Line geometry in three dimensions over GF (3), and the allied geometry of quadrics in four and five dimensions

This paper, a sequel to another, describes the line geometry of a [3] S over the Galois field K of 3 marks; this involves some investigation of the geometry, also over K , of quadrics in [4] and [5]. Sections 2 to 4 introduce th e Plücker co-ordinates and explain how reguli in S can be positive or negative. Section 6 introduces K lein’s m apping of the lines of S on a quadric Q in [5]. The lines of [5] fall into four categories relative to Q, and the num ber in each category is found. The points off Q fall into two oppositely signed batches of 117 each, and it is helpful to use X 2 + Y 2 + Z 2 = U 2 + V 2 + W 2 as a canonical form for th e equation of Q. Section 10 digresses to calculate th e number of quadrics in the [5] which ad m it such an equation. Section 11 introduces the screws of S , each of which consists of 40 lines that can be arranged as 36 decades, and explains how a screw is signed. Section 12 shows that the 40 lines fall into 270 reguli signed oppositely to and 540 reguli signed similarly to the screw. It is in section 12 that the number 27, so familiar in the geometry of the cubic surface, arises. A screw is mapped by a prime section w of Q, and the lines of the prime fall, when regard is had to the signs of tangents, into five categories relative to w . Sections 13 to 15 describe the resulting geometry in [4], account for every linear subspace and display a table of incidences. The numbers in this table are essential to a proper study of the group w (5, 3) of order 51840. Section 16 compares this figure in [4] with that which Baker and Todd elaborated from Burkhardt’s quartic primal. Sections 18 and 19 give the order of PO 2 (6, 3) and describe its involutions, w (5, 3) is a subgroup of index 117 of PO 2 (6, 3) and sections 20 to 22 discuss briefly the involutions in w (5, 3) and mention certain of its permutation representations.