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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
. One needs a definition of intersection number in order to state results like
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degr ...
. 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 In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
.


Definition for Riemann surfaces

Let ''X'' be a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
. 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 c : S^1 \to X), we can associate a
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
\eta_c of compact support with the property that integrals along ''c'' can be calculated by integrals over ''X'': :\int_c \alpha = -\iint_X \alpha \wedge \eta_c = (\alpha, *\eta_c), for every closed (1-)differential \alpha on ''X'', where \wedge is the
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
of differentials, and * is the
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
. Then the intersection number of two closed curves, ''a'' and ''b'', on ''X'' is defined as :a \cdot b := \iint_X \eta_a \wedge \eta_b = (\eta_a, -*\eta_b) = -\int_b \eta_a. The \eta_c have an intuitive definition as follows. They are a sort of
dirac delta In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
along the curve ''c'', accomplished by taking the differential of a
unit step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
that drops from 1 to 0 across ''c''. More formally, we begin by defining for a simple closed curve ''c'' on ''X'', a function ''fc'' by letting \Omega be a small strip around ''c'' in the shape of an annulus. Name the left and right parts of \Omega \setminus c as \Omega^ and \Omega^. Then take a smaller sub-strip around ''c'', \Omega_0, with left and right parts \Omega_0^ and \Omega_0^. Then define ''fc'' by :f_c(x) = \begin 1, & x \in \Omega_0^ \\ 0, & x \in X \setminus \Omega^ \\ \mbox, & x \in \Omega^ \setminus \Omega_0^ \end. The definition is then expanded to arbitrary closed curves. Every closed curve ''c'' on ''X'' is homologous to \sum_^N k_i c_i for some simple closed curves ''ci'', that is, :\int_c \omega = \int_ \omega = \sum_^N k_i \int_ \omega, for every differential \omega. Define the \eta_c by :\eta_c = \sum_^N k_i \eta_.


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
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s 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''''1'', ..., ''Z''''n'' which have local equations ''f''''1'', ..., ''f''''n'' near ''x'' for polynomials ''f''''i''(''t''''1'', ..., ''t''''n''), such that the following hold: * n = \dim_k X. * f_i(x) = 0 for all ''i''. (i.e., ''x'' is in the intersection of the hypersurfaces.) * \dim_x \cap_^n Z_i = 0 (i.e., the divisors are in general position.) * The f_i are nonsingular at ''x''. Then the intersection number at the point ''x'' (called the intersection multiplicity at ''x'') is :(Z_1 \cdots Z_n)_x := \dim_k \mathcal_ / (f_1, \dots, f_n), where \mathcal_ 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 Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
k , where \mathfrak_x 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''''i''. 2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection. :(Z_1 \cdots Z_n) = \sum_ (Z_1 \cdots Z_n)_x 3. Extend the definition to ''effective'' divisors by linearity, i.e., :(n Z_1 \cdots Z_n) = n(Z_1 \cdots Z_n) and ((Y_1 + Z_1) Z_2 \cdots Z_n) = (Y_1 Z_2 \cdots Z_n) + (Z_1 Z_2 \cdots Z_n). 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''''i'' = ''P''''i'' - ''N''i, and use rules of the form :((P_1 - N_1) P_2 \cdots P_n) = (P_1 P_2 \cdots P_n) - (N_1 P_2 \cdots P_n) to transform the intersection. 5. The intersection number of arbitrary divisors is then defined using a "
Chow's moving lemma In algebraic geometry, Chow's moving lemma, proved by , states: given algebraic cycles ''Y'', ''Z'' on a nonsingular quasi-projective variety ''X'', there is another algebraic cycle ''Z' '' on ''X'' such that ''Z' '' is rationally equivalent to '' ...
" 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 In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
''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 mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
in a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
. 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to a plane curve ''W''. The first problem requires the machinery of
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
, 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
described how to find the multiplicity of each intersection point by methods of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
and
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. This connection between a geometric notion of intersection and a homological notion of a
derived tensor product In algebra, given a differential graded algebra ''A'' over a commutative ring ''R'', the derived tensor product functor is :- \otimes_A^ - : D(\mathsf_A) \times D(_A \mathsf) \to D(_R \mathsf) where \mathsf_A and _A \mathsf are the categories of ri ...
has been influential and led in particular to several
homological conjectures in commutative algebra In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a ...
. 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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
\mathcal_ 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 :e(X; V, W; x) = \sum_^ (-1)^i \mathrm_A(\operatorname_i^A(A/I, A/J)) where length is the
length of a module In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It ...
over a local ring, and Tor is the
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to constr ...
. 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 In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
''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 sequence In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions Fo ...
s, for example if they are
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
, 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 In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection n ...
.


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 In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
. Specifically, if two manifolds, ''X'' and ''Y'', intersect transversely in a manifold ''M'', the homology class of the intersection is the
Poincaré dual Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * L ...
of the cup product D_M X \smile D_M Y 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 In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of ''X'' and ''G'' the Grothendieck group of the category of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on ''X'' whose support is
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
over an Artinian subscheme of ''S''. For each ''L'' in Pic(''X''), define the endomorphism ''c''1(''L'') of ''G'' (called the
first Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
of ''L'') by :c_1(L)F= F - L^ \otimes F. It is additive on ''G'' since tensoring with a line bundle is exact. One also has: *c_1(L_1)c_1(L_2) = c_1(L_1) + c_1(L_2) - c_1(L_1 \otimes L_2); in particular, c_1(L_1) and c_1(L_2) commute. *c_1(L)c_1(L^) = c_1(L) + c_1(L^). *\dim \operatorname c_1(L)F \le \dim \operatorname F - 1 (this is nontrivial and follows from a dévissage argument.) The intersection number :L_1 \cdot \cdot L_r of line bundles ''L''''i'''s is then defined by: :L_1 \cdot \cdot L_r \cdot F = \chi(c_1(L_1) \cdots c_1(L_r) F) where χ denotes the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
. Alternatively, one has by induction: :L_1 \cdot \cdot L_r \cdot F = \sum_0^r (-1)^i \chi(\wedge^i (\oplus_0^r L_j^) \otimes F). Each time ''F'' is fixed, L_1 \cdot \cdot L_r \cdot F is a symmetric functional in ''L''''i'''s. If ''L''''i'' = ''O''''X''(''D''''i'') for some
Cartier divisor In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
s ''D''''i'''s, then we will write D_1 \cdot \cdot D_r for the intersection number. Let f:X \to Y be a morphism of ''S''-schemes, L_i, 1 \le i \le m line bundles on ''X'' and ''F'' in G with m \ge \dim \operatornameF. Then :f^*L_1 \cdots f^* L_m \cdot F = L_1 \cdots L_m \cdot f_* F.


Intersection multiplicities for plane curves

There is a unique function assigning to each triplet (P,Q,p) consisting of a pair of projective curves, P and Q, in K ,y/math> and a point p \in K^2, a number I_p(P,Q) called the ''intersection multiplicity'' of P and Q at p that satisfies the following properties: # I_p(P,Q) = I_p(Q,P) # I_p(P,Q) = \infty if and only if P and Q have a common factor that is zero at p # I_p(P,Q) = 0 if and only if one of P(p) or Q(p) is non-zero (i.e. the point p is off one of the curves) # I_p(x,y) = 1 where p = (0,0) # I_p(P,Q_1Q_2) = I_p(P,Q_1) + I_p(P,Q_2) # I_p(P + QR,Q) = I_p(P,Q) for any R \in K ,y/math> 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 K x,y. By making a change of variables if necessary, we may assume that p = (0,0). Let P(x,y) and Q(x,y) 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 z = 1. Let I = (P,Q) denote the ideal of K x,y generated by P and Q. The intersection multiplicity is the dimension of K x,y/I as a vector space over K. Another realization of intersection multiplicity comes from the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
of the two polynomials P and Q. In coordinates where p = (0,0), the curves have no other intersections with y = 0, and the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of P with respect to x is equal to the total degree of P, I_p(P,Q) can be defined as the highest power of y that divides the resultant of P and Q (with P and Q seen as polynomials over K /math>). Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if P and Q define curves which intersect only once in the closure of an open set U, then for a dense set of (\epsilon,\delta) \in K^2, P - \epsilon and Q - \delta are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number n points in U. We say then that I_p(P,Q) = n.


Example

Consider the intersection of the ''x''-axis with the parabola :y = x^2.\ Then :P = y,\ and :Q = y - x^2,\ so : I_p(P,Q) = I_p(y,y - x^2) = I_p(y,x^2) = I_p(y,x) + I_p(y,x) = 1 + 1 = 2.\, Thus, the intersection degree is two; it is an ordinary
tangency In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
.


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
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of some specific kind, two representatives are intersected that are in
general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...
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 Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degr ...
. The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
s with a
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s. Calculating the intersection numbers at the fixed points counts the fixed points ''with multiplicity'', and leads to the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
in quantitative form.


Notes


References

* * Appendix A. * * ''Algebraic Curves: An Introduction To Algebraic Geometry'', by William Fulton with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989.
Full text online
* * *{{Citation , last1=Kollár , first1=János , author1-link=Janos Kollar , title=Rational Curves on Algebraic Varieties , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, location=Berlin, Heidelberg , isbn=978-3-642-08219-1 , doi=10.1007/978-3-662-03276-3 , mr=1440180 , year=1996 Algebraic geometry