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Normal cone

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</style><div role="note" class="hatnote navigation-not-searchable">For normal cones in real vector spaces, see <a href="Convex_cone" title="Convex cone">convex cone</a>.</div>

In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the <a href="Normal_bundle" title="Normal bundle">normal bundle</a> or <a href="Tubular_neighborhood" title="Tubular neighborhood">tubular neighborhood</a> in differential geometry.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The normal cone CXY or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X/Y}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X/Y}}</annotation>

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</math><img src="./2a26e1f264bdf87622683f572c00ce1943ee560b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.37ex; height:3.009ex;" alt="{\displaystyle C_{X/Y}}" loading="lazy"> of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the <a href="Relative_Spec" class="mw-redirect" title="Relative Spec">relative Spec</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} _{X}\left(\bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}\right).}</annotation>

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When the embedding i is <a href="Regular_embedding" title="Regular embedding">regular</a> the normal cone is the normal bundle, the vector bundle on X corresponding to the dual of the sheaf I/I2.

If X is a point, then the normal cone and the normal bundle to it are also called the <a href="Tangent_cone" title="Tangent cone">tangent cone</a> and the <a href="Tangent_space" title="Tangent space">tangent space</a> (<a href="Zariski_tangent_space" title="Zariski tangent space">Zariski tangent space</a>) to the point. When Y = Spec R is affine, the definition means that the normal cone to X = Spec R/I is the Spec of the <a href="Associated_graded_ring" title="Associated graded ring">associated graded ring</a> of R with respect to I.

If Y is the product X × X and the embedding i is the <a href="Diagonal_embedding" class="mw-redirect" title="Diagonal embedding">diagonal embedding</a>, then the normal bundle to X in Y is the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> to X.

The normal cone (or rather its projective cousin) appears as a result of <a href="Blowing_up" title="Blowing up">blow-up</a>. Precisely, let

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<annotation encoding="application/x-tex">{\displaystyle \pi :\operatorname {Bl} _{X}Y=\operatorname {Proj} _{Y}\left(\bigoplus _{n=0}^{\infty }I^{n}\right)\to Y}</annotation>

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be the blow-up of Y along X. Then, by definition, the <a href="Exceptional_divisor" title="Exceptional divisor">exceptional divisor</a> is the pre-image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\pi ^{-1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle E=\pi ^{-1}(X)}</annotation>

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</math><img src="./4106784003a283ba8ecb160d9892da00ecf0f63f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.33ex; height:3.176ex;" alt="{\displaystyle E=\pi ^{-1}(X)}" loading="lazy">; which is the <a href="Cone_(algebraic_geometry)" title="Cone (algebraic geometry)">projective cone</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigoplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}}">

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<annotation encoding="application/x-tex">{\textstyle \bigoplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}}</annotation>

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</math><img src="./2bc193c0ed17c541fd2ffbf04f57e17d994cbee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.4ex; height:3.176ex;" alt="{\textstyle \bigoplus _{0}^{\infty }I^{n}\otimes _{{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}}" loading="lazy">. Thus,

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<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {P} (C_{X}Y).}</annotation>

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The global sections of the normal bundle classify embedded infinitesimal deformations of Y in X; there is a natural bijection between the set of closed subschemes of Y ×k D, flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NX Y).<a href="#cite_note-FOOTNOTEHartshorne1977Ch._III,_Exercise_9.7.-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Compositions_of_regular_embeddings">Compositions of regular embeddings</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}">

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<annotation encoding="application/x-tex">{\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}</annotation>

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</math><img src="./9d942772bc12c9259023f24eb50e36315784b944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.076ex; height:2.509ex;" alt="{\displaystyle i:X\hookrightarrow Y,\,j:Y\hookrightarrow Z}" loading="lazy"> are <a href="Regular_embedding" title="Regular embedding">regular embeddings</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\circ i}">

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<annotation encoding="application/x-tex">{\displaystyle j\circ i}</annotation>

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</math><img src="./a6855dab5dac9d02c1f01f1c1f0d70d1792aa5ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.982ex; height:2.509ex;" alt="{\displaystyle j\circ i}" loading="lazy"> is a regular embedding and there is a natural exact sequence of vector bundles on X:<a href="#cite_note-FOOTNOTEFulton1998Appendix_B.7.4.-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle 0\to N_{X/Y}\to N_{X/Z}\to i^{*}N_{Y/Z}\to 0.}</annotation>

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}\hookrightarrow X}">

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</math><img src="./cab27876d38eb6cf175f5e1078cf1da497881662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.037ex; height:2.509ex;" alt="{\displaystyle Y_{i}\hookrightarrow X}" loading="lazy"> are regular embeddings of codimensions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

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</math><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}">

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<annotation encoding="application/x-tex">{\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}</annotation>

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</math><img src="./357bff28e8044c8fbf6f458a58d8ed2dc7607e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.34ex; height:3.009ex;" alt="{\textstyle W:=\bigcap _{i}Y_{i}\hookrightarrow X}" loading="lazy"> is a regular embedding of codimension <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum c_{i}}">

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</math> then<a href="#cite_note-FOOTNOTEFulton1998Appendix_B.7.4.-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle N_{W/X}=\bigoplus _{i}N_{Y_{i}/X}|_{W}.}</annotation>

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In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to S}">

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</math><img src="./3496d15a6102a8883ca2f5d33afd25d2d7dc9f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle X\to S}" loading="lazy"> is a <a href="Smooth_morphism" title="Smooth morphism">smooth morphism</a>, then the normal bundle to the <a href="Diagonal_embedding" class="mw-redirect" title="Diagonal embedding">diagonal embedding</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :X\hookrightarrow X\times _{S}\cdots \times _{S}X}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta :X\hookrightarrow X\times _{S}\cdots \times _{S}X}</annotation>

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</math> (r-fold) is the direct sum of r − 1 copies of the <a href="Relative_tangent_bundle" class="mw-redirect" title="Relative tangent bundle">relative tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X/S}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{X/S}}</annotation>

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</math><img src="./b072e9ba1086923b654033497abe6559b83fb3f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.872ex; height:3.009ex;" alt="{\displaystyle T_{X/S}}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow X'}">

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</math><img src="./de97a2c0db7d9446cc24cd915c2baf4631651dfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.568ex; height:2.509ex;" alt="{\displaystyle X\hookrightarrow X'}" loading="lazy"> is a closed immersion and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y'\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y'\to Y}</annotation>

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</math><img src="./8c8a8e0534828c0a470841b66611daa2c98a0d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.972ex; height:2.509ex;" alt="{\displaystyle Y'\to Y}" loading="lazy"> is a flat morphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'=X\times _{Y}Y'}">

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<annotation encoding="application/x-tex">{\displaystyle X'=X\times _{Y}Y'}</annotation>

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</math><img src="./a4290f26cb3e5df66186ebc154611c94dfdb9664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.672ex; height:2.843ex;" alt="{\displaystyle X'=X\times _{Y}Y'}" loading="lazy">, then<a href="#cite_note-FOOTNOTEFulton1998The_first_part_of_the_proof_of_Theorem_6.5.-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle C_{X'/Y'}=C_{X/Y}\times _{X}X'.}</annotation>

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to S}">

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</math><img src="./3496d15a6102a8883ca2f5d33afd25d2d7dc9f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle X\to S}" loading="lazy"> is a <a href="Smooth_morphism" title="Smooth morphism">smooth morphism</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow Y}">

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</math><img src="./a5d77a675aa10c404233db9c24ab7b9dfbcde726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.66ex; height:2.176ex;" alt="{\displaystyle X\hookrightarrow Y}" loading="lazy"> is a regular embedding, then there is a natural exact sequence of vector bundles on X:<a href="#cite_note-FOOTNOTEFulton1998Appendix_B_7.1.-4">[4]</a>

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(which is a special case of an exact sequence for <a href="Cotangent_sheaf" title="Cotangent sheaf">cotangent sheaves</a>.)

<div class="mw-heading mw-heading3"><h3 id="Cartesian_square">Cartesian square</h3></div>

For a Cartesian square of schemes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}X'&\to &Y'\\\downarrow &&\downarrow \\X&\to &Y\end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}X'&\to &Y'\\\downarrow &&\downarrow \\X&\to &Y\end{matrix}}}</annotation>

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</math><img src="./9454ea66877cf385ab67de95634a63d7588f9ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.491ex; height:2.843ex;" alt="{\displaystyle f:X'\to X}" loading="lazy"> the vertical map, there is a closed embedding <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X'/Y'}\hookrightarrow f^{*}C_{X/Y}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X'/Y'}\hookrightarrow f^{*}C_{X/Y}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Dimension_of_components">Dimension of components</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a scheme of finite type over a field and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subset X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is of <a href="Pure_dimension" class="mw-redirect" title="Pure dimension">pure dimension</a> r; i.e., every irreducible component has dimension r, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{W/X}}">

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</math><img src="./9653cdcc3e41fa0b9b93e28f15a118d56a1ce3ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.838ex; height:3.009ex;" alt="{\displaystyle C_{W/X}}" loading="lazy"> is also of pure dimension r.<a href="#cite_note-FOOTNOTEFulton1998Appendix_B._6.6.-5">[5]</a> (This can be seen as a consequence of <a href="#Deformation_to_the_normal_cone">#Deformation to the normal cone</a>.) This property is a key to an application in intersection theory: given a pair of closed subschemes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,X}">

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</math><img src="./a993dff62521aebb4c9a345c27e8d960acabb25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.801ex; height:2.509ex;" alt="{\displaystyle V,X}" loading="lazy"> in some ambient space, while the <a href="Scheme-theoretic_intersection" title="Scheme-theoretic intersection">scheme-theoretic intersection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\cap X}">

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</math><img src="./4fcea381b740092f05a11cae35be769a26160415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.35ex; height:2.176ex;" alt="{\displaystyle V\cap X}" loading="lazy"> has irreducible components of various dimensions, depending delicately on the positions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,X}">

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</math><img src="./a993dff62521aebb4c9a345c27e8d960acabb25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.801ex; height:2.509ex;" alt="{\displaystyle V,X}" loading="lazy">, the normal cone to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\cap X}">

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<annotation encoding="application/x-tex">{\displaystyle V\cap X}</annotation>

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</math><img src="./4fcea381b740092f05a11cae35be769a26160415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.35ex; height:2.176ex;" alt="{\displaystyle V\cap X}" loading="lazy"> is of pure dimension.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\hookrightarrow X}">

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<annotation encoding="application/x-tex">{\displaystyle D\hookrightarrow X}</annotation>

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</math><img src="./70addbec0fff30ae1f932e16d4091ac11590a282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.811ex; height:2.176ex;" alt="{\displaystyle D\hookrightarrow X}" loading="lazy"> be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is<a href="#cite_note-FOOTNOTEFulton1998Appendix_B.6.2.-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}.}">

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<annotation encoding="application/x-tex">{\displaystyle N_{D/X}={\mathcal {O}}_{D}(D):={\mathcal {O}}_{X}(D)|_{D}.}</annotation>

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</math>

<div class="mw-heading mw-heading3"><h3 id="Non-regular_Embedding">Non-regular Embedding</h3></div>

Consider the non-regular embedding<a href="#cite_note-:0-7">[7]</a>: 4–5

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<annotation encoding="application/x-tex">{\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)\to \mathbb {A} ^{3}}</annotation>

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</math>

then, we can compute the normal cone by first observing

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I&=(xz,yz)\\I^{2}&=(x^{2}z^{2},xyz^{2},y^{2}z^{2})\\\end{aligned}}}</annotation>

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</math>

If we make the auxiliary variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=xz}">

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<annotation encoding="application/x-tex">{\displaystyle a=xz}</annotation>

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</math><img src="./09873d1c8908667fa9e2e3625113f5a74abbcc07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.746ex; height:1.676ex;" alt="{\displaystyle a=xz}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=yz}">

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<annotation encoding="application/x-tex">{\displaystyle b=yz}</annotation>

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</math><img src="./54ca17feeb0a688fea1c2536a2185e5706aeffa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.34ex; height:2.509ex;" alt="{\displaystyle b=yz}" loading="lazy"> we get the relation

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<annotation encoding="application/x-tex">{\displaystyle ya-xb=0.}</annotation>

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We can use this to give a presentation of the normal cone as the relative spectrum

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<annotation encoding="application/x-tex">{\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}_{X}\left({\frac {{\mathcal {O}}_{X}[a,b]}{(ya-xb)}}\right)}</annotation>

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</math>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{3}}</annotation>

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</math><img src="./d1f9ca25a690f22a1ea88900611588eb384fc90c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{3}}" loading="lazy"> is affine, we can just write out the relative spectrum as the affine scheme <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z][a,b]}{(xz,yz,ya-xb)}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X}\mathbb {A} ^{3}={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z][a,b]}{(xz,yz,ya-xb)}}\right)}</annotation>

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</math> giving us the normal cone.

<div class="mw-heading mw-heading4"><h4 id="Geometry_of_this_normal_cone">Geometry of this normal cone</h4></div>

The normal cone's geometry can be further explored by looking at the fibers for various closed points of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Note that geometrically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the union of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

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</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy">-plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-axis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=H\cup L}">

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<annotation encoding="application/x-tex">{\displaystyle X=H\cup L}</annotation>

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</math> so the points of interest are smooth points on the plane, smooth points on the axis, and the point on their intersection. Any smooth point on the plane is given by a map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x\mapsto z_{1}&y\mapsto z_{2}&z\mapsto 0\end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x\mapsto z_{1}&y\mapsto z_{2}&z\mapsto 0\end{matrix}}}</annotation>

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</math> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1},z_{2}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle z_{1},z_{2}\in \mathbb {C} }</annotation>

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</math><img src="./a8222f3146c75ff25584472f414669d569bc6138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.823ex; height:2.509ex;" alt="{\displaystyle z_{1},z_{2}\in \mathbb {C} }" loading="lazy"> and either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}\neq 0}</annotation>

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</math><img src="./5f5f47c07a9f808f1d385c2415142187802b741b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.396ex; height:2.676ex;" alt="{\displaystyle z_{1}\neq 0}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle z_{2}\neq 0}</annotation>

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</math><img src="./40b6e97e9bb66c69d7c181365c7ce75103b5b058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.396ex; height:2.676ex;" alt="{\displaystyle z_{2}\neq 0}" loading="lazy">. Since it is arbitrary which point we take, for convenience let us assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}\neq 0,z_{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}\neq 0,z_{2}=0}</annotation>

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</math><img src="./6d7821ee687685f7ff19d2042b8d4868b3f51772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.827ex; height:2.676ex;" alt="{\displaystyle z_{1}\neq 0,z_{2}=0}" loading="lazy">. Hence the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}\mathbb {A} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X}\mathbb {A} ^{3}}</annotation>

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</math><img src="./7e59be439ecd5d62b46bb9a031257162aff9b55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.026ex; height:3.009ex;" alt="{\displaystyle C_{X}\mathbb {A} ^{3}}" loading="lazy"> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=(z_{1},0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle p=(z_{1},0,0)}</annotation>

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</math><img src="./4aca149102f7aac60a3c12f0b4e8ad7d9defb76c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.695ex; height:2.843ex;" alt="{\displaystyle p=(z_{1},0,0)}" loading="lazy"> is isomorphic to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}\mathbb {A} ^{3}|_{p}\cong {\frac {\mathbb {C} [a,b]}{(z_{1}b)}}\cong \mathbb {C} [a]}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X}\mathbb {A} ^{3}|_{p}\cong {\frac {\mathbb {C} [a,b]}{(z_{1}b)}}\cong \mathbb {C} [a]}</annotation>

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</math> giving the normal cone as a one dimensional line, as expected. For a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> on the axis, this is given by a map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}x\mapsto 0&y\mapsto 0&z\mapsto z_{3}\end{matrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x\mapsto 0&y\mapsto 0&z\mapsto z_{3}\end{matrix}}}</annotation>

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</math> hence the fiber at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=(0,0,z_{3})}">

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<annotation encoding="application/x-tex">{\displaystyle q=(0,0,z_{3})}</annotation>

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</math><img src="./264fe323a0beb80632c31ee604418713e69fc658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.506ex; height:2.843ex;" alt="{\displaystyle q=(0,0,z_{3})}" loading="lazy"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}\mathbb {A} ^{3}|_{q}\cong {\frac {\mathbb {C} [a,b]}{(0)}}\cong \mathbb {C} [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X}\mathbb {A} ^{3}|_{q}\cong {\frac {\mathbb {C} [a,b]}{(0)}}\cong \mathbb {C} [a,b]}</annotation>

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</math> which gives a plane. At the origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=(0,0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle r=(0,0,0)}</annotation>

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</math><img src="./9720f52c6e72a6d5c6f40df018f11f35efd0460c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.512ex; height:2.843ex;" alt="{\displaystyle r=(0,0,0)}" loading="lazy">, the normal cone over that point is again isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [a,b]}</annotation>

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</math><img src="./ecc7268e484802789c6c7852059db2230b378757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.233ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [a,b]}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Nodal_cubic">Nodal cubic</h3></div>

For the nodal cubic curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> given by the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}+x^{2}(x-1)}">

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<annotation encoding="application/x-tex">{\displaystyle y^{2}+x^{2}(x-1)}</annotation>

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</math><img src="./16ffdb9f94193bfcb9c6b40c8125427a36bce110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.581ex; height:3.176ex;" alt="{\displaystyle y^{2}+x^{2}(x-1)}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> the point at the node, the cone has the isomorphism <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X/Y}\cong {\text{Spec}}\left(\mathbb {C} [x,y]/\left(y^{2}-x^{2}\right)\right)}">

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<annotation encoding="application/x-tex">{\displaystyle C_{X/Y}\cong {\text{Spec}}\left(\mathbb {C} [x,y]/\left(y^{2}-x^{2}\right)\right)}</annotation>

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</math> showing the normal cone has more components than the scheme it lies over.

<div class="mw-heading mw-heading2"><h2 id="Deformation_to_the_normal_cone">Deformation to the normal cone</h2></div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle i:X\to Y}</annotation>

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</math><img src="./976e2f3fea37ed92af6613865d979dc22b272d31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.107ex; height:2.176ex;" alt="{\displaystyle i:X\to Y}" loading="lazy"> is an embedding. This can be deformed to the embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> inside the normal cone <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{X/Y}}</annotation>

</semantics>

</math><img src="./2a26e1f264bdf87622683f572c00ce1943ee560b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.37ex; height:3.009ex;" alt="{\displaystyle C_{X/Y}}" loading="lazy"> (as the zero section) in the following sense:<a href="#cite_note-:0-7">[7]</a>: 6 there is a flat family <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :M_{X/Y}^{o}\to \mathbb {P} ^{1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :M_{X/Y}^{o}\to \mathbb {P} ^{1}}</annotation>

</semantics>

</math> with generic fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> and special fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{X/Y}}</annotation>

</semantics>

</math><img src="./2a26e1f264bdf87622683f572c00ce1943ee560b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.37ex; height:3.009ex;" alt="{\displaystyle C_{X/Y}}" loading="lazy"> such that there exists a family of closed embeddings <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {P} ^{1}\hookrightarrow M_{X/Y}^{o}}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {P} ^{1}\hookrightarrow M_{X/Y}^{o}}</annotation>

</semantics>

</math> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./5e38808c13ecc929e128079b70c1642554e2a4b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{1}}" loading="lazy"> such that

<ol><li>Over any point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {P} ^{1}-\{0\}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {P} ^{1}-\{0\}}</annotation>

</semantics>

</math><img src="./59d3b53c6b2e8c59b7847d66eab500f88f0c223a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:3.176ex;" alt="{\displaystyle t\in \mathbb {P} ^{1}-\{0\}}" loading="lazy"> the associated embeddings are an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \{t\}\hookrightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times \{t\}\hookrightarrow Y}</annotation>

</semantics>

</math><img src="./a0174ce201701762bfd99223098fc039274e72ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.665ex; height:2.843ex;" alt="{\displaystyle X\times \{t\}\hookrightarrow Y}" loading="lazy"></li>

<li>The fiber over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\in \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./918936b88e52f3353f18fc16d4944572adc17f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 0\in \mathbb {P} ^{1}}" loading="lazy"> is the embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow C_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\hookrightarrow C_{X/Y}}</annotation>

</semantics>

</math><img src="./3e3916e9a9ea56934855953c5cd6c66aeea0c057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.257ex; height:3.009ex;" alt="{\displaystyle X\hookrightarrow C_{X/Y}}" loading="lazy"> given by the zero section.</li></ol>

This construction defines a tool analogous to differential topology where non-transverse intersections are performed in a tubular neighborhood of the intersection. Now, the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with a cycle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> can be given as the pushforward of an intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{X/Y}}</annotation>

</semantics>

</math><img src="./2a26e1f264bdf87622683f572c00ce1943ee560b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.37ex; height:3.009ex;" alt="{\displaystyle C_{X/Y}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Construction">Construction</h3></div>

One application of this is to define intersection products in the <a href="Chow_ring" class="mw-redirect" title="Chow ring">Chow ring</a>. Suppose that X and V are closed subschemes of Y with intersection W, and we wish to define the intersection product of X and V in the Chow ring of Y. Deformation to the normal cone in this case means that we replace the embeddings of X and W in Y and V by their normal cones CY(X) and CW(V), so that we want to find the product of X and CWV in CXY.

This can be much easier: for example, if X is <a href="Regularly_embedded" class="mw-redirect" title="Regularly embedded">regularly embedded</a> in Y then its normal cone is a vector bundle, so we are reduced to the problem of finding the intersection product of a subscheme CWV of a vector bundle CXY with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to CWV.

Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :M\to Y\times \mathbb {P} ^{1}}</annotation>

</semantics>

</math>

be the blow-up of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\times \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\times \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./16c7a04f20e62766d30c808bf0e2117ed7122fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.088ex; height:2.676ex;" alt="{\displaystyle Y\times \mathbb {P} ^{1}}" loading="lazy"> along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times 0}</annotation>

</semantics>

</math><img src="./002973271af61a1878f58ac166f22aed9cda17c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle X\times 0}" loading="lazy">. The exceptional divisor is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C_{X}Y}}=\mathbb {P} (C_{X}Y\oplus 1)}">

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {C_{X}Y}}=\mathbb {P} (C_{X}Y\oplus 1)}</annotation>

</semantics>

</math><img src="./b849d878bd1fa8c1b5586119cba9e59cac2f94fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.856ex; height:3.509ex;" alt="{\displaystyle {\overline {C_{X}Y}}=\mathbb {P} (C_{X}Y\oplus 1)}" loading="lazy">, the projective completion of the normal cone; for the notation used here see <a href="Cone_(algebraic_geometry)#Properties" title="Cone (algebraic geometry)">Cone (algebraic geometry) § Properties</a>. The normal cone <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}Y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{X}Y}</annotation>

</semantics>

</math><img src="./1df2acce4454d39caee92d406f8bee54eff37085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.067ex; height:2.509ex;" alt="{\displaystyle C_{X}Y}" loading="lazy"> is an open subscheme of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C_{X}Y}}}">

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {C_{X}Y}}}</annotation>

</semantics>

</math><img src="./9dfff6a55b5941db25135797b40b41b627d38ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.457ex; height:3.343ex;" alt="{\displaystyle {\overline {C_{X}Y}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is embedded as a zero-section into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}Y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{X}Y}</annotation>

</semantics>

</math><img src="./1df2acce4454d39caee92d406f8bee54eff37085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.067ex; height:2.509ex;" alt="{\displaystyle C_{X}Y}" loading="lazy">.

Now, we note:

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho :M\to \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./27cb058c358bdf34aa8de597ee2e47c42534df07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.67ex; height:3.176ex;" alt="{\displaystyle \rho :M\to \mathbb {P} ^{1}}" loading="lazy">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> followed by projection, is flat.</li>

<li>There is an induced closed embedding <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {i}}:X\times \mathbb {P} ^{1}\hookrightarrow M}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {i}}:X\times \mathbb {P} ^{1}\hookrightarrow M}</annotation>

</semantics>

</math> that is a morphism over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./5e38808c13ecc929e128079b70c1642554e2a4b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{1}}" loading="lazy">.</li>

<li>M is trivial away from zero; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{-1}(\mathbb {P} ^{1}-0)=Y\times (\mathbb {P} ^{1}-0)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{-1}(\mathbb {P} ^{1}-0)=Y\times (\mathbb {P} ^{1}-0)}</annotation>

</semantics>

</math><img src="./49ccc9f07910b2f72c5b495a329f64c8ad0c91c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.82ex; height:3.176ex;" alt="{\displaystyle \rho ^{-1}(\mathbb {P} ^{1}-0)=Y\times (\mathbb {P} ^{1}-0)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {i}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {i}}}</annotation>

</semantics>

</math><img src="./ca3c4238d511246d8caeb43282344e70345ca401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\widetilde {i}}}" loading="lazy"> restricts to the trivial embedding <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times (\mathbb {P} ^{1}-0)\hookrightarrow Y\times (\mathbb {P} ^{1}-0).}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times (\mathbb {P} ^{1}-0)\hookrightarrow Y\times (\mathbb {P} ^{1}-0).}</annotation>

</semantics>

</math></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ^{-1}(0)}</annotation>

</semantics>

</math><img src="./6b799b82cb8d111e4b2be4238de75901f5197410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.506ex; height:3.176ex;" alt="{\displaystyle \rho ^{-1}(0)}" loading="lazy"> as the divisor is the sum <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C_{X}Y}}+{\widetilde {Y}}}">

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {C_{X}Y}}+{\widetilde {Y}}}</annotation>

</semantics>

</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {Y}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {Y}}}</annotation>

</semantics>

</math><img src="./2b90c60496c04992e056b6e745998ddd3d97ab96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle {\widetilde {Y}}}" loading="lazy"> is the blow-up of Y along X and is viewed as an effective Cartier divisor.</li>

<li>As divisors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C_{X}Y}}}">

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {C_{X}Y}}}</annotation>

</semantics>

</math><img src="./9dfff6a55b5941db25135797b40b41b627d38ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.457ex; height:3.343ex;" alt="{\displaystyle {\overline {C_{X}Y}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {Y}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {Y}}}</annotation>

</semantics>

</math><img src="./2b90c60496c04992e056b6e745998ddd3d97ab96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle {\widetilde {Y}}}" loading="lazy"> intersect at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (C)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (C)}</annotation>

</semantics>

</math><img src="./54309ea51ed1ef8fd6d3cede6fee28bebca689ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.996ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (C)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (C)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (C)}</annotation>

</semantics>

</math><img src="./54309ea51ed1ef8fd6d3cede6fee28bebca689ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.996ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (C)}" loading="lazy"> sits at infinity in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C_{X}Y}}}">

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {C_{X}Y}}}</annotation>

</semantics>

</math><img src="./9dfff6a55b5941db25135797b40b41b627d38ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.457ex; height:3.343ex;" alt="{\displaystyle {\overline {C_{X}Y}}}" loading="lazy">.</li></ol>

Item 1 is clear (check torsion-free-ness). In general, given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subset Y}</annotation>

</semantics>

</math><img src="./69f783c117107950a3a84e75299d8a4dcb342449.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X\subset Y}" loading="lazy">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bl} _{V}X\subset \operatorname {Bl} _{V}Y}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bl} _{V}X\subset \operatorname {Bl} _{V}Y}</annotation>

</semantics>

</math><img src="./344fcde558cce1026b2a0d1183bd80e0d2733a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.203ex; height:2.509ex;" alt="{\displaystyle \operatorname {Bl} _{V}X\subset \operatorname {Bl} _{V}Y}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times 0}</annotation>

</semantics>

</math><img src="./002973271af61a1878f58ac166f22aed9cda17c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle X\times 0}" loading="lazy"> is already an effective Cartier divisor on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./a0cbf323972b185a2d5cabd51b29cfe1525849bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.295ex; height:2.676ex;" alt="{\displaystyle X\times \mathbb {P} ^{1}}" loading="lazy">, we get

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {P} ^{1}=\operatorname {Bl} _{X\times 0}X\times \mathbb {P} ^{1}\hookrightarrow M,}</annotation>

</semantics>

</math>

yielding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {i}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {i}}}</annotation>

</semantics>

</math><img src="./ca3c4238d511246d8caeb43282344e70345ca401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.676ex;" alt="{\displaystyle {\widetilde {i}}}" loading="lazy">. Item 3 follows from the fact the blowdown map π is an isomorphism away from the center <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times 0}</annotation>

</semantics>

</math><img src="./002973271af61a1878f58ac166f22aed9cda17c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle X\times 0}" loading="lazy">. The last two items are seen from explicit local computation. <a href="Q.E.D." title="Q.E.D.">Q.E.D.</a>

Now, the last item in the previous paragraph implies that the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times 0}</annotation>

</semantics>

</math><img src="./002973271af61a1878f58ac166f22aed9cda17c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle X\times 0}" loading="lazy"> in M does not intersect <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {Y}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {Y}}}</annotation>

</semantics>

</math><img src="./2b90c60496c04992e056b6e745998ddd3d97ab96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle {\widetilde {Y}}}" loading="lazy">. Thus, one gets the deformation of i to the zero-section embedding of X into the normal cone.

<div class="mw-heading mw-heading2"><h2 id="Intrinsic_normal_cone">Intrinsic normal cone</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Intrinsic_normal_bundle">Intrinsic normal bundle</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a <a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a> locally of finite type over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {L}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {L}}_{X}}</annotation>

</semantics>

</math><img src="./b5e896b447bdb6635b0405e4930ffd0ebebddc56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.241ex; height:2.509ex;" alt="{\displaystyle {\textbf {L}}_{X}}" loading="lazy"> denotes the <a href="Cotangent_complex" title="Cotangent complex">cotangent complex</a> of X relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, then the intrinsic normal bundle<a href="#cite_note-:1-8">[8]</a>: 27 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Quotient_stack" title="Quotient stack">quotient stack</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{X}:=h^{1}/h^{0}({\textbf {L}}_{X,{\text{fppf}}}^{\vee })}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{X}:=h^{1}/h^{0}({\textbf {L}}_{X,{\text{fppf}}}^{\vee })}</annotation>

</semantics>

</math> which is the stack of fppf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {L}}_{X}^{\vee ,0}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {L}}_{X}^{\vee ,0}}</annotation>

</semantics>

</math><img src="./399de462ccd05c04c6100c8aafadca8c9c6a6af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.216ex; height:3.509ex;" alt="{\displaystyle {\textbf {L}}_{X}^{\vee ,0}}" loading="lazy">-<a href="Torsor_(algebraic_geometry)" title="Torsor (algebraic geometry)">torsors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {L}}_{X}^{\vee ,1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {L}}_{X}^{\vee ,1}}</annotation>

</semantics>

</math><img src="./8bacca79b9ee359d652ba0819a52ad1a4f5edbf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.216ex; height:3.509ex;" alt="{\displaystyle {\textbf {L}}_{X}^{\vee ,1}}" loading="lazy">. A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Properties_of_intrinsic_normal_bundle">Properties of intrinsic normal bundle</h4></div>

More concretely, suppose there is an étale morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to X}</annotation>

</semantics>

</math><img src="./e0e8241962e6e1883ae50b5a471db0ecff377950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.377ex; height:2.176ex;" alt="{\displaystyle U\to X}" loading="lazy"> from an affine finite-type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> together with a locally closed immersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:U\to M}</annotation>

</semantics>

</math><img src="./bb95879a34ba79dcbae9912630ddd5864e5ef340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.055ex; height:2.509ex;" alt="{\displaystyle f:U\to M}" loading="lazy"> into a smooth affine finite-type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Then one can show <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{X}|_{U}=[N_{U/M}/f^{*}T_{M}]}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{X}|_{U}=[N_{U/M}/f^{*}T_{M}]}</annotation>

</semantics>

</math> meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}_{U}\to {\mathcal {T}}_{M}|_{U}\to {\mathcal {N}}_{U/M}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{U}\to {\mathcal {T}}_{M}|_{U}\to {\mathcal {N}}_{U/M}}</annotation>

</semantics>

</math> to be exact on the right hand side. Moreover, for special cases discussed below, we are now considering the quotient as a continuation of the previous sequence as a triangle in some triangulated category. This is because the local stack quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [N_{U/M}/f^{*}T_{M}]}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [N_{U/M}/f^{*}T_{M}]}</annotation>

</semantics>

</math><img src="./4ed6c54772f87e5d19ab9f86314cedcb181ddf30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.055ex; height:3.176ex;" alt="{\displaystyle [N_{U/M}/f^{*}T_{M}]}" loading="lazy"> can be interpreted as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B{\mathcal {T}}_{U}={\mathcal {T}}_{U}[+1]}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B{\mathcal {T}}_{U}={\mathcal {T}}_{U}[+1]}</annotation>

</semantics>

</math> in certain cases.

<div class="mw-heading mw-heading3"><h3 id="Normal_cone">Normal cone</h3></div>

The intrinsic normal cone to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {C}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X}}</annotation>

</semantics>

</math><img src="./5799856ad4d5ef1e8f4285b7bd0585bae6ec076d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {C}}_{X}}" loading="lazy">,<a href="#cite_note-:1-8">[8]</a>: 29 is then defined by replacing the normal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{U/M}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N_{U/M}}</annotation>

</semantics>

</math><img src="./93e6090d91a7a8cfee7382b0a710bed8b1b08cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.908ex; height:3.009ex;" alt="{\displaystyle N_{U/M}}" loading="lazy"> with the normal cone <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{U/M}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{U/M}}</annotation>

</semantics>

</math><img src="./fb23a7feb2e6982c0223b2546474fb1442b51cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.703ex; height:3.009ex;" alt="{\displaystyle C_{U/M}}" loading="lazy">; i.e.,

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X}|_{U}=[C_{U/M}/f^{*}T_{M}].}</annotation>

</semantics>

</math>

Example: One has that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a local complete intersection if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}}</annotation>

</semantics>

</math><img src="./8a46ed8bfc2d6d1af8d4c153e61d8922c88c3c5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.722ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}}" loading="lazy">. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Smooth_stack" class="mw-redirect" title="Smooth stack">smooth</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}=BT_{X}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}=BT_{X}}</annotation>

</semantics>

</math><img src="./c54d8b4b5b5da3f526073dd470e59626e889867a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.574ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {C}}_{X}={\mathfrak {N}}_{X}=BT_{X}}" loading="lazy"> is the <a href="Classifying_stack" class="mw-redirect" title="Classifying stack">classifying stack</a> of the tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation>

</semantics>

</math><img src="./a94e80072681ec0d850d5d10601d58032619803e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}" loading="lazy">, which is a commutative group scheme over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

More generally, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\to Y}</annotation>

</semantics>

</math><img src="./290b16963d52e4a7995aae01ee854b97a6ea10c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\to Y}" loading="lazy"> is a Deligne-Mumford Type (DM-type) morphism of Artin Stacks which is locally of finite type. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {C}}_{X/Y}\subseteq {\mathfrak {N}}_{X/Y}}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X/Y}\subseteq {\mathfrak {N}}_{X/Y}}</annotation>

</semantics>

</math><img src="./e3bc346b5e939eab939df98b4c22a160170e0170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.873ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {C}}_{X/Y}\subseteq {\mathfrak {N}}_{X/Y}}" loading="lazy"> is characterised as the closed substack such that, for any étale map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to X}</annotation>

</semantics>

</math><img src="./e0e8241962e6e1883ae50b5a471db0ecff377950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.377ex; height:2.176ex;" alt="{\displaystyle U\to X}" loading="lazy"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to X\to Y}</annotation>

</semantics>

</math><img src="./9f2fef6d014877bdc25f4ca91403bfb87d2d278f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.764ex; height:2.176ex;" alt="{\displaystyle U\to X\to Y}" loading="lazy"> factors through some smooth map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\to Y}</annotation>

</semantics>

</math><img src="./41fed9a6ea0a33f66cec940cc6c2431a078777be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.83ex; height:2.176ex;" alt="{\displaystyle M\to Y}" loading="lazy"> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{Y}^{n}\to Y}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{Y}^{n}\to Y}</annotation>

</semantics>

</math><img src="./734af127a153bbe23aa393bb5594c2885bf23d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.552ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{Y}^{n}\to Y}" loading="lazy">), the pullback is:

<msub>

</mrow>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {C}}_{X/Y}|_{U}=[C_{U/M}/T_{M/Y}|_{U}].}</annotation>

</semantics>

</math>

<ul><li>Abelian cone</li>

<li><a href="Segre_class" title="Segre class">Segre class</a></li>

<li><a href="Residual_intersection" title="Residual intersection">Residual intersection</a></li>

<li><a href="Virtual_fundamental_class" title="Virtual fundamental class">Virtual fundamental class</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-FOOTNOTEHartshorne1977Ch._III,_Exercise_9.7.-1"><a href="#cite_ref-FOOTNOTEHartshorne1977Ch._III,_Exercise_9.7._1-0">^</a> <a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, p. Ch. III, Exercise 9.7..

</li>

<li id="cite_note-FOOTNOTEFulton1998Appendix_B.7.4.-2">^ <a href="#cite_ref-FOOTNOTEFulton1998Appendix_B.7.4._2-0">a</a> <a href="#cite_ref-FOOTNOTEFulton1998Appendix_B.7.4._2-1">b</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, p. Appendix B.7.4..

</li>

<li id="cite_note-FOOTNOTEFulton1998The_first_part_of_the_proof_of_Theorem_6.5.-3"><a href="#cite_ref-FOOTNOTEFulton1998The_first_part_of_the_proof_of_Theorem_6.5._3-0">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, p. The first part of the proof of Theorem 6.5..

</li>

<li id="cite_note-FOOTNOTEFulton1998Appendix_B_7.1.-4"><a href="#cite_ref-FOOTNOTEFulton1998Appendix_B_7.1._4-0">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, p. Appendix B 7.1..

</li>

<li id="cite_note-FOOTNOTEFulton1998Appendix_B._6.6.-5"><a href="#cite_ref-FOOTNOTEFulton1998Appendix_B._6.6._5-0">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, p. Appendix B. 6.6..

</li>

<li id="cite_note-FOOTNOTEFulton1998Appendix_B.6.2.-6"><a href="#cite_ref-FOOTNOTEFulton1998Appendix_B.6.2._6-0">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, p. Appendix B.6.2..

</li>

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</style><cite id="CITEREFBattistellaCarocciManolache2020" class="citation journal cs1">Battistella, Luca; Carocci, Francesca; Manolache, Cristina (2020-04-09). <a rel="nofollow" class="external text" href="https://doi.org/10.3842%2FSIGMA.2020.026">"Virtual classes for the working mathematician"</a>. Symmetry, Integrability and Geometry: Methods and Applications. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1804.06048">1804.06048</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3842%2FSIGMA.2020.026">10.3842/SIGMA.2020.026</a>.</cite>

</li>

<li id="cite_note-:1-8">^ <a href="#cite_ref-:1_8-0">a</a> <a href="#cite_ref-:1_8-1">b</a> <cite id="CITEREFBehrendFantechi1997" class="citation journal cs1">Behrend, K.; Fantechi, B. (1997-03-19). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/alg-geom/9601010">alg-geom/9601010</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002220050136">10.1007/s002220050136</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18533009">18533009</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBehrendFantechi1997" class="citation journal cs1">Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/alg-geom/9601010">alg-geom/9601010</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002220050136">10.1007/s002220050136</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18533009">18533009</a>.</cite></li>

<li><cite id="CITEREFFulton1998" class="citation cs2">Fulton, William (1998), Intersection theory, <a href="Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete" title="Ergebnisse der Mathematik und ihrer Grenzgebiete">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-62046-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1644323">1644323</a></cite></li>

<li><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <a href="Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90244-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://mathoverflow.net/questions/343625/fibers-of-the-normal-cone?rq=1">Fibers of the normal cone</a></li></ul></div><div><div class="zim-footer">

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