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On the quartic surface

Published online by Cambridge University Press:  24 October 2008

B. Segre
B. Segre
Affiliation:
The UniversityManchester
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Summary

1. The projective transformations of F into itself … 121

2. The flecnodal curve, and the lines of F … 122

3. A geometric characterization of F, and the six different types of F in the real domain … 123

4. The τ-points and τ-planes, and a notation for the lines of F … 124

5. The incidence conditions for the lines of F … 125

6. The tetrads of the first kind … 126

7. The tetrads of the second kind … 126

8. The pairs of lines of F … 127

9. The tetrads of the third kind … 129

10. The 16-tangent quadrics of F … 130

11. The conics of the first kind … 131

12. The conics of the second kind … 132

13. No other irreducible conics lie on F … 133

14. The tangent planes of F of multiplicity greater than 3 … 136

15. On twisted curves, especially cubics and quartics, lying on F … 138

16. The T-transformations … 139

17. Construction of an infinite discontinuous group of birational transformations of F into itself … 142

18. Deduction of an infinity of unicursal curves lying on F … 143

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 40 , Issue 2 , June 1944 , pp. 121 - 145
Copyright
Copyright © Cambridge Philosophical Society 1944

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References

* G. Salmon, Analytic geometry of three dimensions, 2 (5th ed. 1915), 251. Cf. also C. M. Jessop's book Quartic surfaces (Cambridge, 1916), which is almost entirely concerned with surfaces possessing multiple points.

* Cf. e.g. G. Salmon, op. cit. p. 278.

† Cf. B. Segre, ‘The maximum number of lines lying on a quartic surface’, Oxford Quarterly Journ., 14 (1943), pp. 86–96, § 3.

* This follows from § 11, and also from an argument we shall develop later.

* Cf. C. Segre, Atti Acc. Scienze Torino, 31 (1896), § 4.

* Cf. M. Noether, ‘Ueber Flächen, welche Scharen rationaler Kurven besitzen’, Math. Ann. 3 (1871).

† Cf. F. Enriques, ‘Sulle superficie algebriche che ammettono un gruppo continuo di trasformazioni birazionali in se stesse', Rend. Circ. Mat. Palermo, 20 (1905).

‡ F seems to be the first instance of an algebraic surface satisfying this condition and containing an infinity of unicursal curves.

* The symbol [AB] denotes, as usual, the intersection number of two curves A, B on F. It is defined even when A, B have a part in common; in particular, when A and B coincide, the number [AA] = [A 2] is the virtual degree of A on F. When A and B have no common component, (AB) will indicate the set of points of intersection of A and B on F.

† In agreement with the involutive character of T CD, the system (50) is transformed into an equivalent one if we interchange A and A′, as is apparent on writing it in the form hence the equivalence (51) is symmetric in A and A′.

* The above argument shows incidentally that the group of projectivities of F into itself operates transitively on the 48 tetrads of the second kind lying on F, i.e. all these tetrads are equivalent with respect to the transformations (2) of § 1. A similar argument shows that the same is true of the 24 tetrads of the first kind and of the 192 tetrads of the third kind (§§ 6, 9).