Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T12:17:28.166Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant groups associated with an algebraic surface

Published online by Cambridge University Press:  24 October 2008

D. B. Scott
D. B. Scott
Affiliation:
Queen Mary College (University of London) c/o King's College Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 4 , October 1940 , pp. 414 - 423
Copyright
Copyright © Cambridge Philosophical Society 1940

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

REFERENCES

(1)Alexander, J. W.Proc. Nat. Acad. Sci., Wash., 10 (1924), 101–3.Google Scholar
(2)Alexander, J. W.Proc. Nat. Acad. Sci., Wash., 10 (1924), 493–4.Google Scholar
(3)Hodge, W. V. D.Proc. Nat. Acad. Sci., Wash., 17 (1931), 643–50.Google Scholar
(4)Lefschetz, S.L'analysis situs et la géometrie algébrique (Paris, 1924).Google Scholar
(5)Lefschetz, S.Mémor. sci. math., 40 (Paris, 1929), 12.Google Scholar
(6)Lefschetz, S.Topology (New York, 1930).Google Scholar
(7)Lefschetz, S.Algebraic geometry, 2 (1937–1938), 51–2 (Princeton mathematical notes).Google Scholar
(8)van der Waerden, B. L.Math. Ann. 102 (1929), 355.Google Scholar
(9)van der Waerden, B. L.Moderne Algebra, 2 (Berlin, 1931), 128.Google Scholar
(10)Zariski, O. “Algebraic surfaces.” Ergebn. Math. iii, 5 (Berlin, 1935), 140–1.Google Scholar