Algebraic Variety

Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers, though modern definitions generalize the notion in several ways while trying to preserve the original geometric intuition. Some definitions require an algebraic variety to be irreducible, meaning it cannot be written as the union of two smaller sets closed in the Zariski topology, with non-irreducible examples then called algebraic sets instead; other conventions drop this requirement. Algebraic varieties can be characterized by dimension, with varieties of dimension one called algebraic curves and those of dimension two called algebraic surfaces, and in the language of modern scheme theory an algebraic variety over a field is defined as an integral scheme over that field whose structure morphism is separated and of finite type.