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On the cohomology of tautological bundles over Quot schemes of curves

Alina Marian, Dragos Oprea, Steven V Sam

TL;DR

The paper studies cohomology of tautological bundles on the Quot scheme $\mathsf{Quot}_{\bP^1}(\bC^N,n,r)$, focusing on the zero-rank case and proving precise vanishing results for $\bigwedge^k L^{[n]}$ and $\mathrm{Sym}^k L^{[n]}$ when $\deg L\ge n\ge k$, with $H^0$ identified as tautological images of $H^0(L^{\oplus N})$. The authors implement Str\'omme's embedding of the Quot scheme into a product of Grassmannians and construct Koszul-type resolutions that allow them to reduce cohomology computations to Grassmannians via the Borel–Weil–Bott theorem and Littlewood–Richardson rules. They obtain explicit Ext-vanishing statements and Euler-characteristic formulas, and they formulate higher-rank conjectures ($r>0$) with universal generating-series behavior; their results link to analogous phenomena for Hilbert schemes on surfaces. The approach yields a conceptually streamlined alternative to localization for computing cohomology of tautological bundles on $\mathsf{Quot}_{\bP^1}$ and sets the stage for further generalizations and universality results across genera and ranks.

Abstract

We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions. To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Strømme. This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers. We further illustrate our approach with a few additional cohomological calculations.

On the cohomology of tautological bundles over Quot schemes of curves

TL;DR

The paper studies cohomology of tautological bundles on the Quot scheme , focusing on the zero-rank case and proving precise vanishing results for and when , with identified as tautological images of . The authors implement Str\'omme's embedding of the Quot scheme into a product of Grassmannians and construct Koszul-type resolutions that allow them to reduce cohomology computations to Grassmannians via the Borel–Weil–Bott theorem and Littlewood–Richardson rules. They obtain explicit Ext-vanishing statements and Euler-characteristic formulas, and they formulate higher-rank conjectures () with universal generating-series behavior; their results link to analogous phenomena for Hilbert schemes on surfaces. The approach yields a conceptually streamlined alternative to localization for computing cohomology of tautological bundles on and sets the stage for further generalizations and universality results across genera and ranks.

Abstract

We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions. To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Strømme. This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers. We further illustrate our approach with a few additional cohomological calculations.
Paper Structure (23 sections, 17 theorems, 153 equations, 2 figures)

This paper contains 23 sections, 17 theorems, 153 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tautological vector bundles
  3. Proofs
  4. Higher rank
  5. Subsequent developments
  6. Plan of the paper.
  7. Acknowledgements
  8. Grassmannian embedding of the Quot scheme and resolutions
  9. Strømme's embedding
  10. Resolutions
  11. The main theorems
  12. Analysis of the resolutions
  13. Partitions and Cauchy's formula
  14. The Borel--Weil--Bott theorem
  15. Indices of partitions
  16. ...and 8 more sections

Key Result

Theorem 1.1.2

(1) For all line bundles $L\to \mathbb P^1$ with $\deg L\geq n\geq k$, we have and the higher cohomology vanishes (2) More generally, assume $\deg L\geq n\geq k$ and let $p_1, \ldots, p_t$ be nonnegative integers, $0\leq t\leq N-1$. We have for $|p|=p_1+\cdots+p_t\leq k$. If $|p|>k$, all the above Ext groups vanish.

Figures (2)

  • Figure 1: A partition of $2$-index $i=3$
  • Figure 2: Partitions of $(1, 3)$-index $i=2$

Theorems & Definitions (34)

  • Theorem 1.1.2
  • Theorem 1.1.3
  • Theorem 1.1.5
  • Corollary 1.1.7
  • Corollary 1.2.1
  • Conjecture \theparentequationa
  • Conjecture \theparentequationb
  • Conjecture \theparentequationc
  • Remark \theparentequationb
  • Proposition \theparentequationd
  • ...and 24 more