Genus Two Stable Maps, Local Equations and Modular Resolutions
  We geometrically describe a canonical sequence of modular blowups of the relative Picard stack of the Artin stack of pre-stable genus two curves. The final blowup stack locally diagonalizes certain tautological derived objects. This implies a resolution of the primary component of the moduli space of genus two stable maps to projective space and meanwhile makes the whole moduli space admit only normal crossing singularities. Our approach should extend to higher genera. 
Yi Hu, Jun Li, and Jingchen Niu
math.AG