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Linear sections of Grassmannians and resonance of vector bundles

Marian Aprodu, Călin Spiridon

TL;DR

This work investigates when a resonance variety ${\mathbb R}(V,K)$ can be realized as the resonance ${\mathbb R}(E)$ of a vector bundle on a smooth projective variety. It develops a flattening-stratification framework and a natural family of morphisms to Quot schemes to detect obstructions, connecting resonance to saturated subbundles and their subpencils. The authors then study the resonance of restricted universal rank-two quotient bundles on transversal linear sections of Grassmannians, proving that in the transversal setting the resonance arises from the restricted universal bundle and giving complete descriptions in low dimensions. In particular, they show that for $n=4,5,6$ there are positive realizability results, with the $n=6$ case yielding a Mukai-bundle realization of fourteen-line resonances on genus $8$ curves. This culminates in a bridge between resonance geometry and Mukai's bundle theory, illustrating a rich interaction between linear sections of Grassmannians and vector-bundle resonances.

Abstract

This work revolves around the question of whether a given resonance variety is associated with a vector bundle. We show the existence of a family of natural morphisms on a stratification of the resonance variety to a suitable family of a Quot scheme and provide some applications in the curve case. The existence of this family of morphisms represents an obstruction to affirmatively answering the main question. In addition, we study the resonance of restricted universal rank-two quotient bundles over transversal linear sections of the Grassmann varieties $\operatorname{Gr}_2(\mathbb{C}^n)$, with a special attention to low-dimensional Grassmannians. These bundles are among the most natural to consider in this context. The analysis for $\operatorname{Gr}_2(\mathbb{C}^6)$ shows that any resonance variety in $\mathbb{P}^5$ consisting of fourteen disjoint lines is the resonance of some bundle which appeared in the work of Mukai.

Linear sections of Grassmannians and resonance of vector bundles

TL;DR

This work investigates when a resonance variety can be realized as the resonance of a vector bundle on a smooth projective variety. It develops a flattening-stratification framework and a natural family of morphisms to Quot schemes to detect obstructions, connecting resonance to saturated subbundles and their subpencils. The authors then study the resonance of restricted universal rank-two quotient bundles on transversal linear sections of Grassmannians, proving that in the transversal setting the resonance arises from the restricted universal bundle and giving complete descriptions in low dimensions. In particular, they show that for there are positive realizability results, with the case yielding a Mukai-bundle realization of fourteen-line resonances on genus curves. This culminates in a bridge between resonance geometry and Mukai's bundle theory, illustrating a rich interaction between linear sections of Grassmannians and vector-bundle resonances.

Abstract

This work revolves around the question of whether a given resonance variety is associated with a vector bundle. We show the existence of a family of natural morphisms on a stratification of the resonance variety to a suitable family of a Quot scheme and provide some applications in the curve case. The existence of this family of morphisms represents an obstruction to affirmatively answering the main question. In addition, we study the resonance of restricted universal rank-two quotient bundles over transversal linear sections of the Grassmann varieties , with a special attention to low-dimensional Grassmannians. These bundles are among the most natural to consider in this context. The analysis for shows that any resonance variety in consisting of fourteen disjoint lines is the resonance of some bundle which appeared in the work of Mukai.

Paper Structure

This paper contains 18 sections, 36 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Saturation of subsheaves
  4. Two degeneracy loci associated to a vector bundle
  5. Resonance of vector bundles
  6. Generalities
  7. Resonance, Quot schemes and flattening stratifications
  8. Vector bundles on curves
  9. Resonance and Brill-Noether loci
  10. Rank-two bundles on the projective line
  11. Transversal vector bundles
  12. General results
  13. Transversality in families
  14. Universal rank two bundles on linear sections of Grassmannians
  15. Low-dimensional Grassmannians and resonance
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $\mathcal{F} \subseteq \mathcal{G} \subseteq E$ be two subsheaves of $E$. Then $\mathcal{F}^{\text{sat}} \subseteq \mathcal{G}^\text{sat}$. Moreover, if $\mathcal{F}$ and $\mathcal{G}$ have the same rank, then $\mathcal{F}^{\text{sat}} = \mathcal{G}^\text{sat}$.

Theorems & Definitions (76)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Example 3.2
  • proof
  • Theorem 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 66 more