Degrees and directional defects of embedded algebraic vector bundles
Abstract
We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound deg(TV) ≤ deg(V)2 for every smooth irreducible affine variety V ⊆ 𝔸n and TV⊆ 𝔸2n its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.
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BibTeX
@misc{lanciano2026degrees,
title = {{Degrees and directional defects of embedded algebraic vector bundles}},
author = {Leonardo Lanciano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27047},
eprint = {2609.27047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27047},
}
RIS
TY - UNPB
TI - Degrees and directional defects of embedded algebraic vector bundles
AU - Leonardo Lanciano
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27047
UR - https://arxiv.org/abs/2609.27047
AB - We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound deg(TV) ≤ deg(V)^2 for every smooth irreducible affine variety V ⊆ 𝔸^n and TV⊆ 𝔸^2n its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.
ER -