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Stability of natural bundles on curves

Izzet Coskun, Eric Larson, Isabel Vogt

TL;DR

This survey synthesizes stability phenomena for the two de facto deformation bundles, the restricted tangent bundle $TY|_X$ and the normal bundle $N_{X/Y}$, across rational, genus 1, and higher-genus curves in projective ambient spaces and homogeneous varieties. It combines explicit computations for rational curves, degeneration methods to nodal curves, and Brill–Noether theory to derive when these bundles are (semi)stable, including characteristic-dependent behavior and phenomena on Grassmannians and complete intersections. A central thread is identifying when stability persists under specializations and what obstructions arise, with extensive discussion of BN-curves, canonical curves, and the interplay with covers via Tschirnhausen bundles and pushforwards. The work highlights both established stable regimes and notable open questions, such as the full classification of BN-triples not (semi)stable and the precise structure of loci where normal bundles fail stability, as well as Beauville-type conjectures for pushforwards under covers.

Abstract

In this paper, we survey recent developments concerning the stability of naturally defined bundles on curves that play a central role in the deformation theory of the curve.

Stability of natural bundles on curves

TL;DR

This survey synthesizes stability phenomena for the two de facto deformation bundles, the restricted tangent bundle and the normal bundle , across rational, genus 1, and higher-genus curves in projective ambient spaces and homogeneous varieties. It combines explicit computations for rational curves, degeneration methods to nodal curves, and Brill–Noether theory to derive when these bundles are (semi)stable, including characteristic-dependent behavior and phenomena on Grassmannians and complete intersections. A central thread is identifying when stability persists under specializations and what obstructions arise, with extensive discussion of BN-curves, canonical curves, and the interplay with covers via Tschirnhausen bundles and pushforwards. The work highlights both established stable regimes and notable open questions, such as the full classification of BN-triples not (semi)stable and the precise structure of loci where normal bundles fail stability, as well as Beauville-type conjectures for pushforwards under covers.

Abstract

In this paper, we survey recent developments concerning the stability of naturally defined bundles on curves that play a central role in the deformation theory of the curve.
Paper Structure (25 sections, 35 theorems, 85 equations)

This paper contains 25 sections, 35 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability
  4. The invariants
  5. Elementary modifications and normal bundles of nodal curves
  6. Normal bundles of nodal curves
  7. Stability on nodal curves
  8. Rational curves
  9. Bundles on rational curves
  10. The restricted tangent bundle of rational curves in projective space
  11. Normal bundles of rational curves in projective space
  12. Restricted tangent bundles of rational curves in Grassmannaians
  13. Normal bundles of rational curves in Grassmannians
  14. Restricted tangent bundles of rational curves in hypersurfaces
  15. Normal bundles of rational curves in complete intersections in homogeneous varieties
  16. ...and 10 more sections

Key Result

Lemma 2.4

hartshornehirschowitz Let $X=X_1\cup X_2 \in \mathbb{P}^r$ be a connected nodal curve. Let $X_1 \cap X_2= \{ p_1, \dots, p_j\}$. Then

Theorems & Definitions (74)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Definition 3.1
  • Example 3.2
  • Corollary 3.3
  • ...and 64 more