Tangent Spaces and Smoothness

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

As in differential geometry, the tangent space is a fundamental tool for understanding the local structure of a variety in algebraic geometry.

Definition of the Tangent Space

In differential geometry, we verified that for the sheaf \(\mathcal{C}^\infty_M\) of smooth functions on \(M\), the collection of all germs vanishing at a point \(x\in M\),

\[\mathfrak{m}_x=\{\mathbf{f}\in \mathcal{C}^\infty_x\mid \mathbf{f}(x)=0\}\]

is a maximal ideal. We then proved that the tangent space can be identified with

\[(\mathfrak{m}_x/\mathfrak{m}_x^2)^\ast\]

(§Cotangent Space, ⁋Lemma 1). This construction is usually not emphasized in differential geometry, but it is essential for generalization to algebraic varieties. Namely, fixing the affine case for convenience, we already know what functions on algebraic varieties are (§Quasi-Projective Varieties, ⁋Definition 5), and we also know that the set of all functions vanishing at a point \(x\in X\) corresponds to the maximal ideal of \(\mathbb{K}[X]\) at that point. Thus we define

\[\mathfrak{m}_x=\{f\in \mathbb{K}[X]\mid f(x)=0\}\]

and consider the localization \(\mathbb{K}[X]_{\mathfrak{m}_x}=\mathcal{O}_{X,x}\) of \(\mathbb{K}[X]\) at this maximal ideal. (§Localization, ⁋Definition 4) Geometrically, recalling §Affine Varieties, ⁋Definition 14, these may be regarded as germs of regular functions at the point \(x\).

Definition 1 The Zariski tangent space \(T_x X\) of a variety \(X\) at a point \(x\) is defined by

\[T_x X = (\mathfrak{m}_x / \mathfrak{m}_x^2)^\ast\]

where \(\mathfrak{m}_x\) is the unique maximal ideal of the local ring \(\mathcal{O}_{X,x}\) at \(x\).

The heart of this definition is that the quotient \(\mathfrak{m}_x / \mathfrak{m}_x^2\) captures the first-order infinitesimal data at \(x\); we call this the Zariski cotangent space \(T_x^\ast X\). Its dual \(T_x X\) is the space of linear functionals acting on this data, that is, the space of directional derivative operators, and this definition agrees with \(T_xX=\Der_\mathbb{K}(\mathcal{O}_{X,x}, \mathbb{K})\).

We do not use the \(\epsilon\)-\(\delta\) style of differentiation from analysis, but since varieties are defined by polynomials, their differentiation may be understood formally: differentiating \(\x^n\) yields \(n\cdot \x^{n-1}\). In the affine case, this can be made particularly explicit.

Proposition 2 For an affine variety \(X = Z(f_1, \ldots, f_k) \subseteq \mathbb{A}^n\) at a point \(x = (x_1, \ldots, x_n)\),

\[T_x X \cong \{v \in \mathbb{K}^n \mid (df_i)_x(v) = 0 \text{ for all } i\}\]

where \((df_i)_x\) is the differential of \(f_i\) at \(x\),

\[(df_i)_x(v) = \sum_{j=1}^n \frac{\partial f_i}{\partial \x_j}(x) v_j\]

Proof

Consider the coordinate ring \(\mathbb{K}[X] = \mathbb{K}[\x_1, \ldots, \x_n] / (f_1, \ldots, f_k)\) of \(X\). Since \(\mathfrak{m}_x = (\x_1 - a_1, \x_2 - a_2, \ldots, \x_n - a_n) / (f_1, \ldots, f_k)\),

\[\mathfrak{m}_x / \mathfrak{m}_x^2 \cong (\x_1 - a_1, \x_2 - a_2, \ldots, \x_n - a_n) / \left( (\x_1 - a_1, \x_2 - a_2, \ldots, \x_n - a_n)^2 + (f_1, \ldots, f_k) \right)\]

Expanding each \(f_i\) in a Taylor series at \(x\),

\[f_i = \sum_{j=1}^n \frac{\partial f_i}{\partial \x_j}(x) (\x_j - a_j) + \text{higher order terms}\]

and the higher order terms lie in \((\x_1 - a_1, \x_2 - a_2, \ldots, \x_n - a_n)^2\). Hence in \(\mathfrak{m}_x / \mathfrak{m}_x^2\), the linear parts \(\sum_j \frac{\partial f_i}{\partial \x_j}(x) (\x_j - a_j)\) of the \(f_i\) vanish.

On the other hand, \(\mathfrak{m}_x / \mathfrak{m}_x^2\) is generated by linear combinations of the \(\x_j - a_j\), so it may be viewed as a quotient of \(\mathbb{K}^n\). The kernel of the differential \((df_i)_x\) then corresponds exactly to the directions that vanish in \(\mathfrak{m}_x / \mathfrak{m}_x^2\). Taking duals,

\[T_x X = (\mathfrak{m}_x / \mathfrak{m}_x^2)^\ast \cong \{v \in \mathbb{K}^n \mid (df_i)_x(v) = 0 \text{ for all } i\}\]

Although the proof is written in the elaborate language of maximal ideals, its underlying idea is simple: for \(X=Z(f_i)\), the condition \((df_i)_x(v)=0\) is precisely the (ordinary) tangent space of the hypersurface \(Z(f_i)\) in \(\mathbb{A}^n\) (viewing \(\mathbb{K}^n\) as \(\mathbb{A}^n\)). While Proposition 2 is stated only for affine varieties, every point \(x\) of any variety \(X\) has an affine neighborhood, so it applies in essence to all varieties. The same is true of the following proposition on the dimension of the tangent space.

Proposition 3 \(T_x X\) is a \(\mathbb{K}\)-vector space, and its dimension is \(n - \operatorname{rank}(J_x)\), where \(J_x\) is the \(k \times n\) Jacobian matrix

\[J_x = \left(\frac{\partial f_i}{\partial \x_j}(x)\right)_{1 \le i \le k, 1 \le j \le n}\]

Proof

Each \((df_i)_x: \mathbb{K}^n \to \mathbb{K}\) is a linear functional. By Proposition 2, \(T_x X\) is the intersection of their kernels, hence a subspace of \(\mathbb{K}^n\). The rows of the Jacobian matrix \(J_x\) are the coordinate representations of these linear functionals, so

\[T_x X = \ker(J_x) = \{v \in \mathbb{K}^n \mid J_x v = 0\}\]

By the rank-nullity theorem, \(\dim T_x X = n - \operatorname{rank}(J_x)\).

Smooth Points and Singular Points

In differential geometry, the tangent space at any point always has the same dimension as the manifold. This is because the definition of a manifold is rather stringent; in algebraic geometry, even an affine variety defined by a single polynomial need not be a manifold (in the classical sense). (Example 6 (Smooth points), Example 7 (Singular points)) Nevertheless, the dimension of the tangent space and that of the variety are not unrelated.

Proposition 4 For any point \(x\) of an irreducible variety \(X\), we have \(\dim T_x X \ge \dim X\).

Proof

We treat only the affine case. Let \(X = Z(f_1, \ldots, f_k) \subseteq \mathbb{A}^n\) be irreducible with \(\dim X = d\). Consider the local ring \(\mathcal{O}_{X,x} = \mathbb{K}[X]_{\mathfrak{m}_x}\) at the point \(x \in X\). Since localization preserves dimension, \(\dim \mathcal{O}_{X,x} = \dim X = d\). (§Dimension)

In general, for a Noetherian local ring \((R, \mathfrak{m})\), we have \(\dim_{\mathbb{K}}(\mathfrak{m}/\mathfrak{m}^2) \ge \dim R\). (§System of Parameters, ⁋Proposition 2) Therefore,

\[\dim T_x X = \dim_{\mathbb{K}}(\mathfrak{m}_x/\mathfrak{m}_x^2) \ge \dim \mathcal{O}_{X,x} = d = \dim X\]

To sharpen our intuition, we examine when this inequality is strict. Such points are called singular points.

Definition 5 A point \(x \in X\) is a smooth point (or nonsingular point) if \(\dim T_x X = \dim X\). Otherwise (that is, if \(\dim T_x X > \dim X\)), it is a singular point.

Example 6 (Smooth points) 1. Every point of \(\mathbb{A}^n\) is a smooth point. Since \(\mathbb{A}^n\) has no defining equations, \(T_x \mathbb{A}^n = \mathbb{K}^n\), and \(\dim T_x \mathbb{A}^n = n = \dim \mathbb{A}^n\).

1. Every point of the parabola \(Z(\y - \x^2)\) is a smooth point. For \(f = \y - \x^2\), we have \(J_{(x,y)} = (-2x, 1)\), which is nonzero everywhere. Hence \(\dim T_x X = 2 - 1 = 1 = \dim X\).

Example 7 (Singular points) 1. (Node) Consider \(X = Z(\y^2 - \x^2(\x+1)) \subset \mathbb{A}^2\). This curve branches into two at the origin.

![nodal_curve](/assets/images/Math/Algebraic_Varieties/Tangent_Spaces_and_Smoothness-1.svg){:style="width:23.17em" class="invert" .align-center}

The Jacobian of this curve is

$$J_{(x,y)} = \begin{pmatrix} -2x - 3x^2 & 2y \end{pmatrix}$$

so at the origin the Jacobian is $$(0,0)$$, and by [Proposition 3](#prop3) the origin is a singular point. Geometrically, the tangent space being 2-dimensional means that both tangent directions of the two branches are included. Specifically, since $$\y^2 - \x^2(\x+1) \approx \y^2 - \x^2 = (\y-\x)(\y+\x)$$, near the origin the curve looks like the union of the two lines $$\y = \x$$ and $$\y = -\x$$. A node is one of the "mildest" singularities. 2. (Cusp) Now consider $$Z(\y^2 - \x^3)\subset \mathbb{A}^2$$.

![cusp](/assets/images/Math/Algebraic_Varieties/Tangent_Spaces_and_Smoothness-2.svg){:style="width:18.80em" class="invert" .align-center}

In this case, the origin of this curve is a singular point. Computing the Jacobian,

$$J_{(x,y)}=\begin{pmatrix}-3x^2&2y\end{pmatrix}$$

we see that at the origin, $$\nabla f(0,0) = (0, 0)$$. Geometrically, the tangent space being 2-dimensional here means that every direction is "tangent" at the origin, indicating that the curve is too sharp to define a tangent line in any direction.

In the examples above, we naturally used the following proposition.

Proposition 8 (Jacobian Criterion) Let \(X = Z(f_1, \ldots, f_k) \subseteq \mathbb{A}^n\) be irreducible and let \(x \in X\). Then \(x\) is a smooth point if and only if the rank of the Jacobian matrix \(J_x\) is \(n - \dim X\).

Proof

In Proposition 2, we showed that \(\dim T_x X = n - \operatorname{rank}(J_x)\). By Definition 5, \(x\) is a smooth point precisely when \(\dim T_x X = \dim X\). Thus \(x\) is a smooth point if and only if

\[n - \operatorname{rank}(J_x) = \dim X\]

that is, \(\operatorname{rank}(J_x) = n - \dim X\).

Existence of Smooth Points

Any algebraic variety is smooth at most points. To establish this, we need the notion of a generic point.

Definition 9 The generic point \(\eta\) of an irreducible variety \(X\) is the unique point belonging to every nonempty open subset of \(X\).

In the affine case \(X = \operatorname{Spec} A\), \(\eta\) corresponds to the minimal prime ideal of \(A\) (namely, the zero ideal), and the local ring \(\mathcal{O}_{X,\eta}\) is exactly the function field \(\mathbb{K}(X) = \operatorname{Frac}(A)\). Geometrically, the generic point is the “most general point” of \(X\), a point possessing no particular property of \(X\). We exploit this idea in the proof below.

Proposition 10 The set \(X_\sm\) of smooth points of a variety \(X\) is a dense open subset of \(X\). In particular, \(X_\sm \ne \emptyset\).

Proof

Let \(X = Z(f_1, \ldots, f_k) \subseteq \mathbb{A}^n\) have dimension \(\dim X = d\). By the Jacobian criterion of Proposition 8,

\[X_\sm = \{x \in X \mid \operatorname{rank}(J_x) = n - d\}\]

We now show that this set is a dense open subset. First, that \(X_\sm\) is open is relatively straightforward. The rank being exactly \(n-d\) is the simultaneous satisfaction of two conditions. First, the rank being at least \(n-d\) is equivalent to some \((n-d) \times (n-d)\) minor having nonzero determinant, which is an open condition in the Zariski topology. Second, the rank being at most \(n-d\) is equivalent to all \((n-d+1) \times (n-d+1)\) minors having zero determinant, which is a closed condition. Hence the set of points where the rank is exactly \(n-d\) is an open subset of \(X\).

Showing that \(X_\sm\) is nonempty is somewhat more technical. The idea is that a general point should be smooth, so we consider the generic point \(\eta\) of \(X\). Localizing at \(\eta\), the local ring \(\mathcal{O}_{X,\eta} = \mathbb{K}(X)\) is a field, hence a regular local ring. By §System of Parameters, ⁋Proposition 2,

\[\dim_{\mathbb{K}}(\mathfrak{m}_\eta/\mathfrak{m}_\eta^2) \ge \dim \mathcal{O}_{X,\eta} = d\]

while Proposition 4 gives the reverse inequality, so \(\dim T_\eta X = d\). Therefore, \(\eta \in X_\sm\). Since \(X\) is irreducible, any nonempty open subset is dense.

We now make the following definition.

Definition 11 A variety \(X\) is called smooth (or nonsingular) if every point is a smooth point, that is, \(X_\sm = X\). Otherwise (that is, if a singular point exists), it is called singular.

Example 12 The varieties in Example 6 (Smooth points) are all smooth, and all varieties in Example 7 (Singular points) are singular.

Tangent Cones

At a singular point, the tangent space is too large to reflect the local structure of the variety accurately. In this situation, the tangent cone provides finer information. Intuitively, the tangent space is too large because the rank of the Jacobian is too small, which occurs when the first-order approximation of the defining functions yields no information. If we instead consider higher-order approximations, the picture may improve.

To this end, for any polynomial \(f\in \mathbb{K}[\x_1,\ldots, \x_n]\), we define the initial term \(\initial(f)\) of \(f\) to be the homogeneous component of \(f\) of smallest degree. Then for any ideal \(\mathfrak{a}\), we define the initial ideal \(\initial(\mathfrak{a})\) of \(\mathfrak{a}\) to be the homogeneous ideal generated by the \(\initial(f)\).

Definition 13 For any affine variety \(X\subseteq \mathbb{A}^n\), the algebraic variety defined by \(\initial(I(X))\) is called the tangent cone of \(X\) at the origin.

More generally, by writing \(f\) as a polynomial in the \(\x_i-x_i\) and making the analogous definition, one obtains the tangent cone at any point. It is called a cone because, as in §Projective Varieties, ⁋Definition 12, it is the zero set of a homogeneous ideal.

Let us now see how this gives a finer classification of singular points.

Example 14 For the nodal curve \(X = Z(\y^2 - \x^2(\x+1))\) of Example 7 (Singular points), the lowest-degree term of \(f\) is \(\y^2 - \x^2 = (\y-\x)(\y+\x)\), so

\[TC_0 X = Z(\y-\x) \cup Z(\y+\x)\]

This exactly shows that the node branches in the directions of the two lines \(\y = \x\) and \(\y = -\x\).

Example 15 For the curve \(X = Z(\y^2 - \x^3)\) of Example 7 (Singular points), the lowest-degree term of \(f\) is \(\y^2\), so

\[TC_0 X = Z(\y^2)\]

This is the line \(\y = 0\) counted twice, showing that the cusp ends sharply in the \(\x\)-axis direction. By contrast, the tangent space \(T_0 X = \mathbb{K}^2\) is too large, as it includes all directions.

In general, as discussed in §Rational Maps, ⁋Example 12, the singularity of a nodal curve can be resolved by blowup: after blowing up, the two branch lines \(\y-\x\) and \(\y+\x\) at the origin are separated by \(\mathbb{P}^1\). A cusp, however, cannot be resolved in this way, so a cusp is generally regarded as a worse singularity than a node.

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References

[Har] J. Harris, Algebraic Geometry: A First Course, Springer, 1992. [Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer, 2013.