Negative contacts in genus one: a comparison of punctured and root stack Gromov-Witten theories

RESUMO

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 $\Gamma$, 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 $(X,D)$.