We define a Deligne-Mumford stack XD,r which depends on a scheme X, an effective Cartier divisor D ⊂ X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M̄g,n(X,β) corresponding to relative stable maps.

We adapt ideas from Ekedahl [Eke84] to prove a Serre-type duality for Witt-divisorial sheaves of $\mathbb Q$-Cartier divisors on smooth projective variety over perfect field finite characteristic. also explain its relationship Tanaka's vanishing theorems [Tan20].