A geometric object defined as the solution set of polynomial equations over a field and equipped with the Zariski topology; in classical terms often taken to be a reduced, separated scheme of finite type over a field, possibly irreducible (an integral variety) or reducible.

Principle

Capture geometry by polynomial equations: the coordinate ring and its ideals encode geometric properties, dimensions, and maps between varieties via algebraic morphisms.

Demonstration

An affine variety V(I) ⊂ A^n_k defined by an ideal I ⊂ k[x_1,...,x_n] has coordinate ring k[x_1,...,x_n]/I; projective varieties arise by homogeneous equations in projective space and are glued from affine pieces.

Misapplication

Assuming every finite type scheme over a field is a variety (ignoring nonreduced schemes or separatedness), or treating varieties only over algebraically closed fields when arithmetic behavior over general fields matters.

Consequence

Varieties allow use of algebraic tools (coordinate rings, dimension theory, morphisms, rational maps) and support geometric notions like smoothness, singularities, and intersection theory over a field.

Reversal

General scheme or analytic space: relaxing the requirement of reducedness, finiteness, or a field base leads to schemes with nilpotents, formal schemes, or analytic spaces where polynomial equations are replaced by convergent power series.

Boundary

Typically restricted to reduced schemes of finite type over a field and often assumed separated; excludes schemes over general rings, stacks, formal schemes, and most analytic or transcendental spaces.

Semantic Tension

Variety versus scheme: varieties are classical, often reduced and over a field; schemes are broader, admitting nilpotents and arithmetic base rings — choosing one or the other depends on which phenomena one wishes to capture.

Synthesis

An algebraic variety is a geometric locus cut out by polynomials over a field, whose coordinate ring encodes its algebraic structure; as a reduced finite type scheme it provides a flexible but classical setting for studying algebraic geometry.