Minimal log discrepancy and orbifold curves

TL;DR

This paper establishes an upper bound on the minimal log discrepancy of isolated Fano cone singularities, linking it to orbifold rational curves, and proposes a conjecture characterizing weighted projective spaces via these curves.

Contribution

It introduces a bound on minimal log discrepancy for Fano cone singularities and proposes a new conjecture characterizing weighted projective spaces using orbifold rational curves.

Findings

Minimal log discrepancy of Fano cone singularities is at most the dimension.

Relation between minimal log discrepancy and moduli spaces of orbifold rational curves.

Conjecture linking weighted projective spaces to orbifold rational curves.

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.