Negative contacts in genus one: a comparison of punctured and root stack Gromov-Witten theories
Abstract
Let D be a smooth divisor in a smooth projective complex variety X. For connected curves of arithmetic genus one with prescribed signed contact orders, we prove that the refined punctured cycle of Battistella--Nabijou--Ranganathan and the negative-contact cycle of Fan--Wu--You agree after pushforward to the common moduli space of stable maps with divisor evaluations. Thus the pushed-forward refined punctured cycle is the constant coefficient of the pushed-forward root-stack virtual class, normalized by one power of the root order for each negative contact. The key step is a comparison for the universal target. After restricting to finite-type open substacks determined by the fixed pair (X,D) and numerical data Γ, we prove that the positive BNR space maps finitely and with generic degree one onto Crumplin's main component. Using Crumplin's genus-one component description and degree formulas, we identify this component's fundamental cycle with the constant coefficient of the universal orbifold virtual class under comparison of root orders. Refined zero-section pullback recovers the negative contacts, and compatible virtual pullbacks and root-forgetting pushforwards transfer the resulting identity to (X,D).
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BibTeX
@misc{wang2026negative,
title = {{Negative contacts in genus one: a comparison of punctured and root stack Gromov-Witten theories}},
author = {Yu Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08124},
eprint = {2609.08124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08124},
}
RIS
TY - UNPB
TI - Negative contacts in genus one: a comparison of punctured and root stack Gromov-Witten theories
AU - Yu Wang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08124
UR - https://arxiv.org/abs/2609.08124
AB - Let D be a smooth divisor in a smooth projective complex variety X. For connected curves of arithmetic genus one with prescribed signed contact orders, we prove that the refined punctured cycle of Battistella--Nabijou--Ranganathan and the negative-contact cycle of Fan--Wu--You agree after pushforward to the common moduli space of stable maps with divisor evaluations. Thus the pushed-forward refined punctured cycle is the constant coefficient of the pushed-forward root-stack virtual class, normalized by one power of the root order for each negative contact. The key step is a comparison for the universal target. After restricting to finite-type open substacks determined by the fixed pair (X,D) and numerical data Γ, we prove that the positive BNR space maps finitely and with generic degree one onto Crumplin's main component. Using Crumplin's genus-one component description and degree formulas, we identify this component's fundamental cycle with the constant coefficient of the universal orbifold virtual class under comparison of root orders. Refined zero-section pullback recovers the negative contacts, and compatible virtual pullbacks and root-forgetting pushforwards transfer the resulting identity to (X,D).
ER -