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Denominators of Hodge integrals and Minkowski numbers
Abstract
We prove that the least common denominator of all genus g Hodge integrals is the Minkowski number T2g or, equivalently, the Bhargava factorial (2g+1)!ℙ associated to the set of primes ℙ⊂ℕ. As a corollary, we prove a conjecture of Liu-Xu on the multiplicity of $3$ in the least common denominators of certain descendant integrals, and generalise their result on a conjecture of Itzykson-Zuber to Hodge integrals with λ classes. Namely, the order of the automorphism group of any pointed stable curve of genus g'≤ g divides the least common denominator of all genus g Hodge integrals. We also comment on the use of generative AI via IMProofBench, which played a fruitful part in the development of this (entirely human-written) paper.
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BibTeX
@misc{holmes2026denominators,
title = {{Denominators of Hodge integrals and Minkowski numbers}},
author = {Daniel Holmes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30274},
eprint = {2610.30274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30274},
}
RIS
TY - UNPB
TI - Denominators of Hodge integrals and Minkowski numbers
AU - Daniel Holmes
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30274
UR - https://arxiv.org/abs/2610.30274
AB - We prove that the least common denominator of all genus g Hodge integrals is the Minkowski number T_2g or, equivalently, the Bhargava factorial (2g+1)!_ℙ associated to the set of primes ℙ⊂ℕ. As a corollary, we prove a conjecture of Liu-Xu on the multiplicity of $3$ in the least common denominators of certain descendant integrals, and generalise their result on a conjecture of Itzykson-Zuber to Hodge integrals with λ classes. Namely, the order of the automorphism group of any pointed stable curve of genus g'≤ g divides the least common denominator of all genus g Hodge integrals. We also comment on the use of generative AI via IMProofBench, which played a fruitful part in the development of this (entirely human-written) paper.
ER -