W-translated Schubert divisors and transversal intersections

Science China Mathematics - We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety $$F{ell _{{n_1}, ldots,{n_k};n}}$$ via Gelfand-Cetlin polytopes....     

Algebraic Montgomery�CYang problem: the non-cyclic case

Montgomery?CYang problem predicts that every pseudofree circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery?CYang problem: every projective surface S with quotient singularities such that the second Betti number b ₂(S) = 1 has at most 3 singular points if its smooth locus S ⁰ is simply connected. We prove the conjecture under the assumption that S has at least one non-cyclic singularity. In the course of the proof, we classify projective surfaces S with quotient singularities such that (i) b ₂(S) = 1, (ii) ${H_1(S^0, \mathbb{Z}) = 0}$ , and (iii) S has 4 or more singular points, not all cyclic, and prove that all such surfaces have ${\pi_1(S^0)\cong \mathfrak{A}_5}$ , the icosahedral group.