  A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal
sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective
two-dimensional torus action if k <= 3. We show that it does not admit such an
action if k >=4 and epsilon <= 1/(3k 2^{2k}). For the proof, we correspond
between the geometry of a symplectic toric four-manifold and the combinatorics
of its moment map image. We also use techniques from the theory of
J-holomorphic curves.
