Blowing Up Acyclic Graphs and Geometrical Configurations

Abstract

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 .

Supported by DIGICYT PB91 0210 C0201