Hierarchical filtrations of torsion-free sheaves and birational geometry

Abstract: We introduce the notion of \emph{hierarchical filtrations} of torsion-free sheaves on normal projective varieties and define the associated numerical invariant called \emph{hierarchical depth}. This invariant measures the maximal length of filtrations by saturated subsheaves of equal rank whose successive quotients are torsion sheaves supported in codimension one. We establish general bounds for hierarchical depth in terms of the divisor class of the determinant and give exact formulas in several basic geometric situations, including the case of smooth projective curves and varieties of Picard rank one. A key technical ingredient is the study of elementary transforms along effective divisors and their commutativity properties. In dimension two, we analyze the behavior of hierarchical depth under birational morphisms and show that it admits a precise description along the minimal model program. In particular, we prove that hierarchical depth transforms additively with respect to exceptional divisors and is explicitly computable on minimal models. As an application, we relate hierarchical depth to degeneracies in algebraic--geometric codes and show that birational simplification via the minimal model program leads to effective improvements of code parameters. These results demonstrate that hierarchical depth provides a new bridge between the birational geometry of vector bundles and arithmetic applications.