Blowing Up Acyclic Graphs and Geometrical Configurations

Blowing up is a useful technique in algebraic and analytic geometry. In particular, it is the main tool for proving resolution of singularities. Hironaka [2] proved in 1964 that every algebraic variety over a field of characteristic zero admits a resolution of singularities which is obtained by successive blowing ups of certain regular centers. Moreover, he proves the stronger version of embedded resolution of singularities, i.e., for every (singular) subvariety X of a smooth variety Z there exists a sequence of birational morphisms $${Z_N} \to {Z_{N - 1}} \to \cdots \to {Z_1} \to {Z_0} = Z,$$ (1.1) such that π i is the blowing up of Z i−1 at a regular center C i which is transversal to the exceptional divisor E i−1 of π i−1 ο⋯ο π1, and such that the strict transform X N of X at Z N is smooth and transversal (normal crossing) to the exceptional divisor E N .