Normalization and birational maps
Enlarge a domain to its integral closure in the same function field and analyze finite normalization maps at cusps and nodes.
On this page
The question for today
What you’ll understand
- Define normality and normalization for an irreducible affine variety.
- State why finite-type algebras over fields have finite normalization.
- Compute cusp and node normalization maps and distinguish birationality, bijectivity, and isomorphism.
A coordinate ring can omit a rational function even when that function satisfies a monic equation over the ring. Normalization adds all such missing integral functions inside the existing fraction field. The resulting variety remains birational to the original but may separate branches or repair a cusp. Its finiteness requires more than a Noetherian domain.
Integral closure inside the same field
Let A be a finitely generated k-domain and K its fraction field. Its integral closure in K consists of elements satisfying monic polynomial equations with coefficients in A. A is normal if it already contains them all. The normalization of the irreducible affine variety with coordinate ring A has this integral closure as its coordinate ring. Because the closure is taken inside K, the fraction field stays the same; this is different from adjoining an arbitrary algebraic field extension.
For a finitely generated algebra over a field, the integral closure of a domain in its fraction field is a finite A-module. This holds in every characteristic over any field, including for integral closure in finite extensions of K. Arbitrary Noetherian domains do not automatically have finite integral closure, so Noetherianity alone is not a sufficient replacement. Our geometric examples retain algebraic closedness and characteristic zero.
A finite birational map need not be an isomorphism
The inclusion A into its integral closure gives a finite birational normalization map in the opposite direction. It is surjective, but it can have several points over a singular point or just one point with a noninvertible local-ring map. A bijection on ordinary points therefore does not settle whether normalization changes the geometry. The cusp has a single normalization point over its tip, while a node has two, illustrating distinct ways in which a map with the same generic field can modify exceptional local data.
Normality does supply a strong converse: a finite birational morphism between integral varieties with normal target is an isomorphism. On affine coordinate rings both domains lie in the same fraction field, and every element of the source ring is integral over the target ring. Integral closedness forces equality. The target hypothesis is essential: a map from a normal source to a nonnormal target, such as cusp normalization, need not be an isomorphism. Finiteness also matters; an open immersion can be birational without being onto.
What normalization repairs and what it does not
For a reduced reducible affine set, there is no single fraction field. One instead takes integral closures of the component domains and their product, or equivalently works in the total quotient ring. This separates components as well as normalizing each one. Applying the irreducible-domain formula without this adjustment can incorrectly divide by zero divisors. A nonreduced scheme first loses nilpotents in the usual normalization construction, so normalization also must not be advertised as preserving every piece of infinitesimal information.
In dimension one, a normal Noetherian local domain is regular, so normalization repairs the local ring singularities of an integral curve. Over a perfect field this gives a smooth normalized curve; the perfect-field qualification is needed when translating regularity into smoothness. In higher dimensions normal varieties can still be singular, so normalization is not a general resolution of singularities. Finally, Noether normalization from the earlier lesson is a different operation: it finds a finite map to affine space, whereas integral-closure normalization enlarges the coordinate ring within its original function field.
Work it through
A finite bijection that fails to be an isomorphism
In characteristic zero, normalize the cusp A=k[t²,t³] and analyze the map ν:A¹→V(y²−x³), t↦(t²,t³).
The fraction field of A is k(t), since t=t³/t². The element t is integral over A through the monic equation T²−t²=0, whose constant coefficient belongs to A. Moreover k[t] is generated as an A-module by 1 and t, because every monomial of degree at least two already belongs to A.
The polynomial ring k[t] is integrally closed. Any element of k(t) integral over A is also integral over k[t], since its monic A-coefficient equation is a k[t]-coefficient equation. Hence it lies in k[t]; together with the integral inclusion this proves the entire normalization is k[t].
Away from the origin, t=y/x recovers the unique parameter. At the origin, t²=t³=0 forces t=0. Thus ν is a bijection on ordinary points, but its inverse is not regular at the tip because t is not in the cusp local ring. The normalization is finite and birational without being an isomorphism.
The cusp normalization is the affine line with coordinate ring k[t]. Its finite bijective map changes the singular local ring even though it adds no ordinary point over the tip.
The living notebook
Explore the model
Interactive lab · function
Two parameters over a node
Pause and Reset, then set a = 1 and x = 1. Treat the horizontal control as t and the quadratic readout as t². The node parameterization is (t²−1, t(t²−1)), so predict (0,0). Reset and set both a = 1 and x = −1: the same node point occurs from a different parameter. This is a direct evaluation of the two polynomial formulas using the quadratic readout; the instrument itself draws only y=t², not the node, its branches, or a normalization algorithm.
- y at chosen x
- 0.608
- Tangent slope · 2ax
- 1.35
- Signed area · ax³/3
- 0.182
This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forA finite birational normalization can have two distinct points over one singular target point.
Pause & reason
Check your understanding
Your turn
Try it yourself
For the nodal curve in characteristic zero, verify its normalization t↦(t²−1,t(t²−1)). Find the points above the node and justify finiteness and birationality algebraically.
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
On x≠0 use t=y/x. Check the monic relation t²=x+1 and use that k[t] is integrally closed.
Compare with the solution
Substitution gives y²=t²(t²−1)²=x²(x+1), so the map lands on N. Its coordinate ring is the domain k[t²−1,t(t²−1)]; irreducibility also follows because x+1 is not a square in k(x). On x≠0, t=y/x, so its fraction field is k(t). The parameter satisfies T²−(x+1)=0, hence k[t] is finite over the curve ring, generated by 1,t. Being integrally closed in the same field, k[t] is its full integral closure by the same argument as for the cusp. At the node x=y=0, t²−1=0 gives the two distinct parameters t=1 and t=−1. The normalization separates these branches; it is not a bijection on all points.
Keep the idea
Take it with you
- Normalization adds integral elements inside the same function field, with finiteness ensured by the affine finite-type hypothesis.
- Finite birational maps can change singular local rings despite being bijective on points.
- A normal target forces a finite birational map to be an isomorphism; higher-dimensional normality still allows singularities.
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