Optimal algebraic tangent cone of torsion-free sheaves via valuations

TL;DR

This paper introduces a valuation-theoretic approach to study tangent cones of torsion-free sheaves on algebraic varieties, incorporating slope stability and Harder-Narasimhan filtrations to establish a canonical tangent cone concept.

Contribution

It develops a new valuation-based framework and slope stability theory for analyzing tangent cones of torsion-free sheaves, generalizing previous work by Chen-Sun.

Findings

Canonical tangent cone exists for quasi-regular valuations

Framework unifies valuation theory with slope stability

Generalizes Chen-Sun's results on tangent cones

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].