Circle and torus actions on equal symplectic blow-ups of CP²

A manifold obtained by k simultaneous symplectic blow-ups of ℂℙ² of equal sizes ε (where the size of ℂℙ¹ ⊂ ℂℙ² is one) admits an effective two dimensional torus action if k < 3 and admits an effective circle action if ε < 1/(k - 1). We show that these bounds are sharp if ε = 1/n where n is a natural number. Our proof combines "soft" equivariant techniques with "hard" holomorphic techniques.