Dan Abramovich’s question on contractions defined by differential or divisor data

Dan Abramovich’s question asks when contractions specified by differential or divisor data can be described as blowups. The available work treats substantial cases, but presents them as only a partial answer to the general question.

Known results

• A 2023 criterion constructs weighted blowups from suitable Cartier divisors on smooth tame Deligne–Mumford stacks.
• The same work identifies $\mathcal{M}_{1,n}$, for $n \ge 2$, as a weighted blowup of $\mathcal{M}^{\mathrm{ps}}_{1,n}$ with weights $4$ and $6$.
• Related contractions in Smyth’s compactifications are treated for $n=3,4,5,6$.

September 2026 partial answer

A newly reported work proves ordinary blowup descriptions for schemes and representable Deligne–Mumford-stack morphisms, and weighted-blowup descriptions for smooth Deligne–Mumford surfaces. It applies these results to Hassett moduli stacks, but the abstract explicitly leaves the broader question unresolved.

Current status (as of September 2026): important scheme, stack, surface, and Hassett cases are claimed, while the general question remains open.