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Optimal algebraic tangent cone of torsion-free sheaves via valuations

Yohei Hada

TL;DR

This work develops a valuation-theoretic framework to study tangent cones of torsion-free sheaves via slope stability, introducing a generalized Harder–Narasimhan theory for finitely generated $ eal$-graded modules over $ eal_{\ge 0}$-graded algebras. It constructs canonical optimal algebraic tangent cones along quasi-regular valuations by employing Rees modules, $v$-valuative functions, and Hecke transforms, proving existence and (up to grading twists) uniqueness of the associated graded data. The main results show that for an affine domain $R$ and a quasi-regular valuation with index $oldsymbol\delta$, there exists a $v_M$ with $oldsymbol\\Phi(v_M)\in[0,oldsymbol\\delta)$, and the graded Harder–Narasimhan data of the optimal extension is unique. The paper also provides concrete examples on monomial valuations in the affine plane and at singular points to illustrate the optimal tangent cone construction and its uniqueness properties.

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].

Optimal algebraic tangent cone of torsion-free sheaves via valuations

TL;DR

This work develops a valuation-theoretic framework to study tangent cones of torsion-free sheaves via slope stability, introducing a generalized Harder–Narasimhan theory for finitely generated -graded modules over -graded algebras. It constructs canonical optimal algebraic tangent cones along quasi-regular valuations by employing Rees modules, -valuative functions, and Hecke transforms, proving existence and (up to grading twists) uniqueness of the associated graded data. The main results show that for an affine domain and a quasi-regular valuation with index , there exists a with , and the graded Harder–Narasimhan data of the optimal extension is unique. The paper also provides concrete examples on monomial valuations in the affine plane and at singular points to illustrate the optimal tangent cone construction and its uniqueness properties.

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 -graded modules over finitely generated -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].
Paper Structure (11 sections, 18 theorems, 89 equations)

This paper contains 11 sections, 18 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic Riemann-Roch theorem
  4. Slope stability
  5. Extension problem via valuations
  6. Extensions
  7. Hecke transform
  8. Uniqueness of optimal valuative function
  9. Examples
  10. Optimal algebraic tangent cone on affine plane along monomial valuations
  11. Optimal algebraic tangent cone on singular points

Key Result

Theorem 1.2

Let $R$ be an affine domain over an algebraically closed field $\mathbbm{k}$ and $v\in \mathop{\mathrm{Val}}\nolimits_{x,X}$ be a quasi--regular valuation centred at a closed point $x\in X:=\mathop{\mathrm{Spec}}\nolimits(R)$ with index $\delta$, i.e. $v(K(R)-\{0\})=\delta\cdot \mathbb{Z}$. For any Furthermore, the graded module of the Harder--Narasimhan filtration of $\hat{E}|_{\mathfrak{X}_0}$

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Riemann-Roch
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 37 more