Logarithmic Pandharipande--Thomas Spaces and the Secondary Polytope

TL;DR

This paper explores logarithmic stable pairs on toric surfaces, expressing their moduli spaces explicitly via secondary polytopes and vector bundles, and computes their Euler characteristics to understand their complexity.

Contribution

It provides a canonical construction of logarithmic stable pairs spaces on toric surfaces using secondary polytopes and explicit vector bundle sections, improving upon previous non-canonical methods.

Findings

Explicit description of moduli spaces as zero sets of sections

Calculation of Euler-Satake characteristics in examples

Demonstration of the spaces' complexity through computations

Abstract

Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. We examine the theory in the case of toric surfaces, and recast the theory in this case using three ingredients: Gelfand, Kapranov and Zelevinsky secondary polytopes, Hilbert schemes of points, and tautological vector bundles. In particular logarithmic stable pairs spaces are expressed as the zero set of an explicit section of a vector bundle on a logarithmically smooth space, thus providing an explicit description of their virtual fundamental class. A key feature of our construction is that moduli spaces are completely canonical, unlike the existing construction, which is only well-defined up to logarithmic modifications. We calculate the Euler-Satake characteristics of our moduli spaces in a number of basic examples. These computations indicate the complexity of the spaces we construct.