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Intersection Theory of Hyperquot Schemes on curves

Riccardo Ontani, Shubham Sinha, Weihong Xu

TL;DR

The paper develops a comprehensive virtual intersection theory for Hyperquot schemes over curves, generalizing the Vafa–Intriligator framework from Quot schemes to sequences of quotients. It builds a perfect obstruction theory, constructs the virtual fundamental class, and establishes compatibility under twisting and elementary modifications, enabling exact virtual counts via localization. A central achievement is the genus-zero formula that expresses generating series of equivariant virtual integrals as sums over non-degenerate Bethe-system solutions, with a unifying degeneration/Lagrange–Bürmann approach and explicit residue computations; higher genus is treated using theta-classes and symmetric-product geometry, yielding an equivariant formula and explicit closed forms in special cases. The results connect to fixed-domain Gromov–Witten theory, quantum cohomology of partial flags, and quasimap enumerative frameworks, and open avenues for positivity, enumerativity, and potential K-theoretic extensions. Overall, the work provides a computable bridge between virtual intersection theory on Hyperquot schemes and quantum-cohomological invariants of flag varieties.

Abstract

We study the virtual intersection theory of Hyperquot schemes parameterizing sequences of quotient sheaves of a vector bundle on a smooth projective curve. Our results generalize the Vafa--Intriligator formula for Quot schemes and provide a closed formula for virtual counts of maps from the curve to a partial flag variety.

Intersection Theory of Hyperquot Schemes on curves

TL;DR

The paper develops a comprehensive virtual intersection theory for Hyperquot schemes over curves, generalizing the Vafa–Intriligator framework from Quot schemes to sequences of quotients. It builds a perfect obstruction theory, constructs the virtual fundamental class, and establishes compatibility under twisting and elementary modifications, enabling exact virtual counts via localization. A central achievement is the genus-zero formula that expresses generating series of equivariant virtual integrals as sums over non-degenerate Bethe-system solutions, with a unifying degeneration/Lagrange–Bürmann approach and explicit residue computations; higher genus is treated using theta-classes and symmetric-product geometry, yielding an equivariant formula and explicit closed forms in special cases. The results connect to fixed-domain Gromov–Witten theory, quantum cohomology of partial flags, and quasimap enumerative frameworks, and open avenues for positivity, enumerativity, and potential K-theoretic extensions. Overall, the work provides a computable bridge between virtual intersection theory on Hyperquot schemes and quantum-cohomological invariants of flag varieties.

Abstract

We study the virtual intersection theory of Hyperquot schemes parameterizing sequences of quotient sheaves of a vector bundle on a smooth projective curve. Our results generalize the Vafa--Intriligator formula for Quot schemes and provide a closed formula for virtual counts of maps from the curve to a partial flag variety.

Paper Structure

This paper contains 44 sections, 40 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Statement of the main formula
  3. Specializations
  4. Related work and future directions
  5. Sketch of proof
  6. Perfect obstruction theory and virtual fundamental class
  7. Obstruction theory
  8. Virtual fundamental class
  9. Compatibility of virtual classes
  10. Twisting by a point
  11. Elementary Modifications
  12. A vanishing criterion
  13. Torus action and fixed loci
  14. Equivariant formulation
  15. Fixed loci
  16. ...and 29 more sections

Key Result

Theorem 1.2

Fix a tuple $\mathbf{r}=(r_1,\dots,r_k)$ and a vector bundle $V$ on $C$ of rank $n$ with $\deg V= 0$. Then for natural numbers $m_{i,j}$ and a generic $(q_1, \dots, q_k) \in (\mathbb{C}^\ast)^k$, we have where the sum is taken over all non-degenerate solutions $(\boldsymbol{\zeta}_1,\dots,\boldsymbol{\zeta}_k)$ of the equations in eq:Bethe_system_intro and $e_i(\boldsymbol{\zeta}_j)$ denotes the

Theorems & Definitions (94)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 84 more