Hierarchical filtrations of line bundles and optimal algebraic geometry codes

TL;DR

This paper introduces hierarchical depth as a new invariant of line bundles, leading to hierarchical filtrations that improve the construction and analysis of algebraic geometry codes with optimized parameters.

Contribution

It develops the concept of hierarchical depth and filtrations, linking surface geometry with the design of algebraic geometry codes for better asymptotic performance.

Findings

Defined hierarchical depth and proved fundamental properties.

Constructed hierarchical algebraic geometry codes with controlled parameters.

Identified an optimal intermediate code balancing rate and distance.

Abstract

We introduce \emph{hierarchical depth}, a new invariant of line bundles and divisors, defined via maximal chains of effective sub-line bundles. This notion gives rise to \emph{hierarchical filtrations}, refining the structure of the Picard group and providing new insights into the geometry of algebraic surfaces. We establish fundamental properties of hierarchical depth, derive inequalities through intersection theory and the Hodge index theorem, and characterize filtrations that are Hodge-tight. Using this framework, we develop a theory of \emph{hierarchical algebraic geometry codes}, constructed from evaluation spaces along these filtrations. This approach produces nested families of codes with controlled growth of parameters and identifies an optimal intermediate code maximizing a utility function balancing rate and minimum distance. Hierarchical depth thus provides a systematic method to construct AG codes with favorable asymptotic behavior, linking geometric and coding-theoretic perspectives. Our results establish new connections between line bundle theory, surface geometry, and coding theory, and suggest applications to generalized Goppa codes and higher-dimensional evaluation codes.