Bounding minimal log discrepancies of general arrangement varieties
Leandro Meier
TL;DR
The paper proves Shokurov's conjectured bound $\mathrm{mld}_x(X) \le \dim(X)$ for klt general arrangement varieties of arbitrary complexity by exploiting presentations as quotients by diagonalizable groups and applying known $mld$-bounds for local complete intersections via Cox rings. The approach reduces to cone singularities or toroidal-analogue settings and uses quotient techniques to transfer bounds from simpler cases to the general arrangement setting. This extends the class of varieties for which the $mld$-bound holds beyond previously settled cases (dimension $\le 3$, toric, and lci), contributing a new pillar in the birational classification program and the study of singularities in the MMP. The results also highlight the utility of Cox rings and quotient-geometry in controlling singularities of $T$-varieties, offering a framework that may inform future extensions to broader classes of singularities.
Abstract
The minimal log discrepancy is an invariant of singularities that plays an important role in the birational classification of algebraic varieties. Shokurov conjectured that the minimal log discrepancy can always be bounded from above in terms of the dimension of the variety. We prove this conjecture for general arrangement varieties, a particular class of T-varieties, adding to previous results on this conjecture which include threefolds, toric varieties, and local complete intersection varieties.