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Montgomery-Yang problem predicts that every pseudofree differentiable circleaction on the 5-dimensional sphere has at most 3 non-free orbits. Using acertain one-to-one correspondence, Koll\'ar formulated the algebraic version ofthe Montgomery-Yang problem: every projective surface $S$ with the second Bettinumber $b_2(S) = 1$ and with quotient singularities only has at most 3 singularpoints if its smooth locus $S^0$ is simply-connected. In a previous paper, wehave confirmed the conjecture when $S$ has at least one non-cyclic quotientsingularity. In this paper, we prove the conjecture either when $S$ is notrational or when $-K_S$ is ample.

Algebraic Montgomery-Yang Problem: the non-rational case and the del Pezzo case