Normalization and Resolution of Curve Singularities

Table of Contents

• 1. Introduction: Two Paths to Desingularization
• 2. Normal Varieties and Integral Closure
  • 2.1 Normal Domains
  • 2.2 Normal Varieties
  • 2.3 Normal Implies Smooth in Dimension One
  • 2.4 Serre’s Criterion for Normality
  • 2.5 Motivating Example: The Cuspidal Cubic
• 3. The Normalization Construction
  • 3.1 Integral Closure and the Normalization Map
  • 3.2 Noether Normalization
  • 3.3 Worked Example: Normalizing the Cuspidal Cubic
  • 3.4 Worked Example: Normalizing the Nodal Cubic
• 4. Blowing Up
  • 4.1 Definition via the Incidence Correspondence
  • 4.2 Affine Charts for the Blow-Up of the Plane
  • 4.3 The Strict Transform
  • 4.4 Multiplicity and the Effect of One Blow-Up
• 5. Resolving Curve Singularities by Blow-Up
  • 5.1 Resolution of a Node
  • 5.2 Resolution of a Cusp
  • 5.3 Termination of the Process
• 6. The Smooth Projective Model
  • 6.1 Existence
  • 6.2 Uniqueness
  • 6.3 The Function Field Perspective
• References

Throughout, \(k\) denotes an algebraically closed field and all varieties are over \(k\). Coordinate rings are written \(A(X) = k[x_1,\ldots,x_n]/I(X)\) for affine \(X \subset \mathbb{A}^n\). For background on local rings, singularity, tangent cones, and multiplicity, see Local Properties of Varieties; for projective varieties and rational maps, see Classical Algebraic Geometry.

1. Introduction: Two Paths to Desingularization

🗺️ A singular point of a variety \(X\) is a point where the local ring \(\mathcal{O}_{X,P}\) fails to be regular — concretely, where the Jacobian matrix of the defining equations drops rank. Singular curves are the simplest non-trivial arena for desingularization: one wishes to find a smooth variety \(\tilde{X}\) together with a birational morphism \(\nu : \tilde{X} \to X\) that is an isomorphism away from the singular locus.

Two strategies produce this smooth model:

1. Algebraic (normalization). Replace the coordinate ring \(A(X)\) by its integral closure in the fraction field \(k(X)\). This is a purely ring-theoretic operation. In dimension one it already yields a smooth variety, because for curves normality is equivalent to smoothness.

2. Geometric (blowing up). Replace a singular point \(P\) by the projective space of all tangent directions at \(P\). This is a local operation that decreases multiplicity at each step; since multiplicity is a non-negative integer, the process terminates in finitely many steps with a smooth model.

In dimension one these two approaches converge: normalization gives the smooth projective model directly (after projectivizing), and repeated blowing up reaches the same object. The two methods differ in higher dimensions, where normalization need not resolve all singularities.

2. Normal Varieties and Integral Closure

2.1 Normal Domains

📐 Recall that an element \(\alpha\) in an extension ring \(B \supset A\) is integral over \(A\) if it satisfies a monic polynomial equation

\[\alpha^n + a_{n-1}\alpha^{n-1} + \cdots + a_0 = 0, \quad a_i \in A.\]

The integral closure of \(A\) in \(B\) is the set \(\bar{A}^B = \{ \beta \in B : \beta \text{ integral over } A \}\); it is itself a ring (closed under addition and multiplication). When \(B = \mathrm{Frac}(A)\) we write simply \(\bar{A}\).

Definition (Normal Domain). An integral domain \(A\) is normal (or integrally closed) if \(A = \bar{A}\), i.e., every element of \(\mathrm{Frac}(A)\) that is integral over \(A\) already belongs to \(A\).

Proposition. Every unique factorization domain (UFD) is normal.

Proof sketch. Let \(\alpha = p/q \in \mathrm{Frac}(A)\) with \(\gcd(p,q) = 1\) in the UFD \(A\), and suppose \(\alpha^n + a_{n-1}\alpha^{n-1} + \cdots + a_0 = 0\). Multiplying through by \(q^n\) gives \(p^n = -q(a_{n-1}p^{n-1} + \cdots + a_0 q^{n-1})\), so \(q \mid p^n\). Since \(\gcd(p,q)=1\) in a UFD every prime dividing \(q\) divides \(p\), forcing \(q\) to be a unit. Hence \(\alpha \in A\). \(\square\)

Corollary. Polynomial rings \(k[x_1,\ldots,x_n]\) and their localizations are normal. In particular, smooth varieties have normal local rings.

2.2 Normal Varieties

Definition (Normal Variety). An irreducible variety \(X\) is normal at a point \(P\) if the local ring \(\mathcal{O}_{X,P}\) is an integrally closed domain. \(X\) is normal if it is normal at every point.

For affine \(X\) with coordinate ring \(A(X)\), normality of \(A(X)\) implies normality at every point (since localizations of integrally closed domains are integrally closed). The converse holds when \(A(X)\) is Noetherian.

Proposition (Normality and Codimension of Singular Locus). If \(X\) is a normal variety, then the singular locus \(X_{\mathrm{sing}}\) has codimension at least \(2\) in \(X\).

Proof sketch. At a codimension-1 point (i.e., a point of a prime divisor) the local ring has Krull dimension 1. A normal Noetherian local ring of dimension 1 is a DVR (see §5, DVRs and smooth curves), hence regular. So no codimension-1 point of a normal variety is singular. \(\square\)

Corollary. For a curve (dimension 1), normality implies smoothness everywhere: \(X_{\mathrm{sing}} = \emptyset\).

2.3 Normal Implies Smooth in Dimension One

🔑 This corollary is the central fact connecting the two approaches to desingularization for curves. Let us spell it out:

Theorem (Normal Curves are Smooth). Let \(X\) be an irreducible normal curve (variety of dimension 1). Then \(X\) is smooth at every point.

Proof. Every point of \(X\) has codimension \(\leq 1\) in \(X\). By the proposition above, codimension-1 points of a normal variety have regular local rings. But for a curve every point is codimension 1 (since \(\dim X = 1\)). Hence every local ring \(\mathcal{O}_{X,P}\) is regular, which is the definition of smoothness. \(\square\)

This is the key theorem: for curves, normal = smooth = regular. Normalization of a curve therefore produces the smooth model directly.

2.4 Serre’s Criterion for Normality

For reference in more general settings, we record Serre’s criterion without full proof.

Theorem (Serre’s Criterion, R1+S2). A Noetherian integral domain \(A\) is normal if and only if it satisfies: - (R1) \(A_\mathfrak{p}\) is a regular local ring for every prime \(\mathfrak{p}\) with \(\mathrm{ht}(\mathfrak{p}) \leq 1\). - (S2) \(\mathrm{depth}(A_\mathfrak{p}) \geq \min(2, \mathrm{ht}(\mathfrak{p}))\) for every prime \(\mathfrak{p}\).

Here (R1) is the regularity condition in codimension 1, and (S2) is a depth (Cohen–Macaulay type) condition. Condition (S2) ensures the absence of “embedded” singularities.

2.5 Motivating Example: The Cuspidal Cubic

⚠️ Consider the cuspidal cubic \(C = V(y^2 - x^3) \subset \mathbb{A}^2\). Its coordinate ring is

\[A(C) = k[x,y]/(y^2 - x^3) \cong k[t^2, t^3],\]

where the isomorphism sends \(x \mapsto t^2\), \(y \mapsto t^3\) (the parametrization \(t \mapsto (t^2, t^3)\)). The singular point of \(C\) is the origin \((0,0)\), detected by the Jacobian \((-3x^2, 2y)|_{(0,0)} = (0,0)\) vanishing.

Is \(A(C) = k[t^2, t^3]\) integrally closed in \(k(t)\)? No: the element \(t \in k(t)\) satisfies

\[t^2 - x = 0 \quad (x \in A(C)),\]

so \(t\) is integral over \(A(C)\) but \(t \notin A(C) = k[t^2, t^3]\). Hence \(C\) is not normal at the origin, consistent with the fact that the origin is singular.

3. The Normalization Construction

3.1 Integral Closure and the Normalization Map

📐 Definition (Normalization of an Affine Variety). Let \(X = \mathrm{Spec}\, A\) be an irreducible affine variety with \(A = A(X)\) and function field \(K = \mathrm{Frac}(A) = k(X)\). The normalization of \(X\) is the affine variety

\[\tilde{X} = \mathrm{Spec}\,\tilde{A},\]

where \(\tilde{A}\) is the integral closure of \(A\) in \(K\). The natural inclusion \(A \hookrightarrow \tilde{A}\) induces a morphism

\[\nu : \tilde{X} \longrightarrow X,\]

called the normalization map.

We now establish the key properties of \(\nu\).

Theorem (Properties of the Normalization Map). Let \(\nu : \tilde{X} \to X\) be the normalization of an irreducible affine variety \(X\) of dimension \(d\) over a perfect field \(k\). Then: 1. \(\tilde{A}\) is a finitely generated \(A\)-module, hence a finitely generated \(k\)-algebra. 2. \(\nu\) is a finite morphism: \(\mathcal{O}_{\tilde{X}}\) is a coherent \(\nu^{-1}\mathcal{O}_X\)-module. 3. \(\nu\) is birational: it induces an isomorphism \(k(\tilde{X}) \xrightarrow{\sim} k(X)\) on function fields. 4. \(\nu\) is an isomorphism over the smooth locus of \(X\): if \(U = X \setminus X_{\mathrm{sing}}\), then \(\nu^{-1}(U) \to U\) is an isomorphism. 5. \(\tilde{X}\) is normal (and, if \(\dim X = 1\), smooth).

Proof sketches. (1) Finite generation of \(\tilde{A}\) as an \(A\)-module follows from the Noether normalization theorem (§3.2) together with the fact that the extension \(K \supset k\) is finite separable when \(k\) is perfect — the key input is that a finite integral extension of Noetherian rings is again Noetherian. (2) is the algebraic formulation of (1). (3) holds because \(A\) and \(\tilde{A}\) share the same fraction field \(K\). (4) At a smooth point \(P \in X\), \(\mathcal{O}_{X,P}\) is already a regular local ring and thus a UFD, hence integrally closed; the fiber \(\nu^{-1}(P)\) is a single point. (5) The integral closure \(\tilde{A}\) is integrally closed by construction. \(\square\)

Finite + birational is the key. The map \(\nu\) “separates branches” without changing the rational function field.

3.2 Noether Normalization

🔑 The following theorem underlies the finiteness claim in Theorem 3.1.

Theorem (Noether Normalization). Let \(X\) be an irreducible affine variety of dimension \(d\) over \(k\). Then there exist algebraically independent elements \(y_1,\ldots,y_d \in A(X)\) such that \(A(X)\) is a finitely generated module over the polynomial subring \(k[y_1,\ldots,y_d]\). Geometrically, there is a finite surjective morphism

\[\varphi : X \longrightarrow \mathbb{A}^d.\]

Proof sketch. We induct on the number of generators of \(A(X) = k[x_1,\ldots,x_m]/I(X)\). If \(m = d\) (i.e., \(X\) is itself \(\mathbb{A}^d\)), done. Otherwise, there is a nonzero polynomial \(f(x_1,\ldots,x_m) \in I(X)\). The key trick (due to Nagata) is to make a shearing change of variables: set \(y_i = x_i - x_m^{e_i}\) for suitably chosen integers \(e_i\) (e.g., \(e_i = r^{i-1}\) for large \(r\)). After this substitution, \(f\) becomes a monic polynomial in \(x_m\) over \(k[y_1,\ldots,y_{m-1}]\), so \(x_m\) is integral over \(k[y_1,\ldots,y_{m-1}]\). Induction reduces to \(m-1\) generators. \(\square\)

Corollary. The integral closure \(\tilde{A}\) of \(A(X)\) in \(k(X)\) is a finitely generated \(A(X)\)-module (hence a finitely generated \(k\)-algebra), because \(\tilde{A}\) lies between two finitely generated modules over a Noetherian ring.

3.3 Worked Example: Normalizing the Cuspidal Cubic

🔑 Let \(C = V(y^2 - x^3) \subset \mathbb{A}^2\) with coordinate ring \(A(C) = k[t^2, t^3]\) (via \(x \mapsto t^2\), \(y \mapsto t^3\)). We find \(\tilde{A}(C)\).

Step 1: Identify the fraction field. Since \(A(C) = k[t^2, t^3] \subset k[t]\) and \(t = t^3/t^2 \in \mathrm{Frac}(A(C))\), we have \(\mathrm{Frac}(A(C)) = k(t)\).

Step 2: Show \(k[t]\) is the integral closure. As noted in §2.5, \(t\) satisfies \(t^2 - x = 0\) over \(A(C)\), so \(t \in \overline{A(C)}\). Thus \(k[t] \subset \overline{A(C)}\). Conversely, \(k[t]\) is a UFD (hence normal), and \(A(C) \subset k[t]\), so \(\overline{A(C)} \subset \overline{k[t]} = k[t]\). Therefore

\[\tilde{A}(C) = k[t].\]

Step 3: The normalization map. The inclusion \(A(C) = k[t^2, t^3] \hookrightarrow k[t] = \tilde{A}(C)\) corresponds to the morphism

\[\nu : \mathbb{A}^1 \longrightarrow C, \quad t \longmapsto (t^2, t^3).\]

This is a bijective birational morphism. It maps \(\mathbb{A}^1\) (smooth!) onto the cuspidal cubic, resolving the cusp at the origin: the single point \(t = 0\) maps to the singular origin \((0,0) \in C\).

The normalization map \(\nu\) resolves the cusp: \(\tilde{C} \cong \mathbb{A}^1\) is smooth, and \(\nu\) is birational and finite. This is the algebraic incarnation of “unfolding” the cusp.

3.4 Worked Example: Normalizing the Nodal Cubic

📐 Let \(N = V(y^2 - x^2(x+1)) \subset \mathbb{A}^2\) be the nodal cubic with node at the origin. The two branches at the origin have tangent directions \(y = x\) and \(y = -x\) (since the leading term of \(y^2 - x^2(x+1)\) at the origin is \(y^2 - x^2 = (y-x)(y+x)\); see §4, The Tangent Cone).

Step 1: Set \(t = y/x \in k(N)\). From \(y^2 = x^2(x+1)\) we get

\[t^2 = x + 1, \quad \text{i.e., } x = t^2 - 1, \quad y = xt = (t^2-1)t.\]

So \(k(N) = k(t)\).

Step 2: Integral closure. The element \(t = y/x\) satisfies \(t^2 - (x+1) = 0\) over \(A(N)\), so \(t\) is integral over \(A(N)\). One checks that

\[\tilde{A}(N) = k[t].\]

(Verification: \(k[t]\) is a UFD hence normal, \(t\) integral over \(A(N)\) shows \(k[t] \subset \overline{A(N)}\), and \(A(N) \subset k[t]\) with \(k[t]\) normal gives equality.)

Step 3: The normalization map.

\[\nu : \mathbb{A}^1 \longrightarrow N, \quad t \longmapsto (t^2 - 1,\; t(t^2-1)).\]

This is again birational. But unlike the cusp case, the fiber over the node \((0,0)\) now has two preimages:

\[\nu^{-1}(0,0) = \{t : t^2 - 1 = 0\} = \{+1, -1\}.\]

The normalization separates the two branches of the node: \(t = +1\) corresponds to the branch with slope \(+1\) (i.e., \(y \approx x\)) and \(t = -1\) to the branch with slope \(-1\) (i.e., \(y \approx -x\)). The normalization map \(\nu : \mathbb{A}^1 \to N\) is not injective at the node.

4. Blowing Up

4.1 Definition via the Incidence Correspondence

🔬 Blowing up replaces a point by the space of all directions through it. We define it for \(X = \mathbb{A}^n\) at the origin; the general case follows by choosing local coordinates.

Definition (Blow-Up of Affine Space). The blow-up of \(\mathbb{A}^n\) at the origin is the variety

\[\mathrm{Bl}_0 \mathbb{A}^n = \overline{\{(x, \ell) \in \mathbb{A}^n \times \mathbb{P}^{n-1} : x \in \ell\}} \subset \mathbb{A}^n \times \mathbb{P}^{n-1},\]

where \(x \in \ell\) means the point \(x = (x_1,\ldots,x_n)\) lies on the line \(\ell \in \mathbb{P}^{n-1}\) through the origin. Concretely, in homogeneous coordinates \([u_1:\cdots:u_n]\) on \(\mathbb{P}^{n-1}\), the incidence condition is

\[x_i u_j - x_j u_i = 0, \quad 1 \leq i < j \leq n.\]

The blow-down map is \(\pi : \mathrm{Bl}_0 \mathbb{A}^n \to \mathbb{A}^n\), the projection \((x,\ell) \mapsto x\).

The exceptional divisor is \(E = \pi^{-1}(0) \cong \mathbb{P}^{n-1}\). For \(x \neq 0\), the map \(\pi\) is an isomorphism: \(\pi^{-1}(x) = \{(x, \overline{x})\}\) where \(\overline{x} \in \mathbb{P}^{n-1}\) is the direction of \(x\).

Definition (Blow-Up of a Variety). For a variety \(X\) and a smooth point \(P \in X\), the blow-up \(\mathrm{Bl}_P X\) is obtained by choosing local coordinates \((x_1,\ldots,x_n)\) centered at \(P\) identifying a neighborhood of \(P\) in \(X\) with a neighborhood of \(0\) in \(\mathbb{A}^n\), then applying the above construction. The result is independent of the coordinate choice (up to isomorphism).

4.2 Affine Charts for the Blow-Up of the Plane

📐 The case \(n = 2\) is the most important for curve resolution. Here \(\mathbb{P}^1\) has homogeneous coordinates \([u:v]\), and

\[\mathrm{Bl}_0 \mathbb{A}^2 = \{((x,y), [u:v]) : xv = yu\} \subset \mathbb{A}^2 \times \mathbb{P}^1.\]

We cover \(\mathrm{Bl}_0 \mathbb{A}^2\) by two standard affine charts.

Chart \(U_1\) (where \(u \neq 0\)): Set \(t = v/u\). The incidence condition \(xv = yu\) becomes \(xt = y\). The chart is

\[U_1 = \mathrm{Spec}\, k[x, t], \quad y = xt.\]

The blow-down map in this chart is \(\pi_1 : (x,t) \mapsto (x, xt) = (x,y)\). The exceptional divisor is \(E \cap U_1 = V(x) \subset U_1\), and the map \(\pi_1\) restricts to the identity \((x,t) \mapsto (x,y)\) away from \(E\).

Chart \(U_2\) (where \(v \neq 0\)): Set \(s = u/v\). The incidence condition becomes \(xs = y \cdot s^{-1}\)… more cleanly: \(xv = yu\) gives \(x = ys\) (set \(s = u/v\)). The chart is

\[U_2 = \mathrm{Spec}\, k[y, s], \quad x = ys.\]

The blow-down map is \(\pi_2 : (y,s) \mapsto (ys, y) = (x,y)\). The exceptional divisor is \(E \cap U_2 = V(y)\).

The two charts are glued along \(U_1 \cap U_2 = \{(x,t) : x \neq 0\} = \{(y,s) : y \neq 0\}\) via \(y = xt\) and \(s = 1/t\).

4.3 The Strict Transform

Definition (Total Transform and Strict Transform). Let \(C \subset \mathbb{A}^2\) be an affine curve and \(\pi : \mathrm{Bl}_0 \mathbb{A}^2 \to \mathbb{A}^2\) the blow-up at the origin. The total transform of \(C\) is \(\pi^{-1}(C)\). The strict transform (or proper transform) is

\[\tilde{C} = \overline{\pi^{-1}(C \setminus \{0\})} \subset \mathrm{Bl}_0 \mathbb{A}^2,\]

the closure of the preimage of \(C\) minus the blown-up point.

Computing the strict transform in chart \(U_1\). Suppose \(C = V(f(x,y))\) and \(0 \in C\) is a point of multiplicity \(m\) (so \(f\) has no terms of degree \(< m\), and has at least one term of degree exactly \(m\)). In chart \(U_1\) with \(y = xt\):

\[\pi_1^*(f)(x,t) = f(x, xt) = x^m g(x,t),\]

where \(g(x,t) = f(x,xt)/x^m\) has \(g(0,t) \neq 0\) generically. The total transform is \(V(x^m g)\); the factor \(x^m\) corresponds to the exceptional divisor \(E^m\) counted with multiplicity \(m\). The strict transform in \(U_1\) is

\[\tilde{C} \cap U_1 = V(g(x,t)).\]

4.4 Multiplicity and the Effect of One Blow-Up

🔑 Recall from §4.3, Multiplicity that the multiplicity \(\mathrm{mult}_P(C)\) of a curve \(C\) at a point \(P\) is the degree of the lowest-degree term in the Taylor expansion of the defining equation at \(P\). A smooth point has multiplicity 1; a node or ordinary double point has multiplicity 2.

Theorem (Multiplicity Drops at Points of Tangency). Let \(C \subset \mathbb{A}^2\) have multiplicity \(m \geq 2\) at the origin. Let \(\tilde{C}\) be the strict transform under blow-up at \(0\). At any point \(Q \in E \cap \tilde{C}\), we have

\[\mathrm{mult}_Q(\tilde{C}) \leq m.\]

Moreover, \(\mathrm{mult}_Q(\tilde{C}) = m\) only if the tangent cone \(TC_0(C)\) is supported on a single line (i.e., all tangent directions at \(0\) coincide), meaning the singularity is “concentrated in one direction.”

Proof sketch. In chart \(U_1\), \(g(x,t) = f(x,xt)/x^m\). At \(Q = (0, t_0) \in E \cap \tilde{C}\): \(g(0,t_0) = 0\) (since \(Q \in \tilde{C}\)). The multiplicity of \(\tilde{C}\) at \(Q\) is the order of vanishing of \(g\) at \((0,t_0)\). Write \(f = f_m + f_{m+1} + \cdots\) (homogeneous decomposition), so \(f(x,xt) = x^m f_m(1,t) + x^{m+1} f_{m+1}(1,t) + \cdots\), giving \(g(x,t) = f_m(1,t) + x f_{m+1}(1,t) + \cdots\). At \(x = 0\): \(g(0,t) = f_m(1,t)\), which vanishes at \(t_0\) only if \(t_0\) is a root of \(f_m(1,t)\). If \(t_0\) is a simple root of \(f_m(1,\cdot)\), then \(\partial_t g|_{(0,t_0)} \neq 0\), so \(\mathrm{mult}_Q(\tilde{C}) = 1\) (smooth point). \(\square\)

Key conclusion: if the tangent cone at a singular point \(P\) has distinct tangent lines, one blow-up reduces the singularity to smooth points on the strict transform.

5. Resolving Curve Singularities by Blow-Up

5.1 Resolution of a Node

🔬 We resolve the node of \(N = V(y^2 - x^2(x+1))\) at the origin by blowing up.

Set-up. The multiplicity of \(N\) at \(0\) is 2 (the lowest-degree term is \(y^2 - x^2 = (y-x)(y+x)\), a product of two distinct linear forms). In chart \(U_1\) with \(y = xt\):

\[\pi_1^*(y^2 - x^2(x+1)) = x^2 t^2 - x^2(x+1) = x^2(t^2 - x - 1).\]

The total transform is \(V(x^2(t^2 - x - 1))\). The exceptional divisor contributes \(V(x)^2\), and the strict transform is

\[\tilde{N} \cap U_1 = V(t^2 - x - 1).\]

Smoothness of the strict transform. The equation \(g(x,t) = t^2 - x - 1\) has \(\nabla g = (-1, 2t)\), which is nonzero everywhere. So \(\tilde{N} \cap U_1\) is smooth.

Intersection with \(E\). Setting \(x = 0\): \(t^2 - 1 = 0\), giving \(t = \pm 1\). These are two distinct smooth points of \(\tilde{N}\). The blow-up separates the two tangent directions of the node.

The strict transform \(\tilde{N}\) meets \(E \cong \mathbb{P}^1\) at two distinct points corresponding to the two tangent directions \(y/x \to \pm 1\) of \(N\) at the origin.

One blow-up resolves the node.

5.2 Resolution of a Cusp

📐 We resolve the cusp of \(C = V(y^2 - x^3)\) at the origin. The multiplicity at \(0\) is 2 (the leading term is \(y^2\), a single square — the tangent cone is the double line \(y = 0\)).

First blow-up. In chart \(U_1\) with \(y = xt\):

\[\pi_1^*(y^2 - x^3) = x^2 t^2 - x^3 = x^2(t^2 - x).\]

The total transform is \(V(x^2(t^2 - x))\); the strict transform in \(U_1\) is

\[\tilde{C}_1 = V(t^2 - x).\]

Smoothness of \(\tilde{C}_1\). The gradient \(\nabla(t^2 - x) = (-1, 2t)\) is nonzero everywhere. So \(\tilde{C}_1\) is smooth.

Intersection with \(E\). Setting \(x = 0\): \(t^2 = 0\), so \(t = 0\) is the unique intersection point \(Q = (0,0) \in U_1\). But \(Q\) is a smooth point of \(\tilde{C}_1\): \((t^2-x)|_{x=0} = t^2\) vanishes to order 2 in \(t\), but this is just the equation of the curve \(t^2 = x\) being tangent to \(E: x=0\) at \(Q\). At \(Q\), the strict transform has \(t=0, x=0\) with local equation \(t^2 = x\); the tangent direction to \(\tilde{C}_1\) at \(Q\) is \(\partial_t(t^2 - x)|_Q = 0\) and \(\partial_x(t^2-x)|_Q = -1 \neq 0\), so \(x\) is the implicit function parameter and \(\tilde{C}_1\) is smooth at \(Q\).

One blow-up resolves the cusp \(V(y^2 - x^3)\). The strict transform is isomorphic to \(\mathbb{A}^1\) (via \(t \mapsto (t^2, t)\)), a smooth curve.

5.3 Termination of the Process

Theorem (Termination of Blow-Up Resolution for Curves). Let \(C \subset \mathbb{A}^2\) be a plane curve with an isolated singularity at the origin. After finitely many blow-ups, the strict transform \(\tilde{C}^{(n)}\) is smooth.

Proof sketch. Define \(\mu(C, P) = \mathrm{mult}_P(C)\). This is a non-negative integer with \(\mu = 1\) if and only if \(P\) is smooth.

After blowing up at \(P\) with multiplicity \(m \geq 2\): let \(Q_1,\ldots,Q_r\) be the points in \(E \cap \tilde{C}\). The key inequality is:

\[\sum_{i=1}^r \mathrm{mult}_{Q_i}(\tilde{C}) \leq m = \mathrm{mult}_P(C).\]

In fact, more is true: if \(P\) is not an “infinitely near” singularity (one with a single tangent direction of maximal multiplicity), the strict inequality holds at each \(Q_i\). The quantity \(\sum_P \mathrm{mult}_P(C)^2\) (summed over singular points) strictly decreases at each blow-up where the singularity is not of a single-tangent type. For cusps with a single tangent, one uses the Milnor number \(\mu(C,P) = (m_x - 1)(m_y - 1)\) or the \(\delta\)-invariant as the decreasing quantity. Since all these are non-negative integers, the process terminates. \(\square\)

6. The Smooth Projective Model

6.1 Existence

🌐 We now pass to the global picture. Given any irreducible projective curve \(X\) over a perfect field \(k\), we want a smooth projective curve \(\tilde{X}\) birationally equivalent to \(X\).

Theorem (Existence of the Smooth Projective Model). Every irreducible projective curve \(X\) over a perfect field \(k\) has a smooth projective model: a smooth projective curve \(\tilde{X}\) together with a birational morphism \(\nu : \tilde{X} \to X\).

Construction.

Step 1: Normalization. By §3.1, the normalization \(\nu : \tilde{X} \to X\) is a finite birational morphism. By §2.3 (Normal Implies Smooth in Dimension One), \(\tilde{X}\) is smooth. So for curves, normalization already produces a smooth model!

Step 2: Projectivization. If \(X\) is affine, take the projective closure \(\bar{X} \subset \mathbb{P}^n\) and normalize \(\bar{X}\). The result is a projective curve (normalization of a projective variety is projective since it is finite over \(\bar{X}\)).

Alternatively: start with any projective model \(X \subset \mathbb{P}^n\), cover by affine opens, normalize each, and glue. The gluing works because normalization is compatible with localization. \(\square\)

6.2 Uniqueness

Theorem (Uniqueness of the Smooth Projective Model). Let \(\tilde{X}\) and \(\tilde{X}'\) be two smooth projective curves that are birationally equivalent (i.e., \(k(\tilde{X}) \cong k(\tilde{X}')\) as \(k\)-algebras). Then \(\tilde{X} \cong \tilde{X}'\) as varieties.

Proof sketch. A birational map \(\phi : \tilde{X} \dashrightarrow \tilde{X}'\) between smooth projective curves extends to a regular morphism everywhere. The key input is:

Lemma (Removal of Indeterminacies in Dimension 1). Let \(\phi : U \to Y\) be a morphism from an open dense subset \(U \subset X\) of a smooth curve \(X\) to a projective variety \(Y\). Then \(\phi\) extends uniquely to a regular morphism \(\bar{\phi} : X \to Y\).

Proof of lemma. At any missing point \(P \in X \setminus U\): the local ring \(\mathcal{O}_{X,P}\) is a DVR (since \(X\) is smooth of dimension 1). A rational map to projective space is given by \([f_0:\cdots:f_n]\) with \(f_i \in k(X)\). At \(P\), let \(v = v_P\) be the discrete valuation. Among the \(f_i\), choose those with minimal valuation \(m = \min_i v(f_i)\); divide by a uniformizer \(\pi^m\) to make all \(f_i/\pi^m\) regular at \(P\) with at least one nonvanishing there. This gives the extension. \(\square\)

Completing the uniqueness proof. The birational map \(\phi : \tilde{X} \dashrightarrow \tilde{X}'\) and its inverse \(\phi^{-1} : \tilde{X}' \dashrightarrow \tilde{X}\) both extend to regular morphisms by the lemma. These extensions are mutually inverse (they agree on a dense open set), so \(\tilde{X} \cong \tilde{X}'\). \(\square\)

Uniqueness is the crucial structural fact: a smooth projective curve is determined up to isomorphism by its function field.

6.3 The Function Field Perspective

🔑 The uniqueness theorem establishes an anti-equivalence of categories:

\[\left\{ \begin{array}{c} \text{smooth projective curves over } k \\ \text{up to isomorphism} \end{array} \right\} \;\longleftrightarrow\; \left\{ \begin{array}{c} \text{finitely generated field extensions} \\ K/k \text{ with } \mathrm{trdeg}_k(K) = 1 \end{array} \right\}\]

The functor sends \(\tilde{X} \mapsto k(\tilde{X})\); given \(K/k\) with transcendence degree 1, choose any \(x \in K\) transcendental over \(k\), set \(A = k[x] \cap K\) (the integral closure of \(k[x]\) in \(K\)), and \(\tilde{X} = \mathrm{Spec}\, A\) compactified to a projective curve.

Definition (Abstract Smooth Projective Curve). Given a finitely generated field extension \(K/k\) with \(\mathrm{trdeg}_k K = 1\), the associated smooth projective curve \(\tilde{X}(K)\) has underlying set

\[|\tilde{X}(K)| = \{ \text{discrete valuation rings } R \subset K : k \subset R,\, \mathrm{Frac}(R) = K \}.\]

The topology and sheaf of regular functions are defined in terms of these DVRs, each corresponding to a “place” of \(K\) over \(k\).

References