What is a blowup?

A smooth curve has a regular local coordinate: near each point, one number describes position along it, with a nonzero tangent. A singular point is where this local description fails. Crossing branches share a position; tangent branches also share a direction; a cusp has just one tangent but still has a singular local geometry.

A blowup changes the space in which we describe the curve. It keeps a direction coordinate that ordinary position forgets. We will calculate one, inspect what it fixes, and then ask why a finite sequence can resolve every variety in characteristic zero.

1. What makes a point singular?

A plane algebraic curve is the zero set of a polynomial, such as $y^2-x^3=0$. An algebraic variety generalizes this to simultaneous polynomial equations in any number of coordinates. Smoothness asks whether the resulting space has regular local coordinates of its expected dimension.

For a plane curve with a reduced defining equation $f(x,y)=0$, the test is concrete. If $\nabla f\ne0$, the implicit function theorem gives a smooth curve and a tangent line perpendicular to $\nabla f$. At the three singular origins below, $\nabla f=0$. The tangent information no longer gives a regular local coordinate. We use reduced equations because artificially squaring an equation would make its gradient vanish without changing its zero set.

Three prototypical singularities form a natural ladder of difficulty:

• Crossing: $f = y^2 - x^2 = 0$ (the lines $y = \pm x$). Two smooth branches crossing with distinct tangents.
• Touching: $f = y^2 - x^4 = 0$ (the parabolas $y = \pm x^2$). Two smooth branches sharing the same tangent: order of contact 2.
• Cusp: $f = y^2 - x^3 = 0$ (the curve $y = \pm x^{3/2}$). One algebraic branch, parametrized by $(x,y)=(\tau^2,\tau^3)$; its velocity vanishes at $\tau=0$. Having a unique tangent does not make the point smooth.

Figure 1. The black curves are the zero sets $f=0$. Move the probe toward the red-ringed origin: violet shows the tangent at the probe and orange the gradient. Turn on the normal field for the surrounding level-set geometry. At the singular point the gradient is zero; a single regular tangent description fails.

2. A physical analogy: the billiard corner

Consider three balls on a line with gaps $g_1 = x_2 - x_1 - 2r \ge 0$ and $g_2 = x_3 - x_2 - 2r \ge 0$. In mass-weighted coordinates, their elastic motion corresponds to a single point bouncing inside a planar wedge bounded by the walls $g_1 = 0$ and $g_2 = 0$. Rescaling positions by $q_i=\sqrt{m_i}x_i$ makes kinetic energy Euclidean, so impacts become ordinary mirror reflections. Because the middle ball belongs to both gaps, the two contact walls meet at an acute angle $\beta$; Part III derives it. When the point strikes one wall, the reflection is unique. But at the corner where $g_1 = g_2 = 0$, all three balls collide simultaneously.

The usual pairwise impact rule does not prescribe which contact to process first at the corner. Perturb the gaps slightly and follow the pairwise elastic collisions: different ratios can select different limiting outcomes. The colored arc records these limiting approaches. This is a real directional blowup of a corner, a physical analogy for the algebraic operation below. The boundary $g_1g_2=0$ consists of two crossing lines; the admissible configuration region is the wedge between them.

Figure 2. The three-ball system as a point in a planar wedge. Each bounce is a pairwise collision; the corner is all three balls touching. Around the corner, the blowup replaces the singular vertex with an arc of incoming directions, colored by the resulting sequence of bounces.

3. Blowing up, and how it resolves each kind

3.1 The idea: record the direction

Start with two straight paths, $y=x$ and $y=-x$. At $x=0$, both arrive at the same point $(0,0)$. The point $(0,0)$ by itself cannot tell us which path arrived. Keep one more piece of information: the slope of the line from the origin to the approaching point.

Our new point is a record: “old position, together with direction.” For $x\ne0$, the record is forced: $t=y/x$. As $x$ tends to zero along $y=x$, the record tends to “origin, slope $+1$”; along $y=-x$, it tends to “origin, slope $-1$”. These are two different points in the new space, although both project to the same old origin.

The exceptional line is the collection of all these records above the origin. Its points are “origin, slope $t$”, one for each finite $t$. In coordinates $(x,t)$ they form the vertical line $x=0$. It is a line in the new coordinate picture. We have not drawn an extra line through the curve in the old plane.

It is called a divisor because it has one fewer dimension than the new surrounding space: the surface chart has coordinates $(x,t)$, while $x=0$ leaves only $t$ free. On a blown-up three-dimensional space, the corresponding divisor is a surface. The word “line” describes this particular chart, not a rule that exceptional divisors must be straight.

Figure 3. Press Bring the points to the origin. Orange and blue follow the crossing branches; the dashed probe follows a direction you choose. At left they arrive at one old point. At right they retain their separate slopes on the violet line. Then move the direction slider: there are more points on the exceptional line than the two used by this curve.

Why this really changes the space

The projection, or blowdown map, forgets the direction: $\pi(x,t)=(x,xt)$. Away from $x=0$ it has the inverse $(x,y)\mapsto(x,y/x)$. At $x=0$ it has no single-valued inverse: all the different records $(0,t)$ map to $(0,0)$. Thus this is more than renaming coordinates. The new space separates limiting points that the old space identified.

For a compact definition, let $[\ell]$ mean an unoriented line through the origin. The blowup of the plane at the origin is the space of all compatible position–direction pairs:

with canonical projection $\pi(p, [\ell]) = p$. Away from the origin, $\pi$ is a diffeomorphism. Over the origin, the fiber $\pi^{-1}(0) = \{0\} \times \mathbb{RP}^1$ is the exceptional divisor $E \cong \mathbb{RP}^1$.

Now lift the curve. Keep the records of points on the curve away from the origin, and include their limits. This is the strict transform. For our crossing, $y=\pm x$ becomes $t=\pm1$: two disjoint horizontal lines in the chart. The entire violet line belongs to the surrounding blown-up surface; only its two intersection points belong to this strict transform.

When a curve $C = \{f = 0\}$ is pulled back under $\pi$, the total transform $\pi^* C$ contains the entire exceptional curve $E$. Factoring out the maximal power of $E$ ($x^m$, where $m$ is the multiplicity) leaves the strict transform:

The strict transform keeps the curve itself; the total transform also records the exceptional factor and its multiplicity. Distinct slopes separate at this step. Branches sharing a slope can still meet above the origin.

Figure 4. 3D visualization of the lift. Grey is the original curve; the colored curve is lifted by recorded direction. Crossing lines separate immediately. Touching curves cross at an angle on step 1. The cusp becomes smooth, but hugs the exceptional divisor $E$ (the vertical violet line).

3.2 By hand: reading off Taylor coefficients

We cover $\operatorname{Bl}_0(\mathbb{R}^2)$ with two affine charts: the $x$-chart $U_x$ with map $(x, t) \mapsto (x, xt)$ (slope $t = y/x$, $E = \{x = 0\}$) and the $y$-chart $U_y$ with map $(s, y) \mapsto (ys, y)$ ($s = x/y$, $E = \{y = 0\}$). In the $x$-chart, a branch $y(x) = a_1 x + a_2 x^2 + \cdots$ has slope:

At $x = 0$, the branch intersects $E$ at $t = a_1$, its initial slope. If the branch still passes through $(0,a_1)$, translate $t$ by $a_1$ and blow up there: the next slope reads off $a_2$. This Taylor-series picture is useful for smooth branches; a cusp requires changing charts as well. The three rungs resolve as follows:

• Crossing ($y^2 - x^2 = 0$): Setting $y = xt$ gives $x^2(t^2 - 1) = 0$. The strict transform $t^2 - 1 = 0 \implies t = \pm 1$ intersects $E$ transversely at two distinct points. Resolved in 1 step.
• Touching ($y^2 - x^4 = 0$): Setting $y = xt$ gives $x^2(t^2 - x^2) = 0$. On $E$ ($x = 0$) both branches meet at $t = 0$, but their order of contact drops from 2 to 1: $t^2 - x^2 = (t - x)(t + x) = 0$ is now a transverse crossing. A second blowup $t = xw$ gives $w^2 - 1 = 0$, fully separating them. Resolved in 2 steps.
• Cusp ($y^2 - x^3 = 0$): Setting $y = xt$ gives $x^2(t^2 - x) = 0 \implies x = t^2$. The curve is smooth in 1 step! But its tangent $x = 0$ is $E$ itself (a tangency). Two further blowups make the strict curve and exceptional boundary meet with simple normal crossings (SNC), where the components are coordinate hyperplanes locally. Pairwise transversality alone does not permit a triple point on a surface. Resolved in 3 steps.

Figure 5. The ladder step-by-step. Each panel is an affine chart with slope $t$ vertically. Orange and blue trace the branches; violet lines are exceptional divisors $E$. Green dots mark transverse crossings; red rings mark points requiring further resolution.

3.3 Run the resolution procedure, with the boundary kept in view

For the cusp, the operational question is concrete: where does the curve, together with the exceptional boundary, still fail the required normal-crossings condition? Blow up that point, substitute in both charts, factor the total transform, and inspect again. A smooth strict transform is not enough: tangencies and three components through one surface point still need work.

Figure 6. Follow the cusp through three point blowups. The right-hand ledger records the actual equation, the current defect, and the next center. Try the harder cusp to see multiplicity stay at 2 for a step. This is a worked plane-curve trace, not a general-purpose implementation or a graph of the full canonical invariant.

Why this becomes difficult in higher dimensions

The obstacle is choosing centers with guaranteed progress. Blowing up an arbitrary point can leave the defect untouched. Even blowing up the worst multiplicity need not lower that multiplicity immediately: try $y^2=x^5$ in the figure above. New exceptional components also create tangencies and intersections that must be tracked.

In higher dimensions the worst set can itself be singular. A usable algorithm must find smooth centers within it, make the choices agree on overlapping charts, and retain enough information to prove termination. Modern constructive versions use an ordered invariant, lower-dimensional coefficient ideals, and exceptional-boundary data. Their termination proofs control a well-founded complexity; the simplest visible measure need not drop at every blowup.

The basic cycle is: compute the current defect, choose a permissible smooth center, calculate every blowup chart, update both the residual equations and the boundary, and repeat until the required smoothness and normal-crossings tests pass. The cusp lets us see this cycle without pretending to implement the full general algorithm.

The Bierstone–Grigoriev–Milman–Włodarczyk paper^[4] organizes resolution through marked ideals $(X,I,E,\mu)$: a smooth ambient space, equations, an exceptional boundary, and an order threshold. Its support is $\{p:\operatorname{ord}_p I\ge\mu\}$. A permissible center is smooth, lies in this support, and meets the boundary normally. After blowing up, the controlled ideal is $I'=I(E_{\rm new})^{-\mu}\pi^*I$. It may differ from the strict-transform ideal.

For our cusp we can calculate the first recursive reduction by hand:

1. Find the highest order. For $f=y^2-x^3$, it is 2 at the origin. Use the companion marked ideal $(f,2)$ for this order-reduction task.
2. Reduce the dimension. Since $\partial f/\partial y=2y$, the line $H=\{y=0\}$ is a hypersurface of maximal contact here. Expanding $f$ in $y$ leaves the coefficient $-x^3$ with threshold 2, represented on $H$ by $(x^3,2)$. Its support is $x=0$, so the center in the plane is $(0,0)$.
3. Transform the marked ideal. Substitute $y=xt$: $f=x^2(t^2-x)$. Divide by $x^2$ because the mark is 2. The transformed marked ideal $(t^2-x,2)$ has empty support: its generator has order 1 everywhere. That recursive order-reduction task is finished.
4. Return to the embedded problem. The strict curve $x=t^2$ is tangent to $E_1=\{x=0\}$. Blow up their common point with $x=ts$. The total transform becomes $t^3s^2(t-s)$. Three smooth components meet there, so blow up once more, $s=tw$, giving $t^6w^2(1-w)$. Now $w=1$ meets $E_3=\{t=0\}$ transversely, away from $E_1=\{w=0\}$.

In the last complementary chart $t=sv$, the total transform is $s^6v^3(v-1)$: the older divisor $v=0$ and the curve $v=1$ meet the new divisor $s=0$ at different points too. Both charts must be checked. The general procedure also separates exceptional monomial factors from the residual ideal, forms companion ideals, recurses on coefficient ideals, and handles boundary intersections. The scalar $3/2$ above is coefficient data in this example, not the complete invariant for all these tasks. See §3 of the 2012 paper^[4].

3.4 A shortcut: weighted blowups and Newton polygons

Standard blowups assign equal weight $(1, 1)$ to $x$ and $y$. But the cusp $y^2 = x^3$ has anisotropic scaling $y \sim x^{3/2}$. A weighted blowup assigns integer weights $(a, b)$ via $x = u^a, y = u^b v$. For $(a, b) = (2, 3)$, both terms match at $u^6$:

For $(2,3)$, the strict equation on this chart cover is $v^2-1=0$, transverse to $u=0$. The map $x=u^2$, $y=u^3v$ has the identification $(u,v)\sim(-u,-v)$: the two displayed lifts do not mean that the cusp has two algebraic branches. A weighted blowup includes these quotient identifications and its other charts. This substitution alone is not an ordinary birational chart of a smooth surface.

The balancing weights come from the compact edge joining $(3,0)$ and $(0,2)$ of the Newton polygon: $3a=2b$. This explains the shortcut in this example; general resolution requires more than choosing weights from one picture.

Figure 7. A weighted chart cover. For $(a,b)=(2,3)$, both monomials have weight 6, and the strict equation becomes $v=\pm1$. Read these as two lifts on the cover, with the identification described above. Try a = 1, b = 1, then a = 2, b = 3: each dot is a monomial, and its label is the power of u it acquires. The weight line touches the least-power terms; only those terms survive at u = 0 after factoring.

The theorem, its constructive proofs, and the Lean formalization

Hironaka proved the characteristic-zero theorem in 1964. Later proofs made center selection more explicit and compatible across charts. The Resolution team’s October 2026 report^[14] describes a Lean library for algebraic and analytic resolution, drawing on Hironaka, Kollár, Włodarczyk, Bravo–Villamayor, and Bierstone–Milman. The library checks the general theorem and its proof; the figures here let us work through particular examples.

For integration, the useful consequence is monomialization: after resolution, a complicated vanishing function becomes a product of coordinate powers times a nonvanishing factor. The Jacobian has coordinate powers too. Part IV uses the very cusp charts above to compute a statistical exponent.

4. What tangent directions can resolve

Can the geometry itself prescribe a modification, without a separate center-selection invariant? The Nash blowup attaches the tangent $d$-plane at each smooth point, then takes the closure of this graph. On a plane curve, an affine chart records the tangent slope $dy/dx$.

On the cusp $(x,y)=(\tau^2,\tau^3)$, the slope is $3\tau/2$. The lift $(\tau^2,\tau^3,3\tau/2)$ is smooth: the extra coordinate recovers the missing regular parameter. One Nash step resolves this curve. This does not include the ambient exceptional-boundary condition imposed in the three-step embedded resolution.

The appealing rule has a limit. In characteristic zero, Nash iteration resolves curves, and normalized Nash iteration resolves surfaces. In higher dimensions, a singular open chart can return isomorphic to the input. The process can then repeat inside that chart indefinitely. Part II computes the three-dimensional obstruction and exhibits a different resolution that terminates.

Figure 8. Left: successive point blowups resolve the cusp and its exceptional crossings. Right: Nash resolves the cusp in one step; the lower diagram schematically illustrates a singular affine chart recurring in the four-dimensional example.

5. What survives when the data are noisy?

A scanned surface or point cloud comes without an exact defining ideal. Directions must be estimated. Petrov and colleagues^[21] use tangent lifts in geometry processing; the experiment below compares estimated tangents with directions from a fitted center.

Figure 9. Start with low noise and two crossing strokes. Rotate the lift: different directions separate the strokes above their shared position. Increase noise or reduce the crossing angle; the direction clusters become harder to distinguish. “Nash” estimates local tangents by PCA; “Center-based” fits the crossing and records directions from it. If center fitting fails, the demo reports a fallback to the sample centroid. Black dashed histogram lines mark the true synthetic directions; red rings flag its ambiguity test. These are numerical illustrations, not executions of an exact algebraic resolution algorithm.

What to carry forward: a blowup keeps the old geometry away from its center and records extra information above it. Resolution requires a smooth final space, compatible charts, and a stopping criterion. Next, a cone makes the exceptional fiber visible before we test the Nash rule in higher dimensions.