Using Stacks to Impose Tangency Conditions on Curves

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 also state an enumerative result counting rational plane curves with tangency conditions to a smooth cubic which generalizes Kontsevich’s recursion and is proved using the WDVV equations for Gromov-Witten invariants of a stack.