Prescribing positive curvature with conical singularities on ${\mathbb{S}}^2$

For conformal metrics with conical singularities and positive curvature on ${\mathbb{S}}^2$, we prove a convergence theorem and apply it to obtain a criterion for nonexistence in an open region of the prescribing data. The core of our study is a fine analysis of the bubble trees and an area identity in the convergence process.