Some Algebraic Curves
This chapter discusses a topic which was studied from various points of view throughout the nineteenth century and which presented itself in such different guises as: the 28 bi-tangents to a quartic curve, the study of a Riemann surface of genus 3 and its group of automorphisms, and the reduction of the modular equation of degree 8. These studies, which began separately, were drawn together by Klein in 1878 and proved crucial to his discovery of automorphic functions.
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- 1.The plane is generally the complex rather than the real plane, but early writers e.g. Cayley are ambiguous. Later writers, like Klein, are more careful.Google Scholar
- 2.If 4 or more inflection points were real they would all have to be, since the line joining two inflection points meets the curve again in a third. But then they would all lie on the same line, which is absurd. The configuration of inflection points on a cubic coincides with the points and lines of the affine plane over the field of 3 elements, an observation first made, in its essentials, by Jordan, Traité 302.Google Scholar
- 3.H. J. S. Smith  pointed out that Eisenstein used the Hessian earlier, in 1844. Eisenstein’s ‘Hessian’ is the Hessian of the cubic form ax3 + 3bx2y + 3cxy2 + dy3, i.e. (b2 − ac)x + (be − ad)xy + (c2 − bd)y2, see Eisenstein [1975, I, 10].Google Scholar
- 4.It takes two to tango.Google Scholar
- 5.In a subsequent paper on this topic, , Steiner was led to invent the Steiner triple system. However, these configurations had already been discovered and published by Kirkman , as Steiner may have known, see the sympathetic remarks of Klein, Entwicklung, 129.Google Scholar
- 6.It has proved impossible to survey the literature of these wonderful configurations. For some historical comments see Henderson [1911, 1972], who also describes how models of the 27 lines may be constructed. For modern mathematical treatments see Griffiths and Harris , Hartshorne  and Mumford . For their connection with the Weyl group of E7 see Manin . The existence of a finite number of lines on a cubic surface was discovered by Cayley , and their enumeration is due to Salmon . The American mathematician A. B. Coble  and  seems to have been the first to illuminate the 27 lines and 28 bitangents with the elementary theory of geometries over finite fields. I am grateful to J. W. P. Hirschfeld for drawing his work to my attention.Google Scholar
- 7.See Weierstrass Werke, IV, 9, quoted and discussed in Neuenschwander [1981b, 94].Google Scholar
- 8.Riemann tacitly assumed that the genus was finite. His proof that it was well-defined was imprecise and was improved by Tonelli .Google Scholar
- 10.The simplicity of the group emerges from Klein’s description of its subgroups without Klein remarking on it explicitly. Silvestri’s comments [1979, 336] in this connection are misleading, and he is wrong to say the group is the transformation group of a seventh order modular function: G168 is the quotient of PSL(2; 2ℤ ) by such a group. Ibo.Google Scholar
- 11.In particular, the inflection points of this curve are its Weierstrass points. For example, when the equation of the curve is obtained in the form λ3μ + μ3ʋ + ʋ3λ=0 the line z = 0 is an inflection tangent at [0,1,0] and meets the curve again at [1,0,0]. The function g(λ,μ,ʋ) = μ/ʋ is regular at [1,0,0] but has a triple pole at [0,1,0].Google Scholar