Algebraic Variety

An algebraic variety is the zero locus of a collection of polynomials. An affine variety in is the set of common zeros of a family of polynomials in variables, the points at which every member of the family vanishes. A projective variety in complex projective space is the set of common zeros of a family of homogeneous polynomials, homogeneity being exactly the condition that vanishing is well defined on projective points, since a polynomial takes a consistent value under rescaling of the coordinates only when its terms all share one degree.

The Zariski topology takes the varieties themselves as the closed sets, so a set is closed precisely when it is cut out by polynomial equations. A variety is irreducible when it is not the union of two strictly smaller varieties, equivalently when its ideal is prime. The dimension is the length of the longest chain of nested irreducible subvarieties, matching the intuitive count of independent directions.

A point is smooth when the variety looks locally like a linear space of the expected dimension there, and singular otherwise. The Jacobian criterion makes this concrete: at a point the matrix of first partial derivatives of the defining polynomials has full expected rank exactly at the smooth points, and drops rank at the singular ones.

The simplest case is a hypersurface, cut out by a single polynomial, which has codimension one. A degree- hypersurface in projective space is the running example behind the quintic, the degree-five hypersurface in . This is the primary, algebraic presentation of objects that also carry a manifold structure when smooth, the polynomial equations being the blueprint from which the geometry follows.

See projective-space for the projective case.