Minimal log discrepancy and orbifold curves

TL;DR

The paper establishes a sharp upper bound $\mathrm{mld}(o,X)\le n$ for the minimal log discrepancy at isolated Fano cone singularities of dimension $n$ by connecting $\mathrm{mld}$ to the dimensions of moduli spaces of orbifold rational curves on an associated base orbifold $\mathcal{Y}$. It provides two proofs—an algebraic approach based on Mori theory in the orbifold setting and a symplectic-field-theory perspective—showing that suitable twisted maps from weighted projective lines yield the desired bound. A key contribution is a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply equality in the bound only at smooth points. The work also presents an orbifold Mori analog of a Mori–Mukai-type result and develops a framework linking minimal singularity invariants to lSFT dimensions and orbifold Gromov–Witten theory, with implications for the geometry and classification of Fano orbifolds with ample orbifold tangent bundles.

Abstract

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply the equality holds only for smooth points.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4