A theory of stacks with twisted fields and resolution of moduli of genus two stable maps
Yi Hu, Jingchen Niu
备注：
89 pages, 18 figures; the relative Picard stack is replaced with the stack of nodal curves and simple divisors; rooted trees are considered as posets instead of graphs; new material based on leaf-to-root sequences is incorporated; a new resolution of genus one stable map moduli is provided
发表信息：
Asian Journal of Mathematics, Vol. 29, Issue 3 (2025), pp. 325-422
我们构造了一个光滑代数叠，其参数化对象为由亏格2结点曲线、远离结点的简单有效除子以及扭场组成的组。该叠给出了到射影空间的亏格2稳定映射模空间的一个奇点消解。构造过程系统应用了带扭场的叠（stacks with twisted fields，简称STF）理论，该理论的雏形见于arXiv:1201.2427与arXiv:1906.10527，本文对其进行了完整发展。作为STF理论的副产品，我们还得到了到射影空间的亏格1稳定映射模空间的一种新的奇点消解，它与Vakil-Zinger及Hu-Li所用顺序相反的吹胀同构。本文的结果是高亏格稳定映射模空间消解研究计划的第二步。
英文摘要：
We construct a smooth algebraic stack of tuples consisting of genus two nodal curves, simple effective divisors away from the nodes, and twisted fields. It provides a desingularization of the moduli of genus two stable maps to projective spaces. The construction is based on systematic application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1201.2427 and arXiv:1906.10527 and is fully developed in this article. As a byproduct of the STF theory, we also obtain a novel desingularization of the moduli of genus one stable maps to projective spaces, which is isomorphic to the blowup that reverses the order used by Vakil-Zinger and Hu-Li. The results of this article are the second step of a program toward the resolutions of the moduli of stable maps of higher genera.
We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.
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