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«The geometry of a net of quadries in four-dimensional space».Acta mathematica 64 (1935), 185–242. This paper will be referred to asG.N.Q.
If we have, on a ruled surface of ordern, a curve of orderm meeting each generator ink points and such thats generators pass through each point of the curve, and also a curve of orderm′ meeting each generator ink′ points and such thats′ generators pass through each point of the curve, then the number of intersections of the two curves ismsk′+m′s′k−nkk′
See, for a corresponding argument in [3],A. C. Dixon,Proc. London Math. Soc. (2), 7 (1909), 153.
This follows immediately also from the existence of the line-cones {P}. The tangents ofϑ at its three intersections with a trisecant are cospatial because, itP is any one of the intersections of the trisecant withϑ they lie in the solid which touches {P} along the plane joining the trisecant to the tangent ofϑ atP. The four tacnodes of the plane projection ofϑ are on conic which touches the four tacnodal tangents.
We shall assume as known the properties of the rational quartic curve and of loci associated with it; for example the loci generated by its tangents, by its chords and by its osculating planes. Many of these properties can be obtained very simply either from the projective method of generating the curve or from its parametric representation (equivalent to the above) first given by Clifford.
For the order of a locus given by the vanishing of the determinants of a matrix see Salmon:Higher Algebra (Dublin, 1885), Lesson 19.
See, for example, Veronese,loc. cit., 202.
See, for example, Room:Proc. London Math. Soc. (2) 36, 1934, 12–15.
Proc. London Math. Soc. (2), 30, 1930, 305.
We need not consider the possibility of the trisecant being a generator of more than one of the three cones. The only chords ofϑ which are generators of two cones are the 120 lines which are chords both ofϑ and of the base curve of the net, and none of these is a trisecant ofϑ.
G.N.Q. § 16.
Concerning the Segre primal and its associated hexahedron see Castelnuovo:Atti Ist. Veneto (6), 6 (1888), 547–565.
Math. Annalen 7 (1874), 308.
Proc. London Math. Soc. (2), 36 (1933), 25–26.
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Edge, W.L. Some special nets of quadrics in four-dimensional space. Acta Math. 66, 253–332 (1936). https://doi.org/10.1007/BF02546520
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DOI: https://doi.org/10.1007/BF02546520