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Codimension of jumping loci

Brian Lehmann, Eric Riedl, Sho Tanimoto

TL;DR

The paper develops a general framework for understanding how slope stability of a vector bundle $\mathcal{E}$ on a smooth projective variety $X$ changes when restricted to curves from a dominant family, via the slope panel $\operatorname{SP}_{X,\alpha}(\mathcal{E})$. Under suitable hypotheses, including a connected-fibers condition on the evaluation map, it proves linear-type lower bounds on the codimension of the jumping locus in terms of the slope discrepancy $\mu$ or the curve-degree data, encoded by affine functions $S$ and $L$. The authors then apply these results to rank $2$ bundles on $\mathbb{P}^{2}$, to the singular locus of moduli spaces of curves, and to questions about $m$-free curves on Fano varieties, combining birational geometry of Fitting ideals, log canonical thresholds, Grauert–Mülich-type stability, and accumulating-map techniques. The approach yields explicit obstruction mechanisms: either the jumping is restricted by linear bounds, or one encounters birational transforms that destabilize the pullback bundle, or the family factors through nontrivial covers, with the latter controlled by Fujita/accumulating-map data. Overall, the work provides a versatile toolkit for quantifying restriction-induced stability changes and deriving codimension estimates with broad geometric consequences.

Abstract

Suppose that $\mathcal{E}$ is a vector bundle on a smooth projective variety $X$. Given a family of curves $C$ on $X$, we study how the Harder-Narasimhan filtration of $\mathcal{E}|_{C}$ changes as we vary $C$ in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by $μ$ should have codimension which depends linearly on $μ$. We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank $2$ bundles on $\mathbb{P}^{2}$ and to study singular loci of moduli spaces of curves.

Codimension of jumping loci

TL;DR

The paper develops a general framework for understanding how slope stability of a vector bundle on a smooth projective variety changes when restricted to curves from a dominant family, via the slope panel . Under suitable hypotheses, including a connected-fibers condition on the evaluation map, it proves linear-type lower bounds on the codimension of the jumping locus in terms of the slope discrepancy or the curve-degree data, encoded by affine functions and . The authors then apply these results to rank bundles on , to the singular locus of moduli spaces of curves, and to questions about -free curves on Fano varieties, combining birational geometry of Fitting ideals, log canonical thresholds, Grauert–Mülich-type stability, and accumulating-map techniques. The approach yields explicit obstruction mechanisms: either the jumping is restricted by linear bounds, or one encounters birational transforms that destabilize the pullback bundle, or the family factors through nontrivial covers, with the latter controlled by Fujita/accumulating-map data. Overall, the work provides a versatile toolkit for quantifying restriction-induced stability changes and deriving codimension estimates with broad geometric consequences.

Abstract

Suppose that is a vector bundle on a smooth projective variety . Given a family of curves on , we study how the Harder-Narasimhan filtration of changes as we vary in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by should have codimension which depends linearly on . We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank bundles on and to study singular loci of moduli spaces of curves.
Paper Structure (34 sections, 47 theorems, 147 equations)

This paper contains 34 sections, 47 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Rank 2 bundles on $\mathbb{P}^{2}$
  5. Locus of non-free curves
  6. Strategy
  7. Background
  8. Slope stability
  9. Bounded families of sheaves
  10. Log canonical thresholds
  11. Birational geometry of Fitting ideals
  12. Fitting ideals
  13. Resolving Fitting ideals in families
  14. Sequences of quotients
  15. Birational transforms
  16. ...and 19 more sections

Key Result

Theorem 1.3

Let $X$ be a smooth projective variety and let $\mathcal{E}$ be a non-zero locally free coherent sheaf on $X$. Fix a genus $g$, an ample divisor $H$ on $X$, and a closed cone $\mathcal{C} \subset N_{1}(X)_{\mathbb{R}}$ such that $\mathcal{C} \backslash \{0\}$ is contained in the interior of $\operat where $s: C \to X$ is a general curve parametrized by $W$. Suppose that: Then the codimension of t

Theorems & Definitions (118)

  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • Definition 2.2
  • ...and 108 more