Relative Resolutions of Moduli of Low Genus Stable Maps.

I will present certain 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) lead to partial resolutions in the general case. (Joint work with Jun Li.)