Hierarchical filtrations of vector bundles and birational geometry

TL;DR

This paper introduces hierarchical filtrations of vector bundles on smooth projective varieties, establishing bounds and formulas for their depth, and connects these concepts to birational geometry and algebraic-geometric codes.

Contribution

It defines hierarchical depth for vector bundles, analyzes its behavior under birational transformations, and links it to improvements in algebraic-geometric codes.

Findings

Hierarchical depth is bounded by the determinant class.

Explicit formulas for depth on curves and Picard rank one varieties.

Birational transformations affect depth additively, aiding code optimization.

Abstract

We introduce and systematically study \emph{hierarchical filtrations} of vector bundles on smooth projective varieties. These are filtrations by saturated subsheaves of equal rank whose successive quotients are torsion sheaves supported in codimension one. The associated numerical invariant, called \emph{hierarchical depth}, measures the maximal length of such filtrations. We establish general bounds for hierarchical depth in terms of the determinant class and provide exact formulas for smooth curves and varieties of Picard rank one. A key technical result concerns the commutativity of elementary transforms along disjoint divisors and their role in constructing filtrations. For surfaces, we analyze the behavior of hierarchical depth under birational morphisms and prove that it transforms additively along the steps of the minimal model program. In particular, we obtain an explicit formula relating the depth on a surface to that on its minimal model via exceptional divisor contributions. As an application, we connect hierarchical depth to degeneracies in algebraic--geometric codes and show that birational simplification via the MMP leads to effective improvements of code parameters. This establishes hierarchical depth as a new bridge between birational geometry, vector bundle theory, and coding theory.