Curves in P^ and sjonplectic packings

613

First of all, Г-Г = 0? -kp? > êjk > 0 by our choice of a and \i ondly, if С IS a reduced and irreducible curve with [C] = dH - X),=i ^i^i^ then . .

Since а//г > kly/F^ by definition and Y. mJduk/Vk-l by (a) of Theorem 1, we conclude that Г-С > 0 Therefore Г is represented by a Kahler form, and we deduce from Lemma 2 that

that IS, v(5^Ä:)^l-| D

Acknowledgement I am grateful to Lawrence Ein, Robert Lazarsfeld and Lisa Traynor for very helpful conversations

References

1 Alexander, J, Hirschowitz, A La méthode d'Horace éclatée application a Г interpolation en degre quatre Invent Math 107, 585^02 (1992)

2 Gromov, M Pseudo-holomorphic curves in symplectic manifolds Invent Math 82, 307-347 (1985)

3 McDufiF, D Blow-ups and symplectic embeddmgs m dimension 4 Topology 30, 409-421 (1991)

4 McDuff, D The structure of rational and ruled symplectic 4-manifolds J Am Math Soc 3, 679-712 (1990)

5 McDuff D , Polterovich, L Symplectic packings and algebraic geometry (Preprmt)

6 Nagata, M On the 14-th problem of Hubert Am J Math 81, 766-772 (1959)

7 Nagata, M On rational surfaces, II Mem Coll Sci Univ Kyoto, Ser A Math 33, 271-293 (1960)

8 Xu, G Subvaneties of general hypersurfaces in projective space J Differ Geom 39, 139-172 (1994)