In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a complete algebraic variety is an algebraic variety

Algebraic variety

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

  X, such that for any variety Y the projection

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

 morphism

Regular map (algebraic geometry)

In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials. For example, if X and Y are subvarieties of An resp...

is a closed map, i.e. maps closed set

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

s onto closed sets.

The most common example of a complete variety is a projective variety, but there do exist complete and non-projective varieties in dimensions

Dimension of an algebraic variety

In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...

 2 and higher. The first examples of non-projective complete varieties were given by Masayoshi Nagata

Masayoshi Nagata

Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....

 and Heisuke Hironaka

Heisuke Hironaka

is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....

. An affine space

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 of positive dimension is not complete.

The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of 'complete', in the sense of 'no missing points', can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley

Claude Chevalley

Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...

.