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Intersection theory

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Intersection_(set_theory)" title="Intersection (set theory)">Intersection (set theory)</a> or <a href="Intersectionality" title="Intersectionality">Intersectionality</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, intersection theory is one of the main branches of <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, where it gives information about the <a href="Intersection" title="Intersection">intersection</a> of two <a href="Subvariety_(algebraic_geometry)" class="mw-redirect" title="Subvariety (algebraic geometry)">subvarieties</a> of a given variety.<a href="#cite_note-FOOTNOTEEisenbudHarris201614-1">[1]</a> The theory for varieties is older, with roots in <a href="B%C3%A9zout's_theorem" title="Bézout's theorem">Bézout's theorem</a> on curves and <a href="Elimination_theory" title="Elimination theory">elimination theory</a>. On the other hand, the topological theory more quickly reached a definitive form.

There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, <a href="Gromov%E2%80%93Witten_theory" class="mw-redirect" title="Gromov–Witten theory">Gromov–Witten theory</a> and the extension of intersection theory from <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> to <a href="Stack_(mathematics)" title="Stack (mathematics)">stacks</a>.<a href="#cite_note-FOOTNOTEEisenbudHarris20162-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Topological_intersection_form">Topological intersection form</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="%CE%95-quadratic_form#Manifolds" title="Ε-quadratic form">ε-quadratic form § Manifolds</a>, and <a href="Intersection_form_(4-manifold)" class="mw-redirect" title="Intersection form (4-manifold)">Intersection form (4-manifold)</a></div>

For a <a href="Connected_space" title="Connected space">connected</a> <a href="Orientability" title="Orientability">oriented manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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"> of <a href="Dimension_of_a_manifold" class="mw-redirect" title="Dimension of a manifold">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

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

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</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy"> the intersection form is defined on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th <a href="Cohomology_group" class="mw-redirect" title="Cohomology group">cohomology group</a> (what is usually called the 'middle dimension') by the evaluation of the <a href="Cup_product" title="Cup product">cup product</a> on the <a href="Fundamental_class" title="Fundamental class">fundamental class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M]}">

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

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</math><img src="./e5ca74e595b2281c0aef1897ecafa282d1f182e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.736ex; height:2.843ex;" alt="{\displaystyle [M]}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2n}(M,\partial M)}">

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<annotation encoding="application/x-tex">{\displaystyle H_{2n}(M,\partial M)}</annotation>

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</math><img src="./46e6ecc08b30cb33e161702b7484baed1ae7fb8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.017ex; height:2.843ex;" alt="{\displaystyle H_{2n}(M,\partial M)}" loading="lazy">. Stated precisely, there is a <a href="Bilinear_form" title="Bilinear form">bilinear form</a>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{M}\colon H^{n}(M,\partial M)\times H^{n}(M,\partial M)\to \mathbf {Z} }</annotation>

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</math><img src="./ac049c37858cc3f00ae99c97aff5e45f569ff5d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.172ex; height:2.843ex;" alt="{\displaystyle \lambda _{M}\colon H^{n}(M,\partial M)\times H^{n}(M,\partial M)\to \mathbf {Z} }" loading="lazy"></dd></dl>

given by

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{M}(a,b)=\langle a\smile b,[M]\rangle \in \mathbf {Z} }</annotation>

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</math><img src="./5224441f15083b73af07a158d2c8f3ddc7e2ce70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.379ex; height:2.843ex;" alt="{\displaystyle \lambda _{M}(a,b)=\langle a\smile b,[M]\rangle \in \mathbf {Z} }" loading="lazy"></dd></dl>

with

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{M}(a,b)=(-1)^{n}\lambda _{M}(b,a)\in \mathbf {Z} .}</annotation>

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</math><img src="./13520cffe5226d07c80153cfffe39f9afcae3654.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.988ex; height:2.843ex;" alt="{\displaystyle \lambda _{M}(a,b)=(-1)^{n}\lambda _{M}(b,a)\in \mathbf {Z} .}" loading="lazy"></dd></dl>

This is a <a href="Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric form</a> for n even (so 2n = 4k <a href="Doubly_even" class="mw-redirect" title="Doubly even">doubly even</a>), in which case the <a href="Signature_(topology)" title="Signature (topology)">signature</a> of M is defined to be the signature of the form, and an <a href="Alternating_form" class="mw-redirect" title="Alternating form">alternating form</a> for n odd (so 2n = 4k + 2 is <a href="Singly_even" class="mw-redirect" title="Singly even">singly even</a>). These can be referred to uniformly as <a href="%CE%95-symmetric_form" class="mw-redirect" title="Ε-symmetric form">ε-symmetric forms</a>, where ε = (−1)n = ±1 respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an <a href="%CE%95-quadratic_form" title="Ε-quadratic form">ε-quadratic form</a>, though this requires additional data such as a <a href="Framed_manifold" class="mw-redirect" title="Framed manifold">framing</a> of the tangent bundle. It is possible to drop the orientability condition and work with Z/2Z coefficients instead.

These forms are important <a href="Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariants</a>. For example, a theorem of <a href="Michael_Freedman" title="Michael Freedman">Michael Freedman</a> states that <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> <a href="Compact_space" title="Compact space">compact</a> <a href="4-manifold" title="4-manifold">4-manifolds</a> are (almost) determined by <a href="Intersection_form_(4-manifold)" class="mw-redirect" title="Intersection form (4-manifold)">their intersection forms</a> up to <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a>.

By <a href="Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a>, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then λM (a, b) is the oriented intersection number of A and B, which is well-defined because since dimensions of A and B sum to the total dimension of M they generically intersect at isolated points. This explains the terminology intersection form.

<div class="mw-heading mw-heading2"><h2 id="Intersection_theory_in_algebraic_geometry">Intersection theory in algebraic geometry</h2></div>

<a href="William_Fulton_(mathematician)" title="William Fulton (mathematician)">William Fulton</a> in Intersection Theory (1984) writes

<blockquote>... if A and B are subvarieties of a non-singular variety X, the intersection product A · B should be an equivalence class of algebraic cycles closely related to the geometry of how A ∩ B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e. dim(A ∩ B) = dim A + dim B − dim X, then A · B is a linear combination of the irreducible components of A ∩ B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A · B is represented by the top <a href="Chern_class" title="Chern class">Chern class</a> of the <a href="Normal_bundle" title="Normal bundle">normal bundle</a> of A in X.</blockquote>

To give a definition, in the general case, of the intersection multiplicity was the major concern of <a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a>'s 1946 book Foundations of Algebraic Geometry. Work in the 1920s of <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">B. L. van der Waerden</a> had already addressed the question; in the <a href="Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a> the ideas were well known, but foundational questions were not addressed in the same spirit.

<div class="mw-heading mw-heading3"><h3 id="Moving_cycles">Moving cycles</h3></div>

A well-working machinery of intersecting <a href="Algebraic_cycle" title="Algebraic cycle">algebraic cycles</a> V and W requires more than taking just the set-theoretic intersection V ∩ W of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted V · W, should consist of the set-theoretic intersection of the two subvarieties. However cycles may be in bad position, e.g. two parallel lines in the plane, or a plane containing a line (intersecting in 3-space). In both cases the intersection should be a point, because, again, if one cycle is moved, this would be the intersection. The intersection of two cycles V and W is called proper if the <a href="Codimension" title="Codimension">codimension</a> of the (set-theoretic) intersection V ∩ W is the sum of the codimensions of V and W, respectively, i.e. the "expected" value.

Therefore, the concept of moving cycles using appropriate <a href="Equivalence_relations_on_algebraic_cycles" class="mw-redirect" title="Equivalence relations on algebraic cycles">equivalence relations on algebraic cycles</a> is used. The equivalence must be broad enough that given any two cycles V and W, there are equivalent cycles V′ and W′ such that the intersection V′ ∩ W′ is proper. Of course, on the other hand, for a second equivalent V′′ and W′′, V′ ∩ W′ needs to be equivalent to V′′ ∩ W′′.

For the purposes of intersection theory, rational equivalence is the most important one. Briefly, two r-dimensional cycles on a variety X are rationally equivalent if there is a rational function f on a (r + 1)-dimensional subvariety Y, i.e. an element of the <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a> k(Y) or equivalently a function f  : Y → P1, such that V − W = f −1(0) − f −1(∞), where f −1(⋅) is counted with multiplicities. Rational equivalence accomplishes the needs sketched above.

<div class="mw-heading mw-heading3"><h3 id="Intersection_multiplicities">Intersection multiplicities</h3></div>

The guiding principle in the definition of <a href="Intersection_multiplicity" class="mw-redirect" title="Intersection multiplicity">intersection multiplicities</a> of cycles is continuity in a certain sense. Consider the following elementary example: the intersection of a parabola y = x2 and an axis y = 0 should be 2 · (0, 0), because if one of the cycles moves (yet in an undefined sense), there are precisely two intersection points which both converge to (0, 0) when the cycles approach the depicted position. (The picture is misleading insofar as the apparently empty intersection of the parabola and the line y = −3 is empty, because only the real solutions of the equations are depicted).

The first fully satisfactory definition of intersection multiplicities was given by <a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre</a>: Let the ambient variety X be smooth (or all local rings <a href="Regular_local_ring" title="Regular local ring">regular</a>). Further let V and W be two (irreducible reduced closed) subvarieties, such that their intersection is proper. The construction is local, therefore the varieties may be represented by two ideals I and J in the coordinate ring of X. Let Z be an irreducible component of the set-theoretic intersection V ∩ W and z its <a href="Generic_point" title="Generic point">generic point</a>. The multiplicity of Z in the intersection product V · W is defined by

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<annotation encoding="application/x-tex">{\displaystyle \mu (Z;V,W):=\sum _{i=0}^{\infty }(-1)^{i}{\text{length}}_{{\mathcal {O}}_{X,z}}{\text{Tor}}_{i}^{{\mathcal {O}}_{X,z}}({\mathcal {O}}_{X,z}/I,{\mathcal {O}}_{X,z}/J),}</annotation>

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</math><img src="./bf0d568cba394f0f6da089926da33437a3a2a951.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.496ex; height:6.843ex;" alt="{\displaystyle \mu (Z;V,W):=\sum _{i=0}^{\infty }(-1)^{i}{\text{length}}_{{\mathcal {O}}_{X,z}}{\text{Tor}}_{i}^{{\mathcal {O}}_{X,z}}({\mathcal {O}}_{X,z}/I,{\mathcal {O}}_{X,z}/J),}" loading="lazy"></dd></dl>

the alternating sum over the <a href="Length_of_a_module" title="Length of a module">length</a> over the local ring of X in z of <a href="Tor_functor" title="Tor functor">torsion</a> groups of the factor rings corresponding to the subvarieties. This expression is sometimes referred to as Serre's Tor-formula.

Remarks:

<ul><li>The first summand, the length of

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<annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {O}}_{X,z}/I\right)\otimes _{{\mathcal {O}}_{X,z}}\left({\mathcal {O}}_{X,z}/J\right)={\mathcal {O}}_{Z,z}}</annotation>

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</math><img src="./9a4964c382c94457bbbd496923cd2cfa02537e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.278ex; height:3.176ex;" alt="{\displaystyle \left({\mathcal {O}}_{X,z}/I\right)\otimes _{{\mathcal {O}}_{X,z}}\left({\mathcal {O}}_{X,z}/J\right)={\mathcal {O}}_{Z,z}}" loading="lazy"></dd></dl></dd>

<dd>is the "naive" guess of the multiplicity; however, as Serre shows, it is not sufficient.</dd></dl></li>

<li>The sum is finite, because the regular local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X,z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X,z}}</annotation>

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</math><img src="./e8d491fa73494f70ad2e6151badad1eabfd0f013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.709ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X,z}}" loading="lazy"> has finite Tor-dimension.</li>

<li>If the intersection of V and W is not proper, the above multiplicity will be zero. If it is proper, it is strictly positive. (Both statements are not obvious from the definition).</li>

<li>Using a <a href="Spectral_sequence" title="Spectral sequence">spectral sequence</a> argument, it can be shown that μ(Z; V, W) = μ(Z; W, V).</li></ul>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Chow_ring" class="mw-redirect" title="Chow ring">Chow ring</a></div>

The <a href="Chow_ring" class="mw-redirect" title="Chow ring">Chow ring</a> is the group of algebraic cycles modulo <a href="Equivalence_relations_on_algebraic_cycles" class="mw-redirect" title="Equivalence relations on algebraic cycles">rational equivalence</a> together with the following commutative intersection product:

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<annotation encoding="application/x-tex">{\displaystyle V\cdot W:=\sum _{i}\mu (Z_{i};V,W)Z_{i}}</annotation>

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</math><img src="./3427bb51b9536a82640c67a7d57863e9dde346a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.665ex; height:5.509ex;" alt="{\displaystyle V\cdot W:=\sum _{i}\mu (Z_{i};V,W)Z_{i}}" loading="lazy"></dd></dl>

whenever V and W meet properly, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\cap W=\cup _{i}Z_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle V\cap W=\cup _{i}Z_{i}}</annotation>

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</math><img src="./596600bc8bf745ac871f8f35621c31d9ded29657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.641ex; height:2.509ex;" alt="{\displaystyle V\cap W=\cup _{i}Z_{i}}" loading="lazy"> is the decomposition of the set-theoretic intersection into irreducible components.

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

Given two subvarieties V and W, one can take their intersection V ∩ W, but it is also possible, though more subtle, to define the self-intersection of a single subvariety.

Given, for instance, a curve C on a surface S, its intersection with itself (as sets) is just itself: C ∩ C = C. This is clearly correct, but on the other hand unsatisfactory: given any two distinct curves on a surface (with no component in common), they intersect in some set of points, which for instance one can count, obtaining an intersection number, and we may wish to do the same for a given curve: the analogy is that intersecting distinct curves is like multiplying two numbers: xy, while self-intersection is like squaring a single number: x2. Formally, the analogy is stated as a <a href="Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> (multiplication) and a <a href="Quadratic_form" title="Quadratic form">quadratic form</a> (squaring).

A geometric solution to this is to intersect the curve C not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve C in some direction, but in general one talks about taking a curve C′ that is <a href="Linear_system_of_divisors" title="Linear system of divisors">linearly equivalent</a> to C, and counting the intersection C · C′, thus obtaining an intersection number, denoted C · C. Note that unlike for distinct curves C and D, the actual points of intersection are not defined, because they depend on a choice of C′, but the “self intersection points of C′′ can be interpreted as k <a href="Generic_point" title="Generic point">generic points</a> on C, where k = C · C. More properly, the self-intersection point of C is the generic point of C, taken with multiplicity C · C.

Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of [C] ∪ [C] – this both gives a number, and raises the question of a geometric interpretation. Note that passing to cohomology classes is analogous to replacing a curve by a linear system.

Note that the self-intersection number can be negative, as the example below illustrates.

<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>

Consider a line L in the <a href="Projective_plane" title="Projective plane">projective plane</a> P2: it has self-intersection number 1 since all other lines cross it once: one can push L off to L′, and L · L′ = 1 (for any choice) of L′, hence L · L = 1. In terms of intersection forms, we say the plane has one of type x2 (there is only one class of lines, and they all intersect with each other).

Note that on the <a href="Euclidean_plane" title="Euclidean plane">affine plane</a>, one might push off L to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off. One says that “the affine plane does not have a good intersection theory”, and intersection theory on non-projective varieties is much more difficult.

A line on a P1 × P1 (which can also be interpreted as the non-singular <a href="Quadric" title="Quadric">quadric</a> Q in P3) has self-intersection 0, since a line can be moved off itself. (It is a <a href="Ruled_surface" title="Ruled surface">ruled surface</a>.) In terms of intersection forms, we say P1 × P1 has one of type xy – there are two basic classes of lines, which intersect each other in one point (xy), but have zero self-intersection (no x2 or y2 terms).

A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in <a href="Birational_geometry" title="Birational geometry">birational geometry</a>. Given an <a href="Algebraic_surface" title="Algebraic surface">algebraic surface</a> S, <a href="Blowing_up" title="Blowing up">blowing up</a> at a point creates a curve C. This curve C is recognisable by its <a href="Genus_(mathematics)#Algebraic_geometry" title="Genus (mathematics)">genus</a>, which is 0, and its self-intersection number, which is −1. (This is not obvious.) Note that as a corollary, P2 and P1 × P1 are <a href="Minimal_model_(birational_geometry)" class="mw-redirect" title="Minimal model (birational geometry)">minimal surfaces</a> (they are not blow-ups), since they do not have any curves with negative self-intersection. In fact, <a href="Guido_Castelnuovo" title="Guido Castelnuovo">Castelnuovo</a>’s <a href="Castelnuovo's_contraction_theorem" title="Castelnuovo's contraction theorem">contraction theorem</a> states the converse: every (−1)-curve is the exceptional curve of some blow-up (it can be “blown down”).

<ul><li><a href="Chow_group" title="Chow group">Chow group</a></li>

<li><a href="Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem" title="Grothendieck–Riemann–Roch theorem">Grothendieck–Riemann–Roch theorem</a></li>

<li><a href="Enumerative_geometry" title="Enumerative geometry">Enumerative geometry</a></li></ul>

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

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<li id="cite_note-FOOTNOTEEisenbudHarris201614-1"><a href="#cite_ref-FOOTNOTEEisenbudHarris201614_1-0">^</a> <a href="#CITEREFEisenbudHarris2016">Eisenbud & Harris 2016</a>, p. 14.

</li>

<li id="cite_note-FOOTNOTEEisenbudHarris20162-2"><a href="#cite_ref-FOOTNOTEEisenbudHarris20162_2-0">^</a> <a href="#CITEREFEisenbudHarris2016">Eisenbud & Harris 2016</a>, p. 2.

</li>

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

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

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</style><cite id="CITEREFGathman" class="citation cs2">Gathman, Andreas, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160521223954/http://www.mathematik.uni-kl.de/agag/mitglieder/professoren/gathmann/notes/alggeom/">Algebraic Geometry</a>, archived from <a rel="nofollow" class="external text" href="http://www.mathematik.uni-kl.de/agag/mitglieder/professoren/gathmann/notes/alggeom/">the original</a> on 2016-05-21, retrieved 2018-05-11</cite></li>

<li><cite id="CITEREFEisenbudHarris2016" class="citation book cs1"><a href="David_Eisenbud" title="David Eisenbud">Eisenbud, David</a>; <a href="Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (2016). 3264 and All That: A Second Course in Algebraic Geometry. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-01708-5</bdi>.</cite></li>

<li><cite id="CITEREFFulton1998" class="citation cs2"><a href="William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, William</a> (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. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, 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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98549-7</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="CITEREFFultonSerge" class="citation cs2">Fulton, William; <a href="Serge_Lang" title="Serge Lang">Serge, Lang</a>, Riemann-Roch Algebra, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-3073-6</bdi></cite></li>

<li><cite id="CITEREFSerre1965" class="citation cs2"><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1965), Algèbre locale. Multiplicités, Cours au Collège de France, 1957--1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <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=0201468">0201468</a></cite></li></ul>

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