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Using stacks to impose tangency conditions on curves
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We define a Deligne-Mumford stack X_{D,r} which depends on a scheme X, an effective Cartier divisor D\subset X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into X_{D,r} provides compactifications of the locally closed substacks of \bar{M}_{g,n}(X,\beta) corresponding to relative stable maps.
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Stringy Chow rings and weighted blow ups
An explicit computation of the stringy Chow ring for quotients by diagonalizable groups and for weighted blowups, with finite generation criteria.
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