Genus Two Stable Maps, Local Equations and Modular Resolutions

Abstract

We provide a geometric construction of a sequence of modular blowups of the Artin stack parameterizing pre-stable pairs consisting of a genus-two nodal curve and a smooth divisor. The resulting stack locally diagonalizes the tautological derived objects associated with the moduli of stable maps from genus-two curves to projective space. As a consequence, the singularities of the main component of the moduli space of stable maps are resolved, and the entire space admits only normal crossing singularities. Our approach is expected to generalize to higher genera.

111 pages, 21 figures; the relative Picard stack is replaced with the stack of nodal curves and simple divisors; the local equations of the genus two stable map moduli are summarized in Proposition 2.19; the technical proofs of the desingularization (Section 4) are reorganized