Algebraic Montgomery-Yang problem and cascade conjecture
The conjecture called algebraic Montgomery-Yang problem is still open for rational $\mathbb{Q}$-homology projective planes with cyclic quotient singularities having ample canonical divisor. All known such surfaces have a special birational behavior called a cascade. In this note, we establish algebraic Montgomery-Yang problem assuming the cascade conjecture, which claims that every rational $\mathbb{Q}$-homology projective planes with quotient singularities having ample canonical divisor admits a cascade.