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Nongeneric J-holomorphic curves in symplectic 4-manifolds

This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J\in \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss of transversally intersecting embedded spheres of self-intersection \le -2 is J-holomorphic. Extending work by Biran (in Invent. Math. (1999)), it shows that when (M,\om) is the blow up of a rational or ruled symplectic 4-manifold, any homology class A\in H_2(M;\Z), with nonzero Gromov invariant and nonnegative intersection both with the spheres in \Ss and with the exceptional classes other than A, has an embedded J-holomorphic representative for some J\in \Jj(\Ss$. This result is a key step in some of the arguments in McDuff (Journ. Topology (2009)) on embedding ellipsoids, and also has applications to symplectic 4-orbifolds.