Relative Resolutions of the Moduli of Low Genus Stable Maps

I will present canonical resolutions of the moduli spaces of genus g stable maps to projective spaces for g = 1 and 2. The resolutions are performed relative to the smooth Artin stack of weighted nodal curves in the case of genus 1 and relative to the smooth Artin stack of pairs of nodal curve and line bundle
in the case of genus 2. Some of the results also generalize to high genus cases, thus (hopefully) providing useful partial resolutions in the general case.