Algebraic geometry is an important branch of modern mathematics. Its basic research object is the geometric characteristics of a set composed of common zeros of several algebraic equations in arbitrary dimension, affine or projective space. Such sets are usually called algebraic clusters, and these equations are called the definition equations of this algebraic cluster.

Algebraic cluster is the locus of points determined by one or more algebraic equations of spatial coordinates. For example, algebraic clusters in three-dimensional space are algebraic curves and surfaces, and algebraic geometry studies the geometric properties of general algebraic curves and surfaces.

Algebraic geometry is a branch of mathematics, which combines abstract algebra, especially commutative algebra, with geometry. It can be regarded as a study of the solution set of algebraic equations.

Algebraic geometry takes algebraic clusters as the research object. Algebraic cluster is the locus of points determined by one or more algebraic equations in spatial coordinates. For example, algebraic clusters in three-dimensional space are algebraic curves and algebraic surfaces. Algebraic geometry studies the geometric properties of general algebraic curves and surfaces.

Algebraic geometry has extensive relations with many branches of mathematics, such as complex analysis, number theory, analytic geometry, differential geometry, commutative algebra, algebraic groups, topology and so on. The development of algebraic geometry and these disciplines play a mutually promoting role.

The idea of studying geometry by algebraic method has developed into another branch of geometry after the appearance of analytic geometry, which is algebraic geometry. The objects of algebraic geometry are plane algebraic curves, space algebraic curves and algebraic surfaces.