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Published online by Cambridge University Press: 24 October 2008
A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.
* Cf. Baker, H. F., Principles of geometry, 6 (Cambridge, 1933), 215.Google Scholar
† Enriques-Campedelli, , Lezioni sulla teoria delle superficie algebriche (Padova, 1932), p. 389,Google Scholar footnote.
‡ For the virtual grade p (2) and virtual genus p (1) of the canonical system the relation p (2) = p (1) − 1 holds.
* At the six fundamental points. C n denotes a curve of order n.
† This may be verified by considering a special F 4 which consists of two quadrics through C.
‡ This is proved in Baker, H. F., Principles of geometry, 5 (Cambridge, 1933), 23.Google Scholar
* Segre, B., Rendiconti R. Acc. Bologna, January, 1936.Google Scholar
‡ Campedelli, L., Rendiconti Lincei, 15 (1932), 358–62.Google Scholar
‡ Pedoe, D., Proc. Camb. Phil. Soc. 31 (1935), 536.CrossRefGoogle Scholar
* Castelnuovo, G., Annali di Mat. (2), 25 (1897), 235–318.CrossRefGoogle Scholar
† Being a multiple of the prime sections.
‡ The singularity known as a (3, 3) point, when found on the branch curve of a double plane (cf. F. Enriques, loc. cit.) has this singularity in its first neighbourhood. More generally, any surface possessing a node with successive tacnode yields the above singularity on transformation.