Local Properties of Varieties

Table of Contents

• 1. Singular and Nonsingular Points
  • 1.1 The Local Ring at a Point
  • 1.2 The Cotangent Space and Zariski Tangent Space
  • 1.3 The Jacobian Criterion
  • 1.4 Regular Local Rings
  • 1.5 The Smooth Locus is Open and Dense
  • 1.6 Examples: Nodal and Cuspidal Cubics
• 2. Power Series Expansion
  • 2.1 Local Parameters
  • 2.2 The Formal Completion
  • 2.3 Power Series Expansion of Regular Functions
• 3. Properties of Nonsingular Points
  • 3.1 Analytical Irreducibility
  • 3.2 Smooth Subvarieties and the Normal Bundle
  • 3.3 The Algebraic Implicit Function Theorem
  • 3.4 Local Factoriality
• 4. The Tangent Cone
  • 4.1 Definition via Initial Forms
  • 4.2 The Associated Graded Ring
  • 4.3 Multiplicity
  • 4.4 Examples
• 5. Discrete Valuation Rings and Smooth Curves
  • 5.1 DVRs: Definitions and Characterizations
  • 5.2 Smooth Points of Curves are DVRs
  • 5.3 The Valuation and Order of Vanishing
• References

Throughout, \(k\) denotes an algebraically closed field (e.g. \(k = \mathbb{C}\)) and all varieties are over \(k\), unless otherwise stated. We write \(\mathbb{A}^n = \mathbb{A}^n_k\) for affine \(n\)-space. For background on affine varieties, coordinate rings, rational functions, local rings, DVRs, and intersection numbers, see Classical Algebraic Geometry.

1. Singular and Nonsingular Points

1.1 The Local Ring at a Point

📐 Let \(X \subset \mathbb{A}^n\) be an irreducible affine variety with coordinate ring \(A(X) = k[x_1,\ldots,x_n]/I(X)\). For a point \(P \in X\), the local ring at \(P\) is the localization

\[\mathcal{O}_{X,P} = A(X)_{\mathfrak{m}_P},\]

where \(\mathfrak{m}_P = \{ f \in A(X) : f(P) = 0 \}\) is the maximal ideal of \(P\) in \(A(X)\). Concretely, \(\mathcal{O}_{X,P}\) consists of fractions \(f/g\) with \(f, g \in A(X)\) and \(g(P) \neq 0\), representing germs of regular functions near \(P\).

Definition (Local Ring and Maximal Ideal). The local ring of \(X\) at \(P\) is the pair \((\mathcal{O}_{X,P}, \mathfrak{m}_{X,P})\), where

\[\mathfrak{m}_{X,P} = \{ \phi \in \mathcal{O}_{X,P} : \phi(P) = 0 \}\]

is the unique maximal ideal of \(\mathcal{O}_{X,P}\). The residue field is \(\mathcal{O}_{X,P}/\mathfrak{m}_{X,P} \cong k\).

1.2 The Cotangent Space and Zariski Tangent Space

🔑 The cotangent space of \(X\) at \(P\) is the \(k\)-vector space \(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2\). Its dual encodes the first-order linear geometry of \(X\) at \(P\).

Definition (Zariski Tangent Space). The Zariski tangent space of \(X\) at \(P\) is

\[T_{X,P} = \bigl(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2\bigr)^\vee = \mathrm{Hom}_k\!\bigl(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2,\, k\bigr).\]

The space \(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2\) is the Zariski cotangent space.

Why quotient by \(\mathfrak{m}^2\)? A function \(f \in \mathfrak{m}_{X,P}\) vanishes at \(P\); in a Taylor expansion, \(f = \sum_i (\partial f/\partial x_i)(P)(x_i - P_i) + \text{higher order}\). The class \([f] \in \mathfrak{m}/\mathfrak{m}^2\) retains only the linear part. Indeed, \(\mathfrak{m}^2 = \{fg : f,g \in \mathfrak{m}\}\) corresponds exactly to functions whose first-order Taylor coefficients all vanish at \(P\). Thus \(\mathfrak{m}/\mathfrak{m}^2\) is the space of linear forms on \(X\) at \(P\), and its dual is the space of tangent directions.

Derivation description. Equivalently, \(T_{X,P} \cong \mathrm{Der}_k(\mathcal{O}_{X,P}, k)\), the set of \(k\)-linear maps \(\delta : \mathcal{O}_{X,P} \to k\) satisfying the Leibniz rule \(\delta(fg) = f(P)\delta(g) + g(P)\delta(f)\). Any such derivation is determined by its values on \(\mathfrak{m}_{X,P}\), and since \(\delta(\mathfrak{m}^2) = 0\) (Leibniz with \(f(P) = g(P) = 0\)), it factors through \(\mathfrak{m}/\mathfrak{m}^2\).

1.3 The Jacobian Criterion

📐 For a concrete hypersurface or complete intersection, the tangent space is computed by linearization. We derive the Jacobian criterion from the cotangent-space definition.

Let \(X = V(f_1,\ldots,f_m) \subset \mathbb{A}^n\) be a variety defined by polynomials \(f_1,\ldots,f_m \in k[x_1,\ldots,x_n]\), with \(P \in X\).

Derivation of the Jacobian Criterion. The coordinate ring of \(\mathbb{A}^n\) maps surjectively onto \(A(X)\), hence \(\mathcal{O}_{\mathbb{A}^n, P} \to \mathcal{O}_{X,P}\) is surjective with kernel generated by the images of \(f_1,\ldots,f_m\). Writing \(\mathfrak{m} = \mathfrak{m}_{\mathbb{A}^n,P}\) and \(\mathfrak{n} = \mathfrak{m}_{X,P}\), we get a surjection

\[\mathfrak{m}/\mathfrak{m}^2 \twoheadrightarrow \mathfrak{n}/\mathfrak{n}^2,\]

whose kernel is spanned by the images of \(f_1,\ldots,f_m\) modulo \(\mathfrak{m}^2\). Since each \(f_i\) vanishes at \(P\), we write \(f_i = \sum_j (\partial f_i/\partial x_j)(P)(x_j - P_j) + O(|\mathbf{x}-P|^2)\), so \([f_i]\) maps to the linear form \(\sum_j (\partial f_i / \partial x_j)(P) \cdot [x_j - P_j]\) in \(\mathfrak{m}/\mathfrak{m}^2 \cong k^n\). Therefore

\[\mathfrak{n}/\mathfrak{n}^2 \cong k^n \Big/ \mathrm{span}\!\left\{ \left(\tfrac{\partial f_i}{\partial x_j}(P)\right)_{j=1}^n : i = 1,\ldots,m \right\}.\]

Taking duals:

Theorem (Jacobian Criterion). With notation as above,

\[T_{X,P} = \ker\!\left(\frac{\partial f_i}{\partial x_j}(P)\right)_{i,j} \subset k^n,\]

where the Jacobian matrix \(J(P) = \bigl((\partial f_i/\partial x_j)(P)\bigr) \in \mathrm{Mat}_{m \times n}(k)\) acts on \(k^n = T_{\mathbb{A}^n, P}\).

Remark. The formula shows \(\dim T_{X,P} = n - \mathrm{rank}\, J(P)\). Since \(\dim X \geq n - m\) (by Krull’s principal ideal theorem), we always have \(\dim T_{X,P} \geq \dim X\).

1.4 Regular Local Rings

Definition (Smooth and Singular Points). A point \(P \in X\) is smooth (or nonsingular, or regular) if

\[\dim_k T_{X,P} = \dim X.\]

\(P\) is singular if \(\dim_k T_{X,P} > \dim X\). The singular locus of \(X\) is \(\mathrm{Sing}(X) = \{ P \in X : P \text{ is singular}\}\).

Definition (Regular Local Ring). A Noetherian local ring \((R, \mathfrak{m}, k)\) is regular if

\[\dim_k \mathfrak{m}/\mathfrak{m}^2 = \dim R,\]

where \(\dim R\) is the Krull dimension. The left side is at least \(\dim R\) for any Noetherian local ring (by Nakayama’s lemma and the theory of Hilbert–Samuel polynomials), so regularity is the extremal equality.

Proposition. \(P \in X\) is smooth if and only if \(\mathcal{O}_{X,P}\) is a regular local ring.

Proof sketch. We have \(\dim_k \mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2 = \dim_k T_{X,P}\). By definition of smoothness, this equals \(\dim X = \dim \mathcal{O}_{X,P}\), which is exactly the regularity condition. \(\square\)

1.5 The Smooth Locus is Open and Dense

🔑 Singular points are isolated in a precise topological sense.

Theorem (Smooth Locus). Let \(X\) be an irreducible variety over an algebraically closed field \(k\) of characteristic zero. Then the smooth locus \(X_{\mathrm{sm}} = X \setminus \mathrm{Sing}(X)\) is open and dense in \(X\).

Proof sketch. Openness: By the Jacobian criterion, smoothness at \(P\) is the condition that the Jacobian matrix \(J(P)\) has rank \(\geq \mathrm{codim}(X, \mathbb{A}^n)\). This is an open condition — the locus where a matrix has rank less than a given value is Zariski closed (defined by vanishing of minors). Hence \(\mathrm{Sing}(X)\) is closed, and \(X_{\mathrm{sm}}\) is open.

Density (nonemptiness of \(X_{\mathrm{sm}}\)): It suffices to show \(X\) has at least one smooth point. By Noether normalization, the function field \(k(X)\) is a separable extension of a pure transcendental subfield (in characteristic zero, all finite extensions are separable). One may show the generic point of \(X\) is smooth — i.e., the local ring at the generic point is regular — and since smoothness is an open condition, there is a dense open subset of smooth points.

Remark. In positive characteristic, one must additionally assume \(X\) is geometrically reduced (which holds e.g. for varieties defined over a perfect field). In characteristic zero over an algebraically closed field, irreducibility suffices.

from sympy import symbols, diff, groebner, Poly, S

def singular_locus_hypersurface(f, variables):
    """
    Input: f - polynomial in variables (as sympy expression)
           variables - list of sympy symbols [x1, ..., xn]
    Output: (generators of Sing(X) ideal, is_smooth bool)
    """
    # Build the ideal generators: f and all partial derivatives
    generators = [f] + [diff(f, xi) for xi in variables]

    # Compute Gröbner basis over QQ
    G = groebner(generators, *variables, domain='QQ')

    # X is smooth iff the ideal is the unit ideal, i.e. 1 is in the basis
    is_smooth = (S.One in G)

    return generators, is_smooth

# Example: nodal cubic y^2 - x^2*(x+1)
x, y = symbols('x y')
f = y**2 - x**2 * (x + 1)
gens, smooth = singular_locus_hypersurface(f, [x, y])
# gens = [y^2 - x^3 - x^2, -3x^2 - 2x, 2y]
# Groebner basis will contain {x, y}, not 1, so is_smooth = False

1.6 Examples: Nodal and Cuspidal Cubics

Example (Nodal Cubic). Let \(X = V(y^2 - x^2(x+1)) \subset \mathbb{A}^2\). Setting \(f = y^2 - x^2(x+1) = y^2 - x^3 - x^2\):

\[\frac{\partial f}{\partial x} = -3x^2 - 2x, \qquad \frac{\partial f}{\partial y} = 2y.\]

At \(P = (0,0)\): both partials vanish, so \(J(P) = (0, 0)\) has rank \(0 < 1 = \mathrm{codim}(X)\). Thus \((0,0)\) is singular. For any other point, at least one partial is nonzero, so \(X\) has only one singular point: the node at the origin.

Example (Cuspidal Cubic). Let \(X = V(y^2 - x^3) \subset \mathbb{A}^2\). Setting \(f = y^2 - x^3\):

\[\frac{\partial f}{\partial x} = -3x^2, \qquad \frac{\partial f}{\partial y} = 2y.\]

At \(P = (0,0)\): both vanish, so the origin is singular — a cusp.

Example (Whitney Umbrella). The surface \(X = V(x^2 - y^2 z) \subset \mathbb{A}^3\) is singular along the \(z\)-axis. Setting \(f = x^2 - y^2z\):

\[\frac{\partial f}{\partial x} = 2x, \quad \frac{\partial f}{\partial y} = -2yz, \quad \frac{\partial f}{\partial z} = -y^2.\]

The Jacobian vanishes when \(x = 0\), \(yz = 0\), \(y = 0\), i.e. when \(x = 0\) and \(y = 0\). So \(\mathrm{Sing}(X) = V(x,y) \cap X\) is the \(z\)-axis \(\{x = y = 0\}\), which has dimension \(1 < 2 = \dim X\), confirming the density theorem.

2. Power Series Expansion

2.1 Local Parameters

🗺️ At a smooth point of an \(n\)-dimensional variety, local algebra admits a canonical set of “coordinates.”

Definition (Local Parameters). Let \(X\) be smooth of dimension \(n\) at a point \(P\). A system of local parameters (or regular system of parameters) at \(P\) is a set \(\{t_1,\ldots,t_n\} \subset \mathfrak{m}_{X,P}\) whose images in \(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2\) form a \(k\)-basis.

By Nakayama’s lemma (since \(\mathfrak{m}_{X,P}/\mathfrak{m}_{X,P}^2 \cong k^n\) at a smooth point), the elements \(t_1,\ldots,t_n\) generate \(\mathfrak{m}_{X,P}\) as an ideal. They are the algebraic analogue of local coordinates in differential geometry.

Existence. Choose any \(n\) regular functions on an affine neighborhood \(U\) of \(P\) such that \(t_i(P) = 0\) and the differentials \(dt_i(P) = [t_i] \in \mathfrak{m}/\mathfrak{m}^2\) form a basis. For example, if \(U \hookrightarrow \mathbb{A}^n\) is an open immersion (valid near a smooth point), the shifted coordinate functions \(x_i - P_i\) restrict to local parameters.

Remark. Different choices of local parameters give isomorphic completions — see below. The set of local parameters is far from unique, but the local ring structure is canonical.

2.2 The Formal Completion

📐 The most powerful tool for local analysis is the formal completion.

Definition (Completion). The formal completion of \(\mathcal{O}_{X,P}\) with respect to \(\mathfrak{m}_{X,P}\) is the inverse limit

\[\hat{\mathcal{O}}_{X,P} = \varprojlim_{n \geq 0} \mathcal{O}_{X,P}/\mathfrak{m}_{X,P}^n.\]

Elements of \(\hat{\mathcal{O}}_{X,P}\) are coherent sequences \((\phi_n)_{n \geq 0}\) with \(\phi_n \in \mathcal{O}_{X,P}/\mathfrak{m}^n\) and \(\phi_{n+1} \mapsto \phi_n\) under reduction. There is a canonical ring homomorphism \(\mathcal{O}_{X,P} \to \hat{\mathcal{O}}_{X,P}\), which is injective when \(\mathcal{O}_{X,P}\) is Noetherian (Krull’s intersection theorem: \(\bigcap_n \mathfrak{m}^n = 0\)).

Theorem (Cohen Structure Theorem, smooth case). If \(P \in X\) is a smooth point of dimension \(n\) with local parameters \(t_1,\ldots,t_n\), then

\[\hat{\mathcal{O}}_{X,P} \cong k[\![t_1,\ldots,t_n]\!],\]

the ring of formal power series in \(n\) variables over \(k\).

Proof sketch. Since \(P\) is smooth, \(\mathcal{O}_{X,P}\) is a regular local ring of dimension \(n\). The general Cohen structure theorem states that every complete equicharacteristic Noetherian local ring \((R, \mathfrak{m}, k)\) is a quotient \(k[\![x_1,\ldots,x_N]\!]/I\) for some \(N\) and ideal \(I\). For a regular local ring, the minimal number of generators of \(\mathfrak{m}\) equals \(n = \dim R\), so \(I = 0\) and \(\hat{\mathcal{O}}_{X,P} \cong k[\![t_1,\ldots,t_n]\!]\). \(\square\)

2.3 Power Series Expansion of Regular Functions

Proposition (Unique Power Series Expansion). Let \(P \in X\) be smooth with local parameters \(t_1,\ldots,t_n\), and let \(f \in \mathcal{O}_{X,P}\). Then \(f\) admits a unique expansion

\[f = \sum_{\alpha \in \mathbb{Z}_{\geq 0}^n} c_\alpha t^\alpha \in k[\![t_1,\ldots,t_n]\!], \qquad c_\alpha \in k,\]

where \(t^\alpha = t_1^{\alpha_1}\cdots t_n^{\alpha_n}\), via the injective map \(\mathcal{O}_{X,P} \hookrightarrow \hat{\mathcal{O}}_{X,P} \cong k[\![t_1,\ldots,t_n]\!]\).

Derivation. Since \(t_1,\ldots,t_n\) generate \(\mathfrak{m}_{X,P}\) and \(\mathcal{O}_{X,P}/\mathfrak{m}^{m+1}\) is spanned by monomials \(t^\alpha\) with \(|\alpha| \leq m\), any \(f \in \mathcal{O}_{X,P}\) determines, for each \(m\), a polynomial of degree \(\leq m\) that is the truncation of the expansion. These truncations are compatible, giving the formal series. Injectivity of \(\mathcal{O}_{X,P} \to k[\![t_1,\ldots,t_n]\!]\) follows from \(\bigcap_m \mathfrak{m}^m = 0\).

Geometric interpretation. The local parameters \(t_1,\ldots,t_n\) play the role of “local coordinates” centered at \(P\). The power series expansion of \(f\) is the analogue of the Taylor series expansion in differential geometry. Unlike the complex-analytic setting, this series need not converge in any metric sense — it lives in the formal power series ring — but it contains complete local information about \(f\).

from sympy import symbols, series, Poly, expand, factor

def power_series_expansion(p, q, point, local_params, ambient_vars, degree):
    """
    Input: p, q - polynomials (f = p/q), q(point) != 0
           point - dict {xi: P_i} giving coordinates of P
           local_params - list of expressions for t_i in terms of ambient_vars
           ambient_vars - list of sympy symbols [x1, ..., xn]
           degree - truncation degree d
    Output: polynomial in local_params representing expansion up to degree d
    """
    # Substitute ambient coords in terms of local params centered at P
    # e.g. x_i = P_i + t_i if local_params = [x1-P1, ..., xn-Pn]
    subs_map = {x: p_val + t
                for x, p_val, t in zip(ambient_vars,
                                       [point[x] for x in ambient_vars],
                                       local_params)}
    p_shifted = p.subs(subs_map).expand()
    q_shifted = q.subs(subs_map).expand()

    # q_shifted = q(P) * (1 + r) where r is in m, expand (1+r)^{-1} to degree d
    q0 = q_shifted.subs({t: 0 for t in local_params})  # = q(P) != 0
    r = (q_shifted / q0 - 1).expand()  # r in m

    # Geometric series: 1/(1+r) = sum_{k=0}^{d} (-r)^k mod m^{d+1}
    inv_q_series = sum(
        ((-r)**k).expand() for k in range(degree + 1)
    )
    # Truncate to total degree <= d
    result = (p_shifted * inv_q_series / q0).expand()
    # Drop monomials of degree > d
    result_poly = Poly(result, *local_params)
    truncated = sum(
        coeff * monom
        for coeff, monom in zip(result_poly.coeffs(), result_poly.monoms())
        if sum(monom) <= degree
    )
    return truncated

3. Properties of Nonsingular Points

3.1 Analytical Irreducibility

🔑 At a smooth point, the variety cannot “branch.”

Definition (Analytically Irreducible). A local ring \((R, \mathfrak{m})\) is analytically irreducible if its completion \(\hat{R}\) is an integral domain.

Theorem (Smooth points are analytically irreducible). If \(P \in X\) is a smooth point of an irreducible variety \(X\), then \(\mathcal{O}_{X,P}\) is analytically irreducible.

Proof sketch. By the Cohen structure theorem, \(\hat{\mathcal{O}}_{X,P} \cong k[\![t_1,\ldots,t_n]\!]\), which is an integral domain (formal power series rings over a field are domains: if \(fg = 0\) in \(k[\![t_1,\ldots,t_n]\!]\), compare leading terms). \(\square\)

Geometric meaning. Analytically irreducible means the formal neighborhood of \(P\) in \(X\) is connected: the variety does not locally look like two (or more) branches crossing at \(P\). This fails at singular points like nodes. For example, the nodal cubic \(V(y^2 - x^2(x+1))\) at the origin: the completion of its local ring factors as \(k[\![t_1]\!] \times k[\![t_2]\!]\) (one factor per branch), which is not a domain.

3.2 Smooth Subvarieties and the Normal Bundle

📐 When \(Y \subset X\) is a smooth closed subvariety of a smooth ambient variety, the first-order geometry of the pair \((X,Y)\) is captured by the normal bundle.

Setup. Let \(X\) be a smooth variety of dimension \(n\), and let \(Y \subset X\) be a closed smooth subvariety of dimension \(r\). Denote by \(\mathcal{I}_Y \subset \mathcal{O}_X\) the ideal sheaf of \(Y\) (the sheaf whose sections on an open \(U\) are the regular functions on \(U\) vanishing on \(U \cap Y\)).

Definition (Conormal Sheaf and Normal Bundle). The conormal sheaf of \(Y\) in \(X\) is

\[\mathcal{N}^\vee_{Y/X} = \mathcal{I}_Y / \mathcal{I}_Y^2,\]

viewed as an \(\mathcal{O}_Y\)-module (via the surjection \(\mathcal{O}_X \to \mathcal{O}_Y\)). The normal sheaf (or normal bundle) is its \(\mathcal{O}_Y\)-dual:

\[\mathcal{N}_{Y/X} = \mathcal{H}om_{\mathcal{O}_Y}(\mathcal{I}_Y/\mathcal{I}_Y^2,\, \mathcal{O}_Y).\]

Theorem. If \(Y\) and \(X\) are both smooth, then \(\mathcal{I}_Y/\mathcal{I}_Y^2\) is a locally free \(\mathcal{O}_Y\)-module of rank \(n - r = \mathrm{codim}(Y,X)\).

Proof sketch. Locally at a point \(P \in Y\), choose local parameters \(t_1,\ldots,t_n\) for \(X\) at \(P\) such that \(t_{r+1},\ldots,t_n\) are a system of local parameters for the regular ideal \(\mathcal{I}_{Y,P}\). (This is possible because \(Y\) is smooth and \(\mathcal{I}_{Y,P}\) is generated by a regular sequence in \(\mathcal{O}_{X,P}\).) Then \(\mathcal{I}_{Y,P}/\mathcal{I}_{Y,P}^2\) is freely generated over \(\mathcal{O}_{Y,P}\) by \([t_{r+1}],\ldots,[t_n]\), giving rank \(n - r\). \(\square\)

Algebraic derivation. The conormal sheaf fits into the conormal sequence:

\[0 \to \mathcal{I}_Y/\mathcal{I}_Y^2 \to \Omega_{X/k}|_Y \to \Omega_{Y/k} \to 0,\]

where \(\Omega_{X/k}\) is the sheaf of Kähler differentials. When \(X\) and \(Y\) are smooth, all three terms are locally free, and the sequence is the algebraic version of the split short exact sequence of tangent/cotangent bundles.

Fiber description. At a point \(P \in Y\), the fiber of \(\mathcal{N}_{Y/X}\) is

\[(\mathcal{N}_{Y/X})_P \otimes k(P) = T_{X,P} / T_{Y,P},\]

the quotient of tangent spaces, which is a \(k\)-vector space of dimension \(n - r\).

3.3 The Algebraic Implicit Function Theorem

Theorem (Algebraic Implicit Function Theorem). Let \(X\) be an \(n\)-dimensional variety. Then \(P \in X\) is smooth if and only if there exists a Zariski open neighborhood \(U \ni P\) in \(X\) and a closed immersion \(U \hookrightarrow \mathbb{A}^n\) that is an isomorphism onto its image (i.e., \(U\) is isomorphic to an open subset of \(\mathbb{A}^n\)).

Proof sketch (one direction). If \(U \cong V \subset \mathbb{A}^n\) for some open \(V\), then \(\mathcal{O}_{X,P} \cong \mathcal{O}_{\mathbb{A}^n,P}\), which is a regular local ring (it is a localization of the polynomial ring \(k[x_1,\ldots,x_n]\), hence regular). So \(P\) is smooth.

Other direction. Suppose \(P\) is smooth. If \(X \subset \mathbb{A}^N\) and \(X\) has codimension \(c = N - n\), the Jacobian matrix \(J(P)\) has rank \(c\). By an implicit function theorem argument in the local ring: \(c\) of the defining equations \(f_1,\ldots,f_c\) form a regular sequence in \(\mathcal{O}_{\mathbb{A}^N,P}\), and the quotient \(\mathcal{O}_{\mathbb{A}^N,P}/(f_1,\ldots,f_c) = \mathcal{O}_{X,P}\) is a regular local ring of dimension \(n\) generated by the remaining \(n\) coordinates. These \(n\) coordinates, restricted to \(X\), form an étale (locally biholomorphic in the algebraic sense) map \(X \to \mathbb{A}^n\) on some Zariski open neighborhood of \(P\). \(\square\)

3.4 Local Factoriality

Theorem (Regular local rings are UFDs). If \(P \in X\) is a smooth point, then \(\mathcal{O}_{X,P}\) is a unique factorization domain (UFD).

Proof sketch. A regular local ring is a UFD by a theorem of Auslander–Buchsbaum (1959), which proves that regular local rings have finite global dimension, and Serre’s criterion then gives the UFD property. \(\square\)

Geometric consequence. Every codimension-1 subvariety (Weil divisor) through a smooth point \(P \in X\) is locally a principal divisor — it is the zero locus of a single regular function in \(\mathcal{O}_{X,P}\). This fails at singular points (e.g. the Weil divisors on a cone need not be Cartier).

4. The Tangent Cone

4.1 Definition via Initial Forms

📐 At a singular point, the Zariski tangent space is “too large” — it does not distinguish the local geometry of the singularity. The tangent cone provides a finer first-order invariant.

Setup. Let \(P \in X\) be a (possibly singular) point. We may assume \(P = 0 \in \mathbb{A}^n\) after a translation. Every \(f \in I(X) \subset k[x_1,\ldots,x_n]\) vanishing at \(P\) can be written as \(f = f_m + f_{m+1} + \cdots\), where \(f_d\) is the homogeneous component of degree \(d\) and \(m = \mathrm{ord}_P(f) \geq 1\) is the order of vanishing.

Definition (Initial Form). For \(f \in \mathfrak{m}_{\mathbb{A}^n,P}\), the initial form (or leading form) of \(f\) at \(P\) is the lowest-degree homogeneous component \(f_m\) where \(m = \mathrm{ord}_P(f)\).

Definition (Tangent Cone). The tangent cone \(C_{X,P}\) to \(X\) at \(P\) is the algebraic set

\[C_{X,P} = V\!\left(\mathrm{in}(I(X))\right) \subset \mathbb{A}^n,\]

where \(\mathrm{in}(I(X)) = \{ \mathrm{in}(f) : f \in I(X) \}\) denotes the initial ideal, i.e. the ideal generated by all initial forms of elements of \(I(X)\).

Remark. The tangent cone is a cone in the sense that it is stable under the scaling \(\mathbf{x} \mapsto \lambda \mathbf{x}\) for \(\lambda \in k^*\), because \(\mathrm{in}(f)\) is homogeneous.

4.2 The Associated Graded Ring

🔑 The algebraic description of the tangent cone is via the associated graded ring.

Definition (Associated Graded Ring). For a local ring \((\mathcal{O}_{X,P}, \mathfrak{m})\), the associated graded ring is

\[\mathrm{gr}_\mathfrak{m}\, \mathcal{O}_{X,P} = \bigoplus_{d \geq 0} \mathfrak{m}^d/\mathfrak{m}^{d+1},\]

with multiplication \([\phi][\psi] = [\phi\psi]\) for \(\phi \in \mathfrak{m}^a/\mathfrak{m}^{a+1}\), \(\psi \in \mathfrak{m}^b/\mathfrak{m}^{b+1}\). This is a graded \(k\)-algebra (with degree-0 part \(\mathcal{O}_{X,P}/\mathfrak{m} \cong k\)).

Theorem (Tangent cone as Spec of associated graded). There is a canonical isomorphism

\[C_{X,P} \cong \mathrm{Spec}\!\left(\mathrm{gr}_\mathfrak{m}\, \mathcal{O}_{X,P}\right)\]

as algebraic varieties (or schemes). Precisely: if \(X \subset \mathbb{A}^n\) and \(P = 0\), then

\[\mathrm{gr}_\mathfrak{m}\, \mathcal{O}_{X,P} \cong k[x_1,\ldots,x_n]/\mathrm{in}(I(X)).\]

Proof sketch. The associated graded of the local ring of \(\mathbb{A}^n\) at the origin is \(k[x_1,\ldots,x_n]\) itself (since the polynomial ring is already graded). The ideal \(I(X)\) acts on successive quotients \(\mathfrak{m}^d/\mathfrak{m}^{d+1}\) via its initial forms, giving exactly \(k[x_1,\ldots,x_n]/\mathrm{in}(I(X))\) as the associated graded of the quotient. \(\square\)

Relationship to the tangent space. At a smooth point \(P\), \(\mathrm{gr}_\mathfrak{m}\, \mathcal{O}_{X,P} \cong k[t_1,\ldots,t_n]\) (the polynomial ring in the local parameters), since the associated graded of a regular local ring is a polynomial ring. This means \(C_{X,P} = T_{X,P}\) at smooth points — the tangent cone equals the tangent space.

from sympy import symbols, groebner, Poly, degree

def tangent_cone_ideal(generators, variables):
    """
    Input: generators - list of polynomials generating I
           variables - list of sympy symbols
    Output: generators of the initial ideal in(I) (tangent cone ideal)

    Uses: degree-reverse-lex (grlex) Gröbner basis; leading forms
    give generators of in(I).
    """
    # Compute Gröbner basis with grlex order (graded reverse lex)
    G = groebner(generators, *variables, order='grlex', domain='QQ')

    # For each basis element, extract its lowest-degree homogeneous part
    def leading_form(f, vars):
        poly = Poly(f, *vars, domain='QQ')
        min_deg = min(sum(m) for m in poly.monoms())
        # Sum all monomials of degree min_deg
        leading = sum(
            c * Poly.from_dict({m: 1}, *vars, domain='QQ').as_expr()
            for c, m in zip(poly.coeffs(), poly.monoms())
            if sum(m) == min_deg
        )
        return leading

    initial_gens = [leading_form(g, variables) for g in G]

    # Remove redundant generators
    initial_gens = list(set(initial_gens))
    return initial_gens

# Example: nodal cubic
x, y = symbols('x y')
I_gens = [y**2 - x**2*(x + 1)]  # y^2 - x^3 - x^2
tc_gens = tangent_cone_ideal(I_gens, [x, y])
# Expected: [y^2 - x^2] (the leading form y^2 - x^2)
print(tc_gens)

4.3 Multiplicity

Definition (Multiplicity). The multiplicity of \(X\) at \(P\), denoted \(\mathrm{mult}_P(X)\), is the degree of the Hilbert polynomial of the associated graded ring \(\mathrm{gr}_\mathfrak{m}\,\mathcal{O}_{X,P}\). Equivalently, it is the degree of the tangent cone \(C_{X,P}\) as a projective variety (via the projectivization of the leading ideal).

For a hypersurface \(X = V(f) \subset \mathbb{A}^n\) with \(P = 0\):

\[\mathrm{mult}_P(X) = \mathrm{ord}_P(f) = \min\{ d : f_d \neq 0 \},\]

the order of vanishing of the defining polynomial at \(P\).

Theorem (Multiplicity and smoothness). \(\mathrm{mult}_P(X) = 1\) if and only if \(P\) is a smooth point of \(X\).

Proof sketch. \(\mathrm{mult}_P(X) = 1\) means the leading form \(f_1\) is linear, i.e. \(\nabla f(P) \neq 0\), which is exactly the Jacobian criterion for smoothness of the hypersurface. For the general case, \(\mathrm{mult}_P(X) = 1\) iff \(\mathrm{gr}_\mathfrak{m}\,\mathcal{O}_{X,P}\) is generated in degree 1, iff \(\dim_k(\mathfrak{m}/\mathfrak{m}^2) = \dim X\), i.e. regularity. \(\square\)

Remark (semicontinuity of multiplicity). Multiplicity is upper semicontinuous in families: in a flat family \(\mathcal{X} \to T\), the function \(P \mapsto \mathrm{mult}_P(\mathcal{X}_t)\) can only jump up at special values of \(t\). This is the algebraic version of “singularities can be created but not destroyed by specialization.”

4.4 Examples

Example (Node, revisited). For \(X = V(y^2 - x^2(x+1))\) at \(P = (0,0)\):

The defining polynomial is \(f = y^2 - x^2(x+1) = y^2 - x^2 - x^3\). The leading form is \(f_2 = y^2 - x^2 = (y-x)(y+x)\). So

\[C_{X,(0,0)} = V(y^2 - x^2) = V(y-x) \cup V(y+x),\]

two distinct lines through the origin. This reflects the two smooth branches of the node: \(y \approx x\) and \(y \approx -x\) near \((0,0)\). The multiplicity is \(\mathrm{mult}_{(0,0)}(X) = 2\).

Example (Cusp). For \(X = V(y^2 - x^3)\) at \(P = (0,0)\):

The leading form is \(f_2 = y^2\), so \(C_{X,(0,0)} = V(y^2)\), a double line along the \(x\)-axis. The multiplicity is \(2\). The tangent cone is a non-reduced scheme, reflecting the fact that the cusp has only one branch but it is “tangent to itself.”

Example (Whitney Umbrella). For \(X = V(x^2 - y^2z)\) at \(P = (0,0,0)\):

The polynomial \(f = x^2 - y^2z\) has lowest-degree term \(f_2 = x^2\) (since \(y^2 z\) has degree 3). So \(C_{X,(0,0,0)} = V(x^2)\), a double plane (the \(yz\)-plane with multiplicity 2). The multiplicity is \(2\).

from sympy import symbols, Poly, total_degree

def multiplicity_at_origin(f, variables):
    """
    Input: f - polynomial vanishing at origin (f(0,...,0) = 0)
           variables - list of sympy symbols
    Output: mult_{(0,0)}(V(f)) = order of vanishing of f at origin
    """
    poly = Poly(f, *variables, domain='QQ')
    max_deg = poly.total_degree()

    for d in range(1, max_deg + 1):
        # Check if f has any monomial of degree exactly d
        homog_part = sum(
            coeff * Poly.from_dict({monom: 1}, *variables).as_expr()
            for coeff, monom in zip(poly.coeffs(), poly.monoms())
            if sum(monom) == d and coeff != 0
        )
        if homog_part != 0:
            return d  # smallest degree present = multiplicity

    return max_deg  # fallback (f is homogeneous)

# Example: cuspidal cubic y^2 - x^3
x, y = symbols('x y')
f = y**2 - x**3
print(multiplicity_at_origin(f, [x, y]))  # Output: 2

# Example: x^4 + y^4 - x^2*y^2
g = x**4 + y**4 - x**2*y**2
print(multiplicity_at_origin(g, [x, y]))  # Output: 4

5. Discrete Valuation Rings and Smooth Curves

5.1 DVRs: Definitions and Characterizations

🔑 Discrete valuation rings are the algebraic avatars of smooth points on curves.

Definition (Discrete Valuation). A discrete valuation on a field \(K\) is a surjective homomorphism \(v : K^\times \to \mathbb{Z}\) satisfying the ultrametric inequality:

\[v(a + b) \geq \min(v(a), v(b)) \quad \text{for all } a, b \in K^\times.\]

(Extend to \(K\) by \(v(0) = +\infty\).)

Definition (Discrete Valuation Ring). The ring of integers (or valuation ring) of \(v\) is

\[R = \{ a \in K : v(a) \geq 0 \} \cup \{0\}.\]

A ring \(R\) is a discrete valuation ring (DVR) if it arises this way for some discrete valuation on \(K = \mathrm{Frac}(R)\).

Theorem (Equivalent Characterizations). For a Noetherian local domain \((R, \mathfrak{m})\), the following are equivalent:

1. \(R\) is a DVR.
2. \(R\) is a principal ideal domain with a unique nonzero prime ideal.
3. \(R\) is a regular local ring of Krull dimension 1.
4. \(\mathfrak{m}\) is principal (generated by a single element \(t\), the uniformizer or local parameter).
5. Every nonzero element \(f \in R\) can be written uniquely as \(f = u \cdot t^n\) for \(u \in R^\times\) and \(n \geq 0\).

Proof sketch (3 \(\Rightarrow\) 1). Regularity of dimension 1 means \(\mathfrak{m} = (t)\) for a single generator. By the unique factorization property of regular local rings (which are UFDs), every nonzero \(f \in R\) writes as \(f = u t^{v(f)}\) for a unit \(u\) and \(v(f) \geq 0\). Extending to \(\mathrm{Frac}(R)\) by \(v(f/g) = v(f) - v(g)\) gives a discrete valuation. \(\square\)

5.2 Smooth Points of Curves are DVRs

Theorem. Let \(C\) be an irreducible curve and \(P \in C\) a smooth point. Then \(\mathcal{O}_{C,P}\) is a DVR.

Proof. Since \(\dim C = 1\) and \(P\) is smooth, \(\mathcal{O}_{C,P}\) is a regular local ring of Krull dimension 1. By characterization 3 in the theorem above, \(\mathcal{O}_{C,P}\) is a DVR. The uniformizer is any local parameter \(t \in \mathfrak{m}_{C,P} \setminus \mathfrak{m}_{C,P}^2\). \(\square\)

Explicit construction. In an affine model \(C \subset \mathbb{A}^2\) near \(P\): a local parameter \(t\) is any function in \(A(C)\) that vanishes to exactly first order at \(P\). For example, if \(C = V(f)\) with \((\partial f/\partial y)(P) \neq 0\), one may take \(t = x - P_x\) (restricted to \(C\)).

The formal completion satisfies \(\hat{\mathcal{O}}_{C,P} \cong k[\![t]\!]\).

5.3 The Valuation and Order of Vanishing

Definition (Order of Vanishing). For \(P\) a smooth point of a curve \(C\) and \(f \in \mathcal{O}_{C,P}\), the order of vanishing of \(f\) at \(P\) is

\[v_P(f) = \max\{ n \geq 0 : f \in \mathfrak{m}_{C,P}^n \},\]

i.e. the unique integer \(n\) such that \(f = u t^n\) for a unit \(u \in \mathcal{O}_{C,P}^\times\) and uniformizer \(t\). Extend to rational functions \(\phi = f/g \in k(C)\) by \(v_P(\phi) = v_P(f) - v_P(g)\).

Properties of the valuation:

1. \(v_P(\phi \psi) = v_P(\phi) + v_P(\psi)\)
2. \(v_P(\phi + \psi) \geq \min(v_P(\phi), v_P(\psi))\), with equality when \(v_P(\phi) \neq v_P(\psi)\)
3. \(v_P(\phi) > 0\) iff \(\phi\) has a zero at \(P\); \(v_P(\phi) < 0\) iff \(\phi\) has a pole at \(P\)

Connection to intersection numbers. The intersection number of a curve \(D\) (defined by \(f = 0\)) with \(C\) at \(P\) is \(I_P(C,D) = v_P(f|_C)\), the order of vanishing of \(f\) on \(C\) at \(P\). This connects DVR valuations to intersection theory from Chapter I of Shafarevich (see Classical Algebraic Geometry, §6 on intersection numbers).

from sympy import symbols, Rational, cancel, series, oo

def curve_valuation(f_num, f_den, param_map, param, point_val=0):
    """
    Input: f_num, f_den - numerator and denominator of f in ambient coords
           param_map - dict {ambient_var: expr_in_param}, the parametrization
           param - sympy symbol for the curve parameter s
           point_val - value of param at P (default 0)
    Output: v_P(f) = order of vanishing of f at P
    """
    # Compose f with parametrization: compute f(gamma(s))
    f_num_param = f_num.subs(param_map)
    f_den_param = f_den.subs(param_map)

    # Shift parameter so P corresponds to s = 0
    t = param - point_val
    f_num_shifted = f_num_param.subs(param, t + point_val).expand()
    f_den_shifted = f_den_param.subs(param, t + point_val).expand()

    # Find order of vanishing of numerator and denominator at t = 0
    def order_at_zero(poly, var):
        """Return smallest degree of var in poly."""
        from sympy import Poly
        p = Poly(poly, var)
        nonzero = [i for i, c in enumerate(reversed(p.all_coeffs()))
                   if c != 0]
        return min(nonzero) if nonzero else oo

    v_num = order_at_zero(f_num_shifted, t)
    v_den = order_at_zero(f_den_shifted, t)
    return v_num - v_den

# Example: C = V(y - x^2), param: s -> (s, s^2), f = y/x = s
s, x, y = symbols('s x y')
param_map = {x: s, y: s**2}
print(curve_valuation(y, x, param_map, s))  # v_P(y/x) = 2 - 1 = 1
print(curve_valuation(y, 1, param_map, s))  # v_P(y) = 2

References