Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T05:55:46.958Z Has data issue: false hasContentIssue false
  • English
  • Français

Configurations of 2-spheres in the K3 surface and other 4-manifolds

Published online by Cambridge University Press:  24 October 2008

Daniel Ruberman
Daniel Ruberman
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02254
Get access Rights & Permissions [Opens in a new window]

Extract

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 2 , August 1996 , pp. 247 - 253
Copyright
Copyright © Cambridge Philosophical Society 1996

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]Barth, W.. Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, preprint (1994).Google Scholar
[2]Beauville, A.. Sur le nombre maximum de points double d'une surface dans P3 (µ(5) = 31); in Journées de geometrie algebrique d'Angers 1979. (Alphen aan den Rijn, 1980), pp. 207–215.Google Scholar
[3]Browder, W.. Surgery on simply-connected manifolds (Springer-Verlag, 1972).CrossRefGoogle Scholar
[4]Catanese, E.. Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications. Inventiones Math. 63 (1981), 433–465.Google Scholar
[5]Freedman, M.. The topology of four-dimensional manifolds. J. Diff. Geom. 17 (1982), 357–432.Google Scholar
[6]Gilmer, P.. Configurations of surfaces in 4-manifolds. Trans. A.M.S. 264 (1981), 353–380.Google Scholar
[7]Hirzebruch, F.. Singularities of algebraic surfaces and characteristic numbers; in The Lefschetz centennial conference. Part 1: Proceedings on algebraic geometry (American Mathematical Society, 1986), pp. 141–156.Google Scholar
[8]Jaffe, D. B. and Ruberman, D.. A sextic cannot have 66 nodes. J. Alg. Geom., to appear.Google Scholar
[9]Kauffman, L. H. and Taylor, L. R.. Signature of links. Trans. A.M.S. 216 (1976), 351–365.Google Scholar
[10]Kronheimer, P. B.. On the genus-minimizing property of algebraic curves. Bull. A.M.S. 29 (1993), 63–69.Google Scholar
[11]Kronheimer, P. B. and Mrowka, T. S.. Gauge theory for embedded surfaces. I. Topology 32 (1993), 773–826.CrossRefGoogle Scholar
[12]Kronheimer, P. B. and Mrowka, T. S.. The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), no. 6, 797–808.Google Scholar
[13]Lee, R. and Wilczynski, D.. Locally flat 2-spheres in simply connected 4-manifolds. Comm. Math. Helv. 65 (1990), 388–412.Google Scholar
[14]Miyaoka, Y.. The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268 (1984), 159–171.CrossRefGoogle Scholar
[15]Nikulin, V. V.. On Kummer surfaces. Math. U.S.S.R. Izvestia 9 (1975), 261–275.CrossRefGoogle Scholar