Algebraic Montgomery-Yang problem and cascade conjecture

TL;DR

This paper links the algebraic Montgomery-Yang problem to the cascade conjecture, proposing that all relevant rational surfaces with ample canonical divisors exhibit a cascade, thus advancing understanding of their structure.

Contribution

It establishes the algebraic Montgomery-Yang problem assuming the cascade conjecture, connecting two important conjectures in the classification of rational surfaces.

Findings

Conditional proof of the algebraic Montgomery-Yang problem

Supports the cascade conjecture for certain rational surfaces

Highlights the importance of cascade behavior in surface classification

Abstract

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.