Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T14:11:40.032Z Has data issue: false hasContentIssue false
  • English
  • Français

On the quartic surface

Published online by Cambridge University Press:  24 October 2008

B. Segre
B. Segre
Affiliation:
The UniversityManchester
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).