Logarithmic Pandharipande–Thomas spaces and the secondary polytope

Kennedy–Hunt, Patrick

doi:10.1090/tran/9026

Logarithmic Pandharipande–Thomas spaces and the secondary polytope
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by Patrick Kennedy–Hunt;

Trans. Amer. Math. Soc. 378 (2025), 1-44

DOI: https://doi.org/10.1090/tran/9026

Published electronically: October 31, 2024

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Abstract:

Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. In the case of toric surfaces we recast this theory 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.

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