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On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces

Ana Pavlaković

TL;DR

The paper develops a torus localisation framework to study the moduli space of stable pairs on toric surfaces, and explicitly computes the virtual tangent space at fixed points, decomposing it into vertex and edge contributions that are assembled into the global virtual Euler characteristic. It demonstrates that local partition functions in the toric setting are, in general, non-rational, yet the global partition function appears to be rational in tested cases, notably for $S=\mathbb{P}^2$ with $d=1$, where $Z^{\mathbb{P}^2}_{d=1}(q)=\frac{3}{(1-q)^2}$. A Python program accompanies the theoretical framework to compute invariants for given $n$ and $d$, and the authors provide explicit computations and conjectures for rationality and symmetry in low degrees, with a suggested pathway via capped localisation. The work connects stable pair theory on surfaces to nested Hilbert schemes and box configurations, extending threefold techniques to surface geometry and offering computational tools to probe fundamental questions about rationality of generating series and symmetry properties in toric contexts.

Abstract

We study the stable pair theory on toric surfaces and determine the virtual tangent space over the fixed point loci. Further, we present a program to compute the virtual Euler characteristic, illustrated by the case of the projective plane. As an application, conjectures regarding rationality and symmetry are supported by verification of a special case.

On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces

TL;DR

The paper develops a torus localisation framework to study the moduli space of stable pairs on toric surfaces, and explicitly computes the virtual tangent space at fixed points, decomposing it into vertex and edge contributions that are assembled into the global virtual Euler characteristic. It demonstrates that local partition functions in the toric setting are, in general, non-rational, yet the global partition function appears to be rational in tested cases, notably for with , where . A Python program accompanies the theoretical framework to compute invariants for given and , and the authors provide explicit computations and conjectures for rationality and symmetry in low degrees, with a suggested pathway via capped localisation. The work connects stable pair theory on surfaces to nested Hilbert schemes and box configurations, extending threefold techniques to surface geometry and offering computational tools to probe fundamental questions about rationality of generating series and symmetry properties in toric contexts.

Abstract

We study the stable pair theory on toric surfaces and determine the virtual tangent space over the fixed point loci. Further, we present a program to compute the virtual Euler characteristic, illustrated by the case of the projective plane. As an application, conjectures regarding rationality and symmetry are supported by verification of a special case.

Paper Structure

This paper contains 48 sections, 10 theorems, 194 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Non-Rationality of Local Partition Function
  3. Further Directions
  4. Plan of the Paper
  5. Notation
  6. Acknowledgements
  7. Stable Pairs on Surfaces
  8. Stable Pairs
  9. Nested Hilbert Schemes
  10. Universal Pair and Deformation Theory
  11. $\mathbf{T}$-Fixed Points
  12. Charts
  13. Monomial Ideals, Young Diagrams, and Partitions
  14. Torus Fixed Points of $P_n(S,\beta)$
  15. Virtual Tangent Space
  16. ...and 33 more sections

Key Result

Theorem 1.1

For a non-singular toric surface $S$ and a homology class $\beta\in H_2(S;\mathbb{Z})$, where and $\mathsf{G}_\alpha, \mathsf{F}_\alpha$, and $\mathsf{E}_{\alpha\beta}$ are defined as in eq:Galpha, eq:vertex contribution, and eq:edge contribution, respectively.

Figures (5)

  • Figure 1: Case $(d_1,d_2) = (1,0), \;b=4$
  • Figure 2: Case $(d_1,d_2) = (1,1), \;a=5,\;b=4$
  • Figure 3: Case $(d_1,d_2) = (2,0), \;a=4, \;b=2$
  • Figure 4: Virtual Euler characteristic for $d=1,2$ and low $m$
  • Figure 5: Virtual Euler characteristic for $d=3,4$ and low $m$

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1
  • Conjecture 2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: pt_nested
  • Remark 2.5
  • Theorem 3.1: gp
  • ...and 19 more