting from such a succession of operations is dealt with in Part III, where it is shown to lead to the plane constructions used by Noether in classifying the curves upon the surfaces of the types here considered.
The results of Parts II and III are essentially given by Noether in the paper just referred to. They were collected by the present writer with a view to using the two-three transformation in an attempt to extend the line configuration theorem of Part IV indefinite. For this purpose it was desirable to establish the uniqueness of the surfaces and curves as determined by the given conditions in sections 6-9. Noether shows that in each case the given conditions determine at least one construct of the type considered, but does not prove definitely that this is the only one. For example, consider a surface of order n having a given (n - 2)-ple line. Its equation contains 6n - 3 independent constants, so that the surface may be made to pass through 6n - 3 arbitrary points. If these 6n - 3 points be taken in threes on 2n - 1 lines meeting the given multiple line it appears that there will always be at least one surface of the required type containing the 2n - 1 lines. There may, however, be a linear system of such surfaces, as we have no justification for assuming that the 6n - 3 linear equations to determine the coefficients are linearly independent when the 6n - 3 points lie by threes on straight lines. Noether appears to assume that this is the case. The proof by elementary methods used in sections 6-9 of this paper establishes with certainty not only the existence, but also the uniqueness of the constructs as determined by the configurations considered. It is interesting to note the mutual relations between the curves and surfaces discussed in sections 10, 12.
Part IV deals with a theorem of line configurations. If a straight line I be taken, any four lines meeting I determine another line which meets all four. Again, if we take five lines meeting I, every four of the five lines determines a new line as above. Then, by the well known theorem of the double-six (incidentally established anew in section 15) these five new lines are all met by a straight line, which we may regard as determined by the original set of five. The double-six plays an important part in the geometry of the cubic surface. It has also been discussed in relation to a certain quadric by Mr G T Bennett, and an elementary proof not involving the cubic surface has been given by Prof Baker.
Now suppose we start with six lines meeting I; every five determine in the above manner a new line. The question arises whether these six new lines are all met by a common transversal; and if so is there a similar theorem for seven lines, and so on universally. In section 15 we use the two-three transformation to deduce the five-line theorem from the four-line theorem; and in the succeeding sections 16, 17 we establish the six-line theorem with a single application of the transformation. For the six lines we get in the course of the construction a configuration consisting of two sets of 12 and 32 lines respectively; such that every line of the 32 is met by 6 of the 12, and every line of the 12 by 16 of the 32.
For the next case, that of seven lines with a common transversal, I have so far been unable to establish the theorem. The figure obtained for the construction of the seven new lines determined by every six of the original seven involves 64 lines, and the figures resulting from one or two applications of our transformation are extremely complicated and difficult to visualise.
The theorem for six lines meeting a common transversal is a particular case of the corresponding theorem for six linear complexes having a line in common. Any four of these complexes have another line in common; hence from a set of five of the six complexes we get five lines which determine a linear complex. From the six original complexes we thus get six sets of five, and so obtain six new complexes, which have one common line. This has been proved by Mr J H Grace by considerations of cubic threefolds in four dimensional space. Mr Grace tells me that he has also considered the cases of seven and of eight lines.
In conclusion, I wish to express my thanks to Prof Baker for his kindness in reading through the paper and in suggesting improvements.
JOC/EFR April 2015
The URL of this page is: