arXiv subjects

Rahim Rahmati-asghar

Publications and source records attributed to Rahim Rahmati-asghar.

2 recordsLinked to original sources

arXiv2025

Hierarchical filtrations of vector bundles and birational geometry

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.

math.AG↗

arXiv2025

Hierarchical filtrations of line bundles and optimal algebraic geometry codes

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.

math.AG↗