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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
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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

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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