Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of "x"- and "y"-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example, if a plane is tangent to a surface  along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory. Definition for Riemann surfaces. Let "X" be a Riemann surface. Then the intersection number of two closed curves on "X" has a simple definition in terms of an integral. For every closed curve "c" on "X" (i.e., smooth function formula_1), we can associate a differential form formula_2 of compact support with the property that integrals along "c" can be calculated by integrals over "X": where formula_5 is the wedge product of differentials, and formula_6 is the Hodge star . Then the intersection number of two closed curves, "a" and "b", on "X" is defined as The formula_2 have an intuitive definition as follows. They are a sort of dirac delta along the curve "c", accomplished by taking the differential of a unit step function that drops from 1 to 0 across "c". More formally, we begin by defining for a simple closed curve "c" on "X", a function "f" by letting formula_9 be a small strip around "c" in the shape of an annulus. Name the left and right parts of formula_10 as formula_11 and formula_12. Then take a  smaller sub-strip around "c", formula_13, with left and right parts formula_14 and formula_15. Then define "f" by The definition is then expanded to arbitrary closed curves. Every closed curve "c" on "X" is homologous to formula_17 for some simple closed curves "c", that is, Define the formula_2 by Definition for algebraic varieties. The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of divisors on a nonsingular variety "X". 1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces  (subvarieties of "X" of codimension one) that are in general position at "x". Specifically, assume we have a nonsingular variety "X", and "n" hypersurfaces "Z", ..., "Z" which have local equations "f", ..., "f" near "x" for polynomials "f"("t", ..., "t"), such that the following hold: Then the intersection number at the point "x" (called the intersection multiplicity at "x") is where formula_27 is the local ring of "X" at "x", and the dimension is dimension as a "k"-vector space. It can be calculated as the localization formula_28, where formula_29 is the maximal ideal of polynomials vanishing at "x", and "U" is an  open affine set containing "x" and containing none of the singularities of the "f". 2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection. 3. Extend the definition to "effective" divisors by linearity, i.e., 4. Extend the definition to arbitrary divisors in general position by noticing every divisor has a unique expression as "D" = "P" - "N" for some effective divisors "P" and "N". So let "D" = "P" - "N", and use rules of the form to transform the intersection. 5. The intersection number of arbitrary  divisors is then defined using a "Chow's moving lemma" that guarantees we can find linearly equivalent divisors that are in general position, which we can then intersect. Note that the definition of the intersection number does not depend on the order in which the divisors appear in the computation of this number. Serre's Tor formula. Let "V" and "W" be two subvarieties of a nonsingular projective variety "X" such that dim("V")+dim("W")=dim("X"). Then we expect the intersection "V"∩"W" to be a finite set of points. If we try to count them, two kinds of problems may arise. First, even  if the expected dimension of "V"∩"W" is zero, the actual intersection may be of a large dimension. For example, we could try to find the self-intersection number of a projective line in a projective plane. The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse. For example, "V" can be a tangent line to a plane curve "W". The first problem requires the machinery of intersection theory, discussed above in detail. The essential idea is to replace "V" and "W" by more convenient subvarieties using the moving lemma. On the other  hand, the second problem can be solved directly, without moving "V" or "W". In 1965 Jean-Pierre Serre described how to find the multiplicity of each intersection point by methods of commutative algebra and homological algebra. This connection between a geometric notion of intersection and a homological notion of a derived tensor product has been influential and led in particular to several homological conjectures in commutative algebra. The Serre's Tor formula is the following result. Let "X" be a regular variety, "V" and "W" two subvarieties of complementary dimension such that "V"∩"W" is zero-dimensional. For any point "x"∈"V"∩"W , let A" be the local ring formula_27 of "x". The structure sheaves of "V" and "W" at "x" correspond to ideals "I", "J"⊆"A". Then the multiplicity of "V"∩"W" at the point "x" is where length is the length of a module over a local ring, and Tor is the Tor functor. When "V" and "W" can be moved into a transverse position, this homological formula produces the expected answer. So, for instance, if "V" and "W" meet transversely at "x", the multiplicity is 1. If "V" is a tangent line at a point "x" to a parabola "W" in a  plane at a point "x", then the multiplicity at "x" is 2. If both "V" and "W" are locally cut out by regular sequences, for example if they are nonsingular, then in the formula above all higher Tor's vanish, hence the multiplicity is positive. The positivity in the arbitrary case is one of Serre's multiplicity conjectures. Further definitions. The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties. In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two  manifolds, "X" and "Y", intersect transversely in a manifold "M", the homology class of the intersection is the Poincaré dual of the cup product formula_36 of the Poincaré duals of "X" and "Y". Snapper–Kleiman definition of intersection number. There is an approach to intersection number, introduced by Snapper in 1959-60 and developed later by Cartier and Kleiman, that defines an intersection number as an Euler characteristic. Let "X" be a scheme over a scheme "S", Pic("X") the Picard group of "X" and G the Grothendieck group of the category of coherent sheaves on "X" whose support is proper  over an Artinian subscheme of "S". For each "L" in Pic("X"), define the endomorphism "c"("L") of G (called the first Chern class of "L") by It is additive on G since tensoring with a line bundle is exact. One also has: The intersection number of line bundles "L"'s is then defined by: where χ denotes the Euler characteristic. Alternatively, one has by induction: Each time "F" is fixed, formula_46 is a symmetric functional in "L"'s. If "L" = "O"("D") for some Cartier divisors "D"'s, then we will write formula_47 for the intersection number. Let formula_48 be a  morphism of "S"-schemes, formula_49 line bundles on "X" and "F" in G with formula_50. Then Intersection multiplicities for plane curves. There is a unique function assigning to each triplet formula_52 consisting of a pair of projective curves, formula_53 and formula_54, in formula_55 and a point formula_56, a number formula_57 called the "intersection multiplicity" of formula_53 and formula_54 at formula_60 that satisfies the following properties: Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways. One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring formula_75 . By making a change of variables if necessary, we may assume that formula_71. Let formula_77 and formula_78 be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting formula_79. Let formula_80 denote the ideal of formula_75 generated by formula_53 and formula_54. The intersection multiplicity is the dimension of formula_84 as a vector space over formula_85. Another realization of intersection multiplicity comes from the resultant of the two polynomials formula_53 and formula_54. In coordinates where formula_71, the curves have no other intersections with formula_89, and the  degree of formula_53 with respect to formula_91 is equal to the total degree of formula_53, formula_57 can be defined as the highest power of formula_94 that divides the resultant of formula_53 and formula_54 (with formula_53 and formula_54 seen as polynomials over formula_99). Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if formula_53 and formula_54 define curves which intersect only once in the closure of an open set formula_102, then for a dense set of formula_103, formula_104 and formula_105 are smooth and intersect transversally (i.e. have different  tangent lines) at exactly some number formula_106 points in formula_102. We say then that formula_108. Example. Consider the intersection of the "x"-axis with the parabola Then and so Thus, the intersection degree is two; it is an ordinary tangency. Self-intersections. Some of the most interesting intersection numbers to compute are "self-intersection numbers". This should not be taken in a naive sense. What is meant is that, in an equivalence class of divisors of some specific kind, two representatives are intersected that are in general position with respect to each other. In this way, self-intersection numbers can become  well-defined, and even negative. Applications. The intersection number is partly motivated by the desire to define intersection to satisfy Bézout's theorem. The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function graphs with a diagonals. Calculating the intersection numbers at the fixed points counts the fixed points "with multiplicity", and leads to the Lefschetz fixed-point theorem in quantitative form.