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Rational Surfaces with Exceptional Unodes

Published online by Cambridge University Press:  20 November 2018

Patrick du Val
Patrick du Val*
Affiliation:
University College, London, England
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Many years ago, I defined (8) three types of exceptional unode on an algebraic surface, which I called U*8, U*9, U*10, corresponding, on a non-singular model of the surface, to sets of six, seven, and eight rational curves, each of grade — 2, with the intersection patterns represented by the Coxeter-Dynkin graphs now usually known as E6, E7, E8:

where each dot represents a curve, and linked dots intersecting curves. In each case we shall denote the curves in the horizontal sequence by S1, s2, … from left to right, and the extra curve meeting s3 by s*.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 938 - 951
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Artin, M., Some numerical criteria for contractability of curves on an algebraic surface, Amer. J. Math., 84 (1962), 485–496.Google Scholar
2. Baker, H. F., Principles of geometry (Cambridge, 1933). pp. 121–131.Google Scholar
3. Bertini, E., Ricerche sulle trasformazioni univoche involutorie net piano, Ann. di Mat., 8 (1877), 244–286.Google Scholar
4. Brieskorn, E., Uber die Auflösung gervisser Singularitäten von holomorphen Abbildungen, Math. Ann., 166 (1966), 76–102.Google Scholar
5. Castelnuovo, G., Suite superficie algebriche le cui sezioni sono curve di genere 3, Atti R. 1st. Sci. Torino, 25 (1880), 695–715.Google Scholar
6. Cayley, A., A memoir on cubic surfaces, Philos. Trans. Roy. Soc., 159 (1869), 231–236; Coll. Papers, 6 (1893), 359-455.Google Scholar
7. Conforto, F., Le superficie razionali (Bologna, 1939). pp. 187–193.Google Scholar
8. Du Val, P., On isolated singularities of surfaces which do not affect the conditions of adjunction, Proc. Cambridge Philos. Soc., 30 (1934), 453–459.Google Scholar
9. Geiser, A., Ueber zwei geometrische Probleme, J. für Math., 67 (1876), 78–83.Google Scholar
10. Herszberg, J., Algebraic characterisation of types of unodes of surfaces in S3 , J. London Math. Soc., 32 (1957), 187–198.Google Scholar
11. Hirzebruch, F., The topology of normal singularities of an algebraic surface, Sém. Bourbaki, 15e année, No. 250 (1962-63).Google Scholar
12. Kirby, D., The structure of isolated multiple points of a surface, I, II, III, Proc London Math. Soc (3), 6 (1956), 597–609 and 7 (1957), 1-28.Google Scholar
13. Nöther, M., Uber eine Classe von auf die einfache Ebene abbildbaren Doppelebenen, Math. Ann., 33 (1889), 525–544; Ueber die rationalen Flächen vierter Ordnung, Math. Ann.,33 (1889), 545–571.Google Scholar
14. von Randow, R., Zur Topologie der Baummannigfaltigkeiten, Bonn. Math. Schr., 14 (Bonn, 1962).Google Scholar