In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, a projective space is a fundamental construction from any vector space. ... This article is about algebraic varieties. ...

In linear algebra terms there is for given vector spaces U and V, over the same field, a natural way to map Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

U × V

to the tensor product space W. This not in general injective, because it takes the pair In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

(u,v)

to the pure tensor w formed from u and v. For any non-zero scalar c, the image of This is a glossary of tensor theory. ... The concept of a scalar is used in mathematics, physics, and computing. ...

(cu,c⁻¹v)

will also be w. In co-ordinate terms, w has co-ordinates formed of all products of a co-ordinate of u with a co-ordinate of v.

Considering now the underlying projective spaces P(U) and P(V), the mapping passes to a morphism of varieties In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

s: P(U) × P(V) → P(W).

This is not only injective in the set-theoretic sense: it is a closed immersion in the sense of algebraic geometry. That is, one can give a set of equations for the image. Except for notational trouble, it is easy to say what such equations are: they express two ways of factoring products of co-ordinates from W, obtained in two different ways as something from U times something from V. This is a glossary of scheme theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

This mapping or morphism s is the Segre embedding. Counting dimensions, it shows how the product of projective spaces of dimensions m and n embeds in dimension

(m + 1)(n + 1) − 1 = mn + m + n.

For example with m = n = 1 we get an embedding of the product of the projective line with itself in P³. The image is a quadric, and is easily seen to contain two one-parameter families of lines. Over the complex numbers this is a quite general non-singular quadric. In mathematics, the projective line is a fundamental example of an algebraic curve. ... 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). ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...

Classical terminology calls the co-ordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.