Search

Although all the coefficients in the equation of a plane algebraic curve may be real numbers, it by no means follows that the equations of all its bitangents are real. But Plücker perceived that this could happen for the 28 bitangents of a nonsingular plane quartic. Where can this be observed in a body of 28 explicit linear equations? This modest note affords an example.

SynopsisA certain plane sextic of genus 5 was encountered by Humbert and publicised by him [3] in 1894. Its striking geometrical properties clamour for elucidation; this was eventually supplied in 1951. For the canonical curve of genus 5 is the base curve C of a net N of quadrics in projective space [4], and C models a Humbert curve when all the quadrics of N have a common self-polar simplex [1]. The projection of C from one of its chords onto a plane is a 5-nodal sextic, the nodes all becoming cusps when the chord of C becomes a tangent. The properties to be elucidated become clear visually in the projection.The sextic H described here is a specialisation of the cusped curve; it emerges as linearly dependent on a pair of reducible plane sextics concocted ad hoc.

SynopsisThis note sets out from the observation that it is not, in general, possible to express a homogeneous cubic polynomial in five variables as a sum of cubes of seven linear forms. Some of the geometry, to which particular cubics which do happen to be so expressible give rise, is described. Further particularisations are mentioned, and one such cubic investigated in some detail.

The curve in question is the non-singular intersection Γ of the n − 1 quadric primalswhere it is presumed that no two of the n + 1 numbers aj are equal. Definethen it will be seen that the osculating prime of Γ at x = ξ is

SynopsisAny non-ruled quartic surface with a double line in space of three dimensions has plane sections of genus 2 and so is a projection of a non-singular duodecimic surface in space of eleven dimensions. This was accepted as an established fact by 1890, but there seems not to be any account of such a projection in action. The following pages are submitted to aid the removal of this century-old anomaly.

The aim in this paper is to indicate one way of interpreting covariants of a binary quintic F geometrically, interpretations having been found recently [7] for its quadratic covariant 2C2, called Γ in [7], and its invariant I4. The symbol dCn will denote a covariant of order n in the binary variables x, y and degree d in the coefficients of F, Id being used in preference to dI0 for invariants. The sum d + n is 4 for both 2C2 and I4, and no other covariant affords as small a sum; so it is natural to have begun by interpreting these two and to use them as auxiliaries in interpreting others.

1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not.

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equationsare linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.

SynopsisThe cubic surfaces in, save for the elliptic cone, are, whatever their singularities, projections of del Pezzo's non-singular surface F, of order 9 in. It is explained how, merely by specifying the geometrical relation of the vertex of projection to F, each cubic surface is obtainable “at a stroke”, without using spaces of intermediate dimensions.

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number andso that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics(1.1)in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

A rational normal curve C of order n in [n] has, at each point P, a nest of osculating spacesas P moves on C the [n — 2] generates a primal Dn-1 of order 2n — 2.Hilbert [3] found the multiplicities on Dn-1 not only of the vμ+1 generated by Dμ for each lesser value of μ but also those of all submanifolds common to these various Dμ.A surface Φ in higher space has, as explained [4] by del Pezzo, a nest of tangent spacesof respective dimensionsthey raise the problem of finding the orders of manifolds generated by them and the multiplicity of each on the higher manifolds to which it belongs: the task does not seem to have been attempted, but it may well be eased if Φ is rational and normal.

When n-1 quadrics in projective space [n] of n dimensions have a common self-polar simplex their common curve Γ admits a group of2 n self-projectivities. The consequent properties of Γ are investigated, and further specialisations are imposed which amplify the the group and endow Γ with further properties. There is some reference to the osculating spaces and principal chords of Γ, and some properties of particular curves in four and five dimensions are described.

SynopsisWhen the points of a projective space [9] map ternary cubics the maps of perfect cubes are the points of a del Pezzo surface F. Several manifolds linked with F are listed and the cubics mapped on these are noted. A new approach leads to the first of Aronhold's two invariants.

1. Cremona's projection of 15 lines on a nodal cubic surface onto a plane π is a striking example of the resolution of a highly complicated figure into its constituent simplicities, for it explains, almost at a glance, so many properties of Pascal's mystic hexagram. These may appropriately be called ((12); (10), p. 162) the Veronese properties and have been described on several occasions ((12); (6); (10); (1), pp. 219–236; (2), pp. 349–355).

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.

Wiman, in 1895, found ((5), p. 208) an equation for a 4-nodal plane sextic W that admits a group S of 120 Cremona self-transformations; of these, 24 are projectivities, the other 96 quadratic transformations. S is isomorphic to the symmetric group of degree 5 and Wiman emphasizes that S does permute among themselves 5 pencils (4 pencils of lines and 1 of conics) and 5 nets (4 nets of conics and 1 of lines). But he gives no geometrical properties of W. The omission should be repaired because, as will be explained below, W can be uniquely determined by elementary geometrical conditions. Furthermore: W is only one, though admittedly the most interesting, of a whole pencil P of 4-nodal sextics; every member of P is invariant under S+, the icosahedral subgroup of index 2 in S, while the transformations in the coset S/S+ transpose the members of P in pairs save for two that they leave fixed, W being one of these. When the triangle of reference is the diagonal point triangle of the quadrangle of its nodes the form of W is (7.2) below. Wiman referred his curve to a different triangle.

A binary form of odd degree,has a quadratic covariant Г, (ab2maxbx in Aronhold's notation, and the discriminant Δ of Г is an invariant of ƒ For m = 2Δ was obtained by Cayley in 1856 [3, p. 274]; it was curiosity as to how Δ could be interpreted geometrically that triggered the writing of this note. An interpretation, in projective space [2m + 1], that does not seem to be on record, of Γ and Δ is found below. If m = 1 one has merely the Hessian and discriminant of a binary cubic whose interpretations in the geometry of the twisted cubic are widely known [5, pp. 241–2].

The cubic surface group, of order 51840, has a representation by orthogonal matrices, of 5 rows and determinant + 1, over GF (3). It can be partitioned into conjugate classes on geometrical grounds because each matrix has two skew linear spaces, S + of even and S - of odd dimension, of latent points; the matrices fall into categories A, B, C according as the join of S + and S - has dimension 4, 2, 0. Subdivisions of A, B, C rest on the relation of S + and S - to the invariant quadric of the orthogonal group. A accounts for the identity matrix and the 4 types of involutions. B falls into two parts; one of 4 classes, discussed in §§5 to 8, the other of 9 classes, discussed in §§9 to 14. §§ 15 and 16 mention criteria for checking the number of operations in a conjugate class. Those classes in category C fall into 3 subcategories of 3, 2, 2 classes and are described in §§ 18 to 25.

The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.

SynopsisHumbert's 5-nodal plane sextic first appeared in his 1894 paper. Its canonical curve C was identified in 1951, when it was shown that the sextic is the outcome of projecting C from one of its own chords on to a plane.In this present paper it is remarked that there are 60 chords of C such that the projection has two tacnodes, each a confluence of two of Humbert's 5 nodes, and an equation is found for this tacnodal curve.A certain specialization permits C to be invariant for a group of 32, not merely 16, projectivities. Further specializations, described in the proper place, permit groups of orders 64, 96, 160. The resulting tacnodal sextics have groups of birational self-transformations isomorphic to these.

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.

The object of this paper is to give some account of the geometry of the three-dimensional space S wherein the co-ordinates belong to a Galois field K of 3 marks. A description of the fundamental properties of quadrics is sufficiently long for one paper, and so an account of the line geometry is deferred. The early paragraphs (§§ 1 to 4) are necessarily concerned with geometry on a line or in a plane. A line consists of 4 points; these are self-projective under all 4! permutations. A plane consists of 13 points and has the same number, 234, of triangles, quadrangles, quadri-laterals and non-singular conics. A diagram is helpful, especially when we consider sections by planes in S . The space S has 40 points. Non-singular quadrics are of two kinds: either ruled, when we call them hyperboloids, or non-ruled, when we call them ellipsoids. A hyperboloid H consists of 16 points and has a pair of reguli; the 24 points of S not on H are the vertices of 6 tetra-hedra that form two allied desmic triads. The ellipsoid F is introduced in § 12; it consists of 10 points, the other 30 points of S being separated into two batches of 15 between which there is a symmetrical (3, 3) correspondence. Either batch can be arranged as a set of 6 pentagons, each of the 15 points being the common vertex of 2 of these. The pentagons of either set have all their edges tangents of F and, with their polar pentahedra, have significant properties and interrelations. By no means their least important attribute is that they afford, with F , so apposite a domain of operation for the simple group of order 360. In §§ 23 to 26 are described the operations of the group in this setting. Thereafter the 36 separations of the 10 points of F into complementary pentads are discussed, no 4 of either pentad being coplanar. During the work constructions for an ellipsoid are encountered; one is in § 16, another in § 30.

1. Introduction. The ternary orthogonal group of projectivities over a finite field leaves a non-singular conic ✗ invariant, but the geometry consequent thereupon does not appear to have been investigated. The group is isomorphic to a binary group of fractional substitutions over the same field and this fact may, since these binary groups and their subgroups are so well known, have inhibited projects to embark on a detailed description of the geometry of the ternary group.

The six quaternary quartic formswere first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

A ternary form of degree n can be expressed as a symmetrical determinant, of n rows and columns, whose elements are linear forms; furthermore, not only is such a mode of expression known to be possible, but A. C. Dixon, in 1902, gave* a process by which the determinant can be obtained when the ternary form is given. This process, however, although it admits of such a straightforward theoretical description, cannot be carried through in practice, for a general ternary form, without the introduction of complicated algebraical irrationalities, even if we restrict ourselves to forms of the fourth degree; consequently no application of Dixon's process to an actual example seems to have been published. If then a choice can be made of a quartic form for which the reduction to a symmetrical determinant can be carried out without undue complication, it seems fitting to give some account of it. The following pages are therefore devoted to the study, from this aspect, of the form x4+y4+z4, for which the reduction can be accomplished without introducing any irrationality other than the fourth root of − 1.

The characters of the cubic surface group G are got by using its representation as a group of orthogonal projectivities in a four-dimensional space a over GF (3). In α , G permutes 27 pentagons transitively and 6 other characters, in addition to the permutation character of degree 27, are obtained by noting how the 5 vertices of each invariant pentagon are permuted. Other geometrical objects that afford transitive permutation representations are available, and provide material for similar investigations. The papers that have preceded this, wherein not only is the geometry described in detail but also many 5-rowed matrices of the orthogonal group are explicitly given, are laid under contribution. In particular, the table at the end of the most recent paper provides at sight not only permutation characters but monomial characters as well.

SynopsisThe curve Γ common to two quadric surfaces has 24 principal chords; they are the sides of six skew quadrilaterals each of which has for its two diagonals a pair of opposite edges of that tetrahedron S which is self-polar for both quadrics. These three pairs of quadrilaterals serve to identify the three pairs of quadrics through Γ that are mutually apolar.The vertices of the quadrilaterals lie four on each edge of S. Both nodes of the plane projection of Γ from such a vertex are biflecnodes.

The group of automorphisms of the non-singular quadric consisting of 135 points in a finite projective space[7] is investigated by using the geometry of the figure.

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

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.

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.