01 · The idea
In everyday language
A polynomial equation may define a space with a crossing, cusp or more complicated singularity. A blow-up replaces a chosen centre with extra geometric information that separates directions previously merged there. One modification need not finish the job: the challenge is choosing centres and proving that a controlled sequence eventually produces a nonsingular model. Hironaka established this in characteristic zero and arbitrary dimension. Away from the singular region, the construction preserves the intended birational relationship, providing a smooth space mapped back to the original rather than simply erasing the troublesome points.
Why it matters
Resolution of singularities became a basic reduction tool in algebraic and complex geometry, allowing smooth methods to address spaces originally defined with singularities.
