Normal cone

In algebraic geometry, the normal cone of a closed subscheme Z \subseteq X (or more generally, of a closed immersion i: Z \to X) is a scheme C_Z X over Z that generalizes the normal bundle N_Z X, capturing infinitesimal directions "normal" to Z within X. It is defined as the relative spectrum over Z of the conormal algebra \mathcal{C}_{Z/X,*} = \bigoplus_{n \geq 0} \mathcal{I}^n / \mathcal{I}^{n+1}, where \mathcal{I} \subset \mathcal{O}_X is the quasi-coherent ideal sheaf defining Z as a closed subscheme of X.^[1] When Z is a Cartier divisor or the embedding is regular, the normal cone coincides with the normal bundle, a vector bundle over Z. In general, it is a cone over Z (in the sense of relative Spec of a graded \mathcal{O}_Z-algebra) and fits into a canonical closed immersion C_Z X \hookrightarrow N_Z X. The normal cone plays a central role in deformation theory, where the deformation to the normal cone provides a universal deformation of Z inside X to first order, and in the study of blowups, singularities, and intersection theory.^[2] It also underlies constructions like the intrinsic normal cone in derived algebraic geometry, which addresses issues in non-regular embeddings.^[3]

Definition and Fundamentals

Formal Definition

In algebraic geometry, a closed embedding of schemes is a morphism i: X \hookrightarrow Y that identifies X with a closed subscheme of Y, defined by a quasi-coherent ideal sheaf \mathcal{I} \subset \mathcal{O}_Y such that X = \operatorname{Spec}_Y(\mathcal{O}_Y / \mathcal{I}).^[4] The ideal sheaf \mathcal{I} generates a filtration on \mathcal{O}_Y, leading to the associated graded ring \bigoplus_{n=0}^\infty \mathcal{I}^n / \mathcal{I}^{n+1}, which is a quasi-coherent sheaf of graded \mathcal{O}_X-algebras on X.^[2] The normal cone C_{X/Y} to the closed embedding i: X \hookrightarrow Y is defined as the relative spectrum

C_{X/Y} = \operatorname{Spec}_X \left( \bigoplus_{n=0}^\infty \mathcal{I}^n / \mathcal{I}^{n+1} \right),

where the relative Spec construction yields a scheme over X whose structure sheaf is determined by the graded algebra above.^[4] This construction positions C_{X/Y} as an infinitesimal cone over the base X, with fibers over points of X corresponding to the spectra of the associated graded rings of the completions of the local rings of Y along X.^[2] When the embedding is regular, the graded algebra \bigoplus_{n=0}^\infty \mathcal{I}^n / \mathcal{I}^{n+1} is isomorphic to the symmetric algebra \operatorname{Sym}_{\mathcal{O}_X}(\mathcal{I}/\mathcal{I}^2) on the conormal sheaf \mathcal{I}/\mathcal{I}^2.^[4] In this case, the normal cone C_{X/Y} recovers the normal bundle N_{X/Y} as a special instance.^[4]

Relation to Normal Bundle

In the case of a regular embedding of schemes, the normal cone specializes to the normal bundle, providing a direct link between infinitesimal geometry in algebraic settings and vector bundle structures. Specifically, for a closed immersion i: X \hookrightarrow Y where the conormal sheaf \mathcal{I}/\mathcal{I}^2 is locally free of constant rank equal to the codimension r, the normal cone C_{X/Y} is isomorphic to the total space of the normal bundle N_{X/Y} = \Spec_X \left( \Sym_{\mathcal{O}_X} \left( \mathcal{I}/\mathcal{I}^2 \right) \right), which is thus a vector bundle of rank r over X.^[5] A closed immersion is regular of codimension r if, locally on affine open sets, the ideal sheaf \mathcal{I} of X in Y is generated by r elements forming a regular sequence in the coordinate ring of Y. Under this condition, the associated graded sheaf \bigoplus_{n \geq 0} \mathcal{I}^n / \mathcal{I}^{n+1} is isomorphic to the symmetric algebra \Sym^\bullet (\mathcal{I}/\mathcal{I}^2), ensuring that the relative Spec construction for the normal cone yields precisely the symmetric algebra associated to the normal bundle.^[5] An explicit local criterion for such regularity is that the Koszul complex generated by these r elements provides a free resolution of \mathcal{O}_X as an \mathcal{O}_Y-module, reflecting the local complete intersection nature of the embedding.^[6] This identification bridges algebraic geometry with differential geometry, as the normal cone in the regular case serves an analogous role to the tubular neighborhood of a submanifold, capturing the directions of infinitesimal deformations transverse (normal) to X inside Y while linearizing the embedding structure around X.^[6]

Key Properties

Behavior under Composition of Embeddings

Consider a composition of regular embeddings i: X \hookrightarrow Y and j: Y \hookrightarrow Z. The normal bundles satisfy the exact sequence

$0 \to N_{X/Y} \to N_{X/Z} \to i^* N_{Y/Z} \to 0,

where the maps are induced by the embeddings.^[7] Since the embeddings are regular, the normal cones coincide with the normal bundles as schemes: C_{X/Y} = N_{X/Y}, C_{X/Z} = N_{X/Z}, and i^* C_{Y/Z} = i^* N_{Y/Z}. Thus, the exact sequence extends directly to an exact sequence of normal cones

0 \to C_{X/Y} \to C_{X/Z} \to i^* C_{Y/Z} \to 0.[](https://link.springer.com/book/10.1007/978-3-662-02421-8) From this sequence, the normal cone to the composed embedding $C_{X/Z}$ fibers over the pullback $i^* C_{Y/Z}$ of the second normal cone, with fibers isomorphic to the first normal cone $C_{X/Y}$.[](https://link.springer.com/book/10.1007/978-3-662-02421-8) More generally, for a [pullback](/page/Pullback) diagram of closed [embedding](/page/Embedding)s \[ \begin{CD} W @>k>> Z \\ @VVlV @VVjV \\ Y @>>i> X, \end{CD}

where the square is Cartesian, the normal cone to the pullback embedding C_{W/Y} is isomorphic to the pullback of the normal cone C_{Z/X} along the diagram:

C_{W/Y} \cong C_{Z/X} \times_{C_{Y/X}} C_{W/Y}.

This follows from the fact that the defining ideal sheaf of W in Y is the pullback of the ideal sheaf of Z in X, preserving the associated graded structure.^[8]

Dimension of Components

When the ambient scheme X is pure-dimensional of dimension r, the normal cone C_{W/X} to a closed subscheme W \subset X inherits this purity and is likewise pure-dimensional of dimension r.^[9] This property ensures that the normal cone behaves consistently with the ambient space in terms of overall dimensionality, facilitating its use in intersection-theoretic constructions where dimensional homogeneity is required. The fibers of the projection C_{W/X} \to W exhibit dimensions that relate to local embedding properties along W. Specifically, over a point p \in W, the dimension of the fiber is connected to the embedding dimension of X at p minus the dimension of the Zariski tangent space to W at p, reflecting the expected codimension in regular cases but potentially exceeding it at singular points due to higher-order relations in the defining ideal.^[9] For instance, in the embedding of the scheme defined by (x^2, xy) in \mathbb{A}^2, the fiber over a generic point on the y-axis has dimension 1, matching the expected value, while the fiber over the origin has dimension 2.^[9] The normal cone C_{W/X} may possess multiple irreducible components, arising from the structure of the ideal sheaf defining W. These components often correspond to the primary components in the primary decomposition of the ideal or to associated points of the subscheme, leading to a decomposition of the cone that captures the non-reduced or non-equidimensional features of the embedding.^[9] In the example of a point embedded in the curve V(y^2 + x^2(x-1)), the normal cone features two distinct irreducible components despite the subscheme being a single point.^[9]

Illustrative Examples

Cartier Divisors

A Cartier divisor D \hookrightarrow X defines a regular embedding of codimension one, providing an elementary case for computing the normal cone. In this setting, the normal cone C_{D/X} is isomorphic to the total space of the normal bundle N_{D/X}, which carries the explicit structure of the line bundle \mathcal{O}_D(D) on D.^[10] This line bundle structure arises because the conormal sheaf C_{D/X} = \mathcal{I}_D / \mathcal{I}_D^2 \cong \mathcal{O}_D(-D) is invertible, and its dual yields the normal sheaf N_{D/X} = \mathcal{O}_D(D).^[10] The computation of the normal cone proceeds via the associated graded algebra. Locally, where D = \operatorname{div}(f) for a section f \in \mathcal{O}_X(U), the ideal sheaf is principal, \mathcal{I}_D = (f). The graded algebra is then \bigoplus_{n \geq 0} \mathcal{I}_D^n / \mathcal{I}_D^{n+1}, with each component \mathcal{I}_D^n / \mathcal{I}_D^{n+1} \cong \mathcal{O}_D via the map induced by multiplication by f^n.^[11] This algebra simplifies to the symmetric algebra \Sym_{\mathcal{O}_D}(\mathcal{O}_D \cdot df), where df generates the conormal sheaf as an \mathcal{O}_D-module, reflecting the regularity of the embedding.^[12] Geometrically, if X is a surface, the normal cone over the curve D is a ruled surface, with rulings given by the affine lines in the fibers of the normal bundle \mathcal{O}_D(D).^[13]

Nodal Cubic Curve

The nodal cubic curve provides an illustrative example of the normal cone for a singular closed subscheme embedding into affine space, where the singularity of the subscheme affects the geometry of the cone despite the embedding being locally complete intersection. Consider the embedding X \hookrightarrow Y = \mathbb{A}^2, where X is defined by the equation y^2 = x^2(x + 1), or equivalently, the ideal \mathcal{I} = (y^2 - x^2(x + 1)) in \mathbb{C}[x, y]. This curve has a node at the origin (0, 0), making X singular at that point. Since the embedding is a hypersurface in the smooth variety Y, the ideal \mathcal{I} is principal, and the associated graded ring along \mathcal{I} is \bigoplus_{d \geq 0} \mathcal{I}^d / \mathcal{I}^{d+1} \cong \mathcal{O}_X, where each graded piece is isomorphic to the structure sheaf \mathcal{O}_X. Thus, the normal cone is C_{X/Y} = \Spec_X(\mathcal{O}_X), the total space of the trivial line bundle over X. The singularity of X at the node impacts the normal cone: while the fiber over every point of X, including the node, is an affine line \mathbb{A}^1, the total space C_{X/Y} inherits a singularity along the entire fiber over the node, manifesting as a line of singular points. This reflects the nodal singularity of X, where the local ring at (0, 0) has embedded dimension 2. In contrast, for a smooth cubic curve embedding in \mathbb{A}^2 (such as the affine part of a smooth plane cubic), the normal cone is likewise the total space of a line bundle, but over a smooth base, resulting in a smooth total space without singular loci. This highlights how subscheme singularities propagate to the normal cone's geometry, even in regular embeddings.

Non-Regular Embeddings

In non-regular embeddings, where the ideal sheaf defining the closed subscheme Z \subset Y is not locally generated by a regular sequence, the normal cone C_Z Y fails to be a vector bundle over Z. Instead, it exhibits more complex geometry, reflecting the singularities or irregularities of the embedding. This contrasts with regular embeddings, where the normal cone is locally isomorphic to the total space of the normal bundle. The construction via the associated graded algebra \mathrm{gr}_{\mathcal{I}} \mathcal{O}_Y = \bigoplus_{n \geq 0} \mathcal{I}^n / \mathcal{I}^{n+1} (with \mathcal{I} the ideal sheaf of Z) yields a scheme that is generally not the spectrum of the symmetric algebra of the conormal sheaf \mathcal{I}/\mathcal{I}^2. The geometry of such normal cones often includes embedded components or non-reduced structures, arising from nilpotent elements in the structure sheaf or from the failure of the graded algebra to be generated freely by its linear part. For instance, the normal cone may appear as a thickened line incorporating nilpotents, capturing higher-order infinitesimal directions that are "stuck" due to the embedding's irregularity. These features encode the obstructions to smoothing or resolving the embedding, with the non-reduced components indicating multiplicity in the infinitesimal neighborhood. A concrete example occurs in the embedding of a closed point p (the origin) into the cuspidal curve Y \subset \mathbb{A}^2 defined by y^2 = x^3. Here, the local ring \mathcal{O}_{Y,p} = k[x,y]/(y^2 - x^3) has maximal ideal \mathfrak{m} = (x,y), which is the ideal \mathcal{I} of the point in Y. The associated graded ring \mathrm{gr}_{\mathfrak{m}} \mathcal{O}_{Y,p} is isomorphic to k[u,v]/(v^2), where u and v are the images of x and y in \mathfrak{m}/\mathfrak{m}^2. Thus, the normal cone C_p Y is the spectrum of this ring, a non-reduced scheme consisting of a double line (the line v=0 with nilpotent structure given by v^2 = 0) over the point p. This non-reduced structure arises because the relation y^2 = x^3 lies in \mathfrak{m}^3, so the initial form in the graded ring is v^2 = 0, thickening the tangent direction without higher-dimensional components. Despite the embedding's non-regularity—the ideal \mathfrak{m} requires two generators but the relation introduces dependencies—the normal cone still parametrizes the first-order infinitesimal neighborhood of p in Y, albeit with torsion elements in its structure sheaf reflecting the cusp's singularity. These torsions manifest as nilpotents that prevent the cone from being reduced or a free module over the base, highlighting how the normal cone adapts to capture embedded irregularities even in defective embeddings. In Fulton's framework, such cones are essential for defining refined intersection products on singular ambient spaces, where the non-bundle structure necessitates careful handling of components and multiplicities.

Deformation to the Normal Cone

Motivational Role in Deformation Theory

The normal cone of a closed subscheme X in a scheme Y serves as a fundamental model for the infinitesimal deformations of the embedding X \hookrightarrow Y, capturing the first-order variations in the position of X within Y. This structure is particularly valuable in deformation theory because it encodes the tangent directions to the space of embeddings, allowing geometers to study how subvarieties can be "pushed" or deformed locally without altering the ambient space. In this context, the normal cone facilitates the analysis of stability and obstructions in deforming singular varieties, providing a geometric counterpart to the algebraic tools used in moduli problems. A key application arises in the realm of blow-up algebras and Rees algebras, where the normal cone underpins the construction of associated graded rings that model the blow-up process. Specifically, the Rees algebra of the ideal sheaf defining X in Y realizes the total space of the normal cone as a Proj construction, enabling the study of filtrations and associated graded modules central to deformation invariants. This connection allows for the resolution of embedded singularities by iteratively blowing up along subvarieties while controlling the exceptional loci through the cone's fibers. In intersection theory, the normal cone plays a pivotal role by enabling the computation of refined intersection products through the deformation to the normal cone, which specializes transverse intersections to those along the cone, thereby circumventing issues with excess components in non-proper intersections. This deformation ensures that cycle classes remain constant in the family, justifying the use of the normal cone's geometry to define proper intersection multiplicities on singular schemes. In his seminal work on resolution of singularities over fields of characteristic zero, Heisuke Hironaka introduced the notion of normal flatness, defined via the flatness of the normal cone over the center, to track the multiplicity and ensure equimultiplicity of exceptional divisors arising from blow-ups along singular loci, thereby controlling the behavior of ideals under resolution.^[14]

Explicit Construction

The deformation to the normal cone for a closed subscheme X \subset Y of schemes is constructed as a flat family M_{X/Y}^o \to \mathbb{P}^1, where the generic fiber over \{\infty\} is isomorphic to Y, and the special fiber over \{0\} consists of the union C_{X/Y} \cup (Y \setminus X), with X embedded in the special fiber as the vertex of the normal cone component. This family arises from the Rees algebra associated to the ideal sheaf \mathcal{I} = \mathcal{I}_{X/Y} of X in Y. The graded \mathcal{O}_Y-algebra is \bigoplus_{n=0}^\infty \mathcal{I}^n t^n \subseteq \mathcal{O}_Y, where t is the deformation parameter. The total space is then given by the relative Proj construction

M_{X/Y} = \Proj_Y \left( \bigoplus_{n=0}^\infty \mathcal{I}^n t^n \right),

which realizes the blow-up of Y along X parametrized by t, with the open part M_{X/Y}^o obtained by removing the irrelevant ideal generated by t. The flatness of M_{X/Y}^o \to \mathbb{P}^1 follows from the graded structure of the Rees algebra, which ensures that the fibers vary properly and the morphism is flat over the base. The normal cone C_{X/Y} in the special fiber is the relative spectrum \Spec_Y \left( \bigoplus_{n=0}^\infty \mathcal{I}^n / \mathcal{I}^{n+1} \right).

Intrinsic Normal Cone

Intrinsic Normal Bundle

In the context of Deligne-Mumford stacks, the intrinsic normal sheaf \mathfrak{N}_\mathfrak{X} of a stack \mathfrak{X} is defined as the quotient h^1(\mathfrak{L}^\vee_\mathfrak{X}) / h^0(\mathfrak{L}^\vee_\mathfrak{X}), where \mathfrak{L}^\vee_\mathfrak{X} is the dual of the cotangent complex \mathfrak{L}^\bullet_\mathfrak{X} of \mathfrak{X} in the derived category of quasi-coherent sheaves. This provides an embedding-independent enhancement that captures the stack's infinitesimal normal structure, without reference to any embedding. For local complete intersection (lci) Deligne-Mumford stacks, the intrinsic normal sheaf \mathfrak{N}_\mathfrak{X} coincides with the classical normal sheaf arising from a local presentation of \mathfrak{X} as a closed substack of a smooth stack. In this case, the higher cohomology vanishes appropriately, reducing \mathfrak{N}_\mathfrak{X} to an actual vector bundle stack over \mathfrak{X}. A key property of the intrinsic normal sheaf is its additivity under fiber products: if \mathfrak{X} and \mathfrak{Y} are Deligne-Mumford stacks over a base scheme S, then \mathfrak{N}_{\mathfrak{X} \times_S \mathfrak{Y}} \cong \mathfrak{N}_\mathfrak{X} \times_S \mathfrak{N}_\mathfrak{Y}. This compatibility ensures that the construction behaves well with respect to the stack's presentation and gluing data. The intrinsic normal sheaf plays a central role in obstruction theory for Deligne-Mumford stacks, intrinsically measuring the spaces of infinitesimal automorphisms and deformations of \mathfrak{X}. Specifically, the zeroth cohomology H^0(\mathfrak{N}_\mathfrak{X}) corresponds to the tangent sheaf encoding automorphisms, while the first cohomology H^1(\mathfrak{N}_\mathfrak{X}) captures the obstruction sheaf for first-order deformations. This framework allows for the definition of perfect obstruction theories E^\bullet \to L^\bullet \mathfrak{X} that refine the intrinsic structure, with \mathfrak{N}_\mathfrak{X} embedding into the bundle associated to

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via a closed immersion when the theory is perfect. This notion generalizes the classical normal sheaf for schemes, providing an embedding-independent tool for derived deformations.^[15]

Definition and Properties of the Intrinsic Normal Cone

The intrinsic normal cone of an algebraic stack \mathfrak{X} is constructed from its intrinsic normal sheaf \mathfrak{N}_\mathfrak{X}, defined as the quotient h^1(\mathfrak{L}^\vee_\mathfrak{X})/h^0(\mathfrak{L}^\vee_\mathfrak{X}), where \mathfrak{L}^\vee_\mathfrak{X} denotes the dual of the cotangent complex of \mathfrak{X}. For a Deligne-Mumford stack \mathfrak{X}, the intrinsic normal cone \mathfrak{C}_\mathfrak{X} is a closed subcone stack of \mathfrak{N}_\mathfrak{X}, obtained by gluing local models: over an étale presentation U \to \mathfrak{X} with a local immersion U \to M into a smooth scheme M, it is given by \mathfrak{C}_\mathfrak{X}|_U = [C_{U/M}/f^* T_M], where C_{U/M} is the classical normal cone of the embedding and the quotient accounts for the stack structure.^[15] For l.c.i. stacks, it coincides classically with \mathfrak{N}_\mathfrak{X} itself.^[15] A key property of the intrinsic normal cone is its base change invariance: for a cartesian diagram of stacks with a flat base change morphism v: Y' \to Y, the natural map \mathfrak{C}_{\mathfrak{X}'/\mathfrak{Y}'} \to \mathfrak{C}_{\mathfrak{X}/\mathfrak{Y}} \times_{\mathfrak{Y}} \mathfrak{Y}' is an isomorphism, ensuring that the cone behaves well under pullbacks.^[15] It is also compatible with smooth stacks, where \mathfrak{C}_\mathfrak{X} \cong \mathfrak{N}_\mathfrak{X} \cong B T_\mathfrak{X}, the classifying stack of the tangent sheaf, meaning the cone "trivializes" to the total space of the normal sheaf in this case.^[15] This compatibility highlights its role in deformation theory, as the cone embeds into a vector bundle stack arising from a perfect obstruction theory on \mathfrak{X}, enabling the construction of virtual fundamental classes via intersection with the zero section.^[15] For quotient stacks [X/G], where X is a scheme and G a group scheme acting on X, the intrinsic normal cone \mathfrak{C}_{[X/G]} arises as the quotient of the classical normal cone C_{X/M} (for a local embedding X \to M) by the joint action of G and the stabilizer f^* T_M, yielding a stack that incorporates equivariant structure and reflects the singularities induced by the group action.^[15]