Embeddings and disjunction of Lagrangian pinwheels via rational blow-ups

We use the symplectic rational blow-up to study some Lagrangian pinwheels in symplectic rational manifolds. In particular, we determine which symplectic forms in the threefold blow-up of P² carry Lagrangian projective planes that can be made disjoint by a Hamiltonian isotopy. In addition, we show that such a disjunction is not possible in del Pezzo surfaces with Euler characteristic between $4$ and $7$. Finally, we determine which symplectic forms on S²× S² carry a Lagrangian L3,1 pinwheel, answering a question of J. Evans.