In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand the topological theory more quickly reached a definitive form. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. ... This article refers to Bézouts theorem in algebraic geometry. ... In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. ...

For example, for a connected oriented manifold M of dimension 2n the intersection form is defined on the cohomology group Hⁿ(M, Z) in the 'middle dimension', by intersection in the form of the cap product. This is a quadratic form for n even, and an alternating form for n odd, because of the graded-commutative nature of the cohomology ring. These forms are important topological invariants. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...

Intersection theory in algebraic geometry

William Fulton in Intersection Theory (1984) writes

... 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 Chern class of the normal bundle of A in X.

To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not addressed in the same spirit. This article needs to be cleaned up to conform to a higher standard of quality. ... In the mathematical field of differential geometry, a normal bundle is a particular kind of vector bundle. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. ...

Self-intersection is a key idea, for example in birational geometry. There on an algebraic surface S, blowing up creates from a point a curve C recognisable by its genus, which is 0, and its self-intersection number, which is −1. There is no paradox, since while C∩C is C as a set, C.C does not mean that set-theoretic intersection. Consider a line L in the projective plane: it has self-intersection number 1 since all other lines cross it once. A line on a non-singular quadric Q in P³ has self-intersection 0, since Q is also P¹×P¹ and a line P¹ can be moved off itself. The quadric Q projects to the plane by means of lines through a fixed point on it. In terms of intersection forms the plane has one of type x² while the quadric one of type XY. This happens on the addition of a negative term to x² − y², and then a change of basis. In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... In mathematics, blowing up is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...