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

TL;DR

This paper investigates the stable pair theory on toric surfaces, providing a method to compute the virtual Euler characteristic and supporting conjectures on rationality and symmetry through specific case verification.

Contribution

It introduces a computational program for the virtual Euler characteristic of stable pairs on surfaces, with explicit calculations on the projective plane.

Findings

Determined the virtual tangent space over fixed point loci.

Developed a computational approach for the virtual Euler characteristic.

Supported conjectures on rationality and symmetry with case verification.

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.