Hironaka: What Can One Blow-Up Chart Show?

The global question behind the local picture

Hironaka’s result concerns algebraic varieties over characteristic-zero fields in arbitrary dimension. The IMU memorial describes a proper modification that leaves the smooth part unchanged.

Kollár’s selected lecture distinguishes resolution from stronger normal-crossing requirements and explains why local constructions must fit together globally. His later exposition is not the original 1964 proof.

The exercise below works over the complex numbers. It tests selected equations in one chart, not the existence or compatibility of all the modifications required by a general theorem.

Sources for this section: [1] · [2]

Define the chart and the objects being compared

In one blow-up chart at the plane origin, write x = u and y = uv. The exceptional divisor in this chart is u = 0. Substitution gives a total transform; removing the common exceptional factor gives the strict transform.

For this exercise, use the plane-curve gradient test: a point on F(u, v) = 0 is smooth when the two first partial derivatives do not both vanish. Check only the origin of this chart. Other charts and intersections still matter to a global or embedded resolution.

Sources for this section: [2]

An original pair of curve calculations

Take curve A: y² = x³. After substitution, u²v² = u³, so the total transform is u²(v²−u) = 0 and the strict transform is v²−u = 0. Find its gradient at (u, v) = (0, 0).

Take curve B: y² = x⁷. The same substitution gives u²(v²−u⁵) = 0 and strict transform v²−u⁵ = 0. Calculate the gradient at the origin. Which strict transform is smooth there? Does smoothness alone establish a transverse intersection with the exceptional divisor?

Sources for this section: [2]

Reference answer: one improvement and two limits

For A the gradient is (−1, 2v), hence (−1, 0) at the origin: the strict transform is smooth there. For B it is (−5u⁴, 2v), hence (0, 0): a singularity remains in this chart.

A still meets the exceptional line u = 0 tangentially: writing u = v² gives tangent line u = 0 at the origin. Thus smoothness of that strict transform is not by itself a normal-crossing conclusion. B demonstrates that the same substitution need not finish even the local smoothing task.

Neither calculation checks the other chart, a complete sequence, global patching or the arbitrary-dimensional theorem. They are useful because they expose those separate obligations, not because they replace Hironaka’s proof.

Sources for this section: [2]