Consider the symplectic manifold $\mathbb P^n$ equipped with the FubiniStudy symplectic form $\omega$. Given $n+1$ "generic" points $z_0,\dots,z_n$ on $\mathbb P^n$, is there an effective Hamiltonian action of the $n$torus $T=(S^1)^n$ on $\mathbb P^n$ whose fixed points are $z_0,\dots,z_n$? The notion of genericity can be part of the answer.
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4$\begingroup$ Use symplectomorphisms to move the points around, so they can be any points you like. Then use the standard action, from the diagonal matrices in the unitary group. $\endgroup$ – Ben McKay Jun 22 '16 at 12:43

$\begingroup$ @benmckay Do we always have a globally defined symplectomorphism that will move a set of points to another set of prescribed points? $\endgroup$ – Anon Jun 22 '16 at 13:38

3$\begingroup$ Compactly supported Hamiltonian functions allow you to move any one point, with the others fixed, in some set. The Sussmann orbit theorem and the transitivity of the linear symplectic group on nonzero vectors tells you that the orbit is open. But it is open around any other point too, so is the whole space (minus the fixed points). $\endgroup$ – Ben McKay Jun 22 '16 at 13:43
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There is an effective Hamiltonian $T=(S^1)^r$ action on $\mathbb{P}^n$, with $2^r=n+1$. Such actions appear in quantum information theory! Please see Lemma (3.1) in http://arxiv.org/abs/1506.05516 for more details.