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We establish various stability results for symplectic surfaces in symplectic$4-$manifolds with $b^+=1$. These results are then applied to prove theexistence of representatives of Lagrangian ADE-configurations as well as toclassify negative symplectic spheres in symplectic $4-$manifolds with$\kappa=-\infty$. This involves the explicit construction of spheres inrational manifolds via a new construction technique called the tiltedtransport.

Stability and Existence of Surfaces in Symplectic 4-Manifolds with $b^+=1$