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

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

2009-06-03

arxiv.org/abs/0906.0633v3

Mathematics

Algebraic Geometry

41 pages. The proof of Proposition 4.2 and Section 8 are simplified

Scientific paper

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Koll\'ar formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with the second Betti number $b_2(S) = 1$ and with quotient singularities only has at most 3 singular points if its smooth locus $S^0$ is simply-connected. In a previous paper, we have confirmed the conjecture when $S$ has at least one non-cyclic quotient singularity. In this paper, we prove the conjecture either when $S$ is not rational or when $-K_S$ is ample.

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