Characterization of contractions as regular weighted blowups

Determine whether every contraction f: X [?] Z between smooth separated DeligneMumford stacks, with smooth irreducible Cartier exceptional divisor E and a smooth closed irreducible DeligneMumford substack Y [?] Z satisfying (f^{-1}Y) [?] E set-theoretically and f: X \ E [?] Z \ Y an isomorphism, is a regular weighted blowup with reduced center Y.

Background

The paper formulates a question attributed to Dan Abramovich concerning the classification of contractions between smooth separated DeligneMumford stacks. The hypotheses require that the exceptional locus be a smooth irreducible Cartier divisor and that the contraction be an isomorphism away from this divisor and a corresponding smooth center in the target.

The paper proves the proposed characterization in two cases: representable contractions are ordinary blowups along smooth centers, and contractions between smooth DeligneMumford surfaces are regular weighted blowups. Thus, the general higher-dimensional nonrepresentable case posed in the question remains unresolved.

References

More precisely, this led Dan Abramovich to the following question.

Suppose $X$ and $Z$ are smooth separated Deligne--Mumford stacks, and there is a contraction $f : X \to Z$ where the exceptional $E$ is a smooth irreducible Cartier divisor. Moreover, suppose there is a smooth closed irreducible Deligne--Mumford substack $Y \subseteq Z$ such that $(f{-1}Y) \simeq E$ set-theoretically, and $f : X \setminus E \xrightarrow{\sim} Z \setminus Y.$ Then is $f$ a regular weighted blowup with reduced center $Y$?

— Characterizing Contractions and Weighted Blowdowns  (2609.04023 - Ghosh et al., 3 Sep 2026) in Section 1, Introduction and Statement of Result, Question 1 (label conje)