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arXiv · 2602.01112

Optimal algebraic tangent cone of torsion-free sheaves via valuations

Yohei Hada
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Abstract

We develop a valuation-theoretic framework for studying tangent cones of torsion-free sheaves on algebraic varieties. To analyze these objects, we introduce a slope stability theory, including the Harder-Narasimhan filtrations, for finitely generated $\mathbb{R}$-graded modules over finitely generated $\mathbb{R}_{\geq 0}$-graded algebras. Using it, we show that there is a canonically determined tangent cone of torsion-free sheaves, up to the expected equivalence ambiguity, for quasi-regular valuations, which generalize Chen-Sun [3].

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Yohei Hada. 2026-02-01. Optimal algebraic tangent cone of torsion-free sheaves via valuations. https://arxiv.org/abs/2602.01112

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