Resolving singular points by replacing them with projective lines — watch singularities untangle
A singularity of an algebraic variety is a point where the variety fails to be smooth — where the tangent space has the wrong dimension, or equivalently, where all partial derivatives of the defining equations vanish simultaneously. Nodes, cusps, and self-intersections are all singularities, and they are ubiquitous: the intersection of two curves is often singular, and projections of smooth varieties typically introduce singularities.
The blow-up is the fundamental tool for resolving singularities. It replaces a singular point with a copy of projective space — the exceptional divisor — whose points parameterize the directions approaching the singularity. The strict transform of the original variety in the blown-up space may be smoother, and iterating this process always terminates in a smooth variety (in characteristic zero), by Hironaka's celebrated 1964 theorem.
Watch the node of y² = x²(x+1) resolve as the blow-up parameter increases. The two branches, which cross at the origin, separate as the exceptional divisor E ≅ P¹ (orange circle) inflates from the singular point. At t = 1 the curve is fully resolved — smooth, with no self-intersection.
The blow-up of k2 at the origin replaces (0, 0) with a copy of P1, the projective line. Formally, the blow-up is the subvariety Bl0(k2) ⊂ k2 × P1 defined by xv = yu, where [u : v] are homogeneous coordinates on P1. Away from the origin, the projection π : Bl0(k2) → k2is an isomorphism. Over the origin, π-1(0, 0) = P1 — the exceptional divisor E.
Each point of E corresponds to a direction (slope) of approach to the origin. If two branches of a curve pass through the origin with different tangent directions, their strict transforms in the blow-up are separated — they pass through different points of E. This is why blow-ups resolve singularities: they "pull apart" branches that were tangled together at the singular point.
A resolution of singularities of a variety X is a proper birational morphism π : X̃ → X where X̃ is smooth. In dimension 1 (curves), resolution is straightforward — normalize the curve. In dimension 2 (surfaces), repeated blow-ups at singular points always work. In arbitrary dimension over fields of characteristic zero, Hironaka (1964) proved that resolution is always possible, earning him the Fields Medal.
The resolution problem in positive characteristic remains one of the major open problems in algebraic geometry. For threefolds in characteristic p > 5, resolution was established by Abhyankar and later refined by Cossart and Piltant. The general case — arbitrary dimension in arbitrary characteristic — is still unsolved and is the subject of active research.