Hironaka's Theorem states that every algebraic variety over a field of characteristic zero has a resolution of singularities. This means that for any given algebraic variety, it is possible to replace it with another variety that is smooth and behaves nicely, thus making the study of its geometric properties more manageable. This theorem is pivotal in understanding the structure of algebraic varieties and their singular points, connecting deeply with methods of blowing up and classifying algebraic surfaces.
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