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On the Virtual Grade of Curves on an Algebraic Surface

Published online by Cambridge University Press:  24 October 2008

D. Pedoe
D. Pedoe
Affiliation:
Magdalene College
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Extract

The theoretical method for computing the grade of a curve on an algebraic surface is well known. In practice difficulties arise which are not considered in the theory; so that it seems worth while to describe a practical method. This is done in § 1 of this paper. The method is then applied to some examples with the object of discovering whether Noether exceptional curves are necessarily exceptional curves†. In particular, a certain quintic surface with three tacnodes is studied, and our examination leads us to results which differ from those which have been accepted up till now. Another example illustrates the limitations of a practical method for computing grades, because of the possible presence of infinitesimal curves, and leads to the transformation of the quintic surface with two tacnodes into a double plane of order ten of a certain type, which has the singularity known as a (5, 5) point on its branch curve. Light is thrown on the Noether composition of this singularity by the transformation, which also shows the relation between two well-known types of surface for which the Noether relation p(2) = p(1) − 1 does not hold.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 4 , October 1935 , pp. 536 - 542
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* See Enriques-Campedelli, , Lezioni sulla teoria delle superficie algebriche (Padova, 1932), p. 61Google Scholar. We refer to this book subsequently as E.C.

† These terms are defined in § 2.

‡ Enriques' principle: a curve varying in a continuous system cannot degenerate without acquiring at least one more double point.

* Compare E.C. p. 57.

† Noether, , Math. Annalen, 8 (1875), 495–533.CrossRefGoogle Scholar

‡ Castelnuovo-Enriques, , Annali di Mat. (3), 6 (1901), 165–225CrossRefGoogle Scholar, § 8.

§ Enriques, , Mem. Soc. Ital. d. Scienze, 10 (1896), 72.Google Scholar

∥ Simart, Picard et, Théorie des fonctions algébriques de deux variables 1 (Paris, 1897), p. 223Google Scholar. This example may possibly be due to Picard originally. See Rendic. Palermo, 36 (1913), p. 277CrossRefGoogle Scholar, for a note by Picard on bibliographical errors.

* But being composite, the theorem referred to in § 2 does not apply. For a composite curve to be exceptional, the conditions π = 0, n = − 1 are necessary but not sufficient. See Zariski, , Algebraic Surfaces (Berlin, 1935), p. 36.Google Scholar

† The conic δ = F 2 = 0 can be transformed into a simple point on a F 15 in S 7 by means of the linear system cut in F 5 by cubic surfaces containing the cubic curve δ = F 3 = 0 and touching the three tacnodal planes α = β = γ = 0. This linear system is the sum of the conic and the regular system cut on F 6 by quadrics passing through the tacnodes.

‡ Campedelli, , Rendiconti Lancei (6), 18 (1933), 380.Google Scholar

* Castelnuovo-Enriques, , Annali di Mat. (3), 6 (1901), 165–225CrossRefGoogle Scholar, § 5.

† Ibid. § 20.

‡ See the chapter on double planes in E.C. for this and other results assumed in this paragraph.

* This verifies the remark in E.C. p. 389.

† See preceding paragraph.

‡ This is verified by Zeuthen's formula.

§ See § 8.

∥ This result was given in a lecture by Prof. H. F. Baker, who proved it by algebraic methods which show that the double plane obtained from the quintic surface has a certain speciality: but this does not affect our argument.

* See E.C. p. 367.

* See § 5.