A theory of stacks with twisted fields and resolution of moduli of genus two stable maps

Abstract

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.

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