The Stack of Similarity Classes of Triangles

The paper proves that the map ϕ: D → Bl[H](P(X)) given by ϕ([a,b,c];(α,β,γ))=([a,b,c];[0,−γ,β]) is a bijection with inverse ψ([a,b,c];[ξa,ξb,ξc])=([a,b,c];(ξb−ξc,ξc−ξa,ξa−ξb)); it then identifies Bl[H](P(X)) with Dyck’s surface (#3P2(R)) via the blow-up of P1(C) at three real points. These are exactly the constructions and identities used in the candidate solution, including the “difference map” that kills the diagonal and lands in the 2-torus of angle triples summing to zero. The key lemmas and the theorem in the paper match the candidate’s steps one-for-one, including behavior on the exceptional divisors and the topological identification of the blow-up. Consequently, both arguments are correct and essentially the same. See the paper’s Lemma 5.1.2 and Definition 5.1.1 for the blow-up set description and angle coordinates and Theorem 5.1.3 for ϕ, ψ, and Dyck’s surface; the 2-torus angle relation is summarized in Lemma 7.2.1.