Classification of surfaces with rational polyhedral Mori cones

Classify all smooth projective surfaces whose closed Mori cone is rational polyhedral.

Background

The paper describes the classification of surfaces with rational polyhedral Mori cones as a fundamental problem because rational polyhedrality implies the Bounded Negativity Conjecture. Existing boundedness results restrict the possible surfaces under fixed numerical data, and complete classifications are known only in limited cases.

In particular, the paper notes that a complete classification remains largely unresolved when the maximal negative self-intersection of curves is at least three.