A projective variety is a subset X of projective space P^n over a field (typically algebraically closed) defined as the common zero locus of a collection of homogeneous polynomials; it is endowed with the Zariski topology and schemes or coordinate ring structures refine its algebraic structure.

Principle

The organizing idea is homogeneous polynomial vanishing: projective varieties are the geometric objects invariant under scalar multiplication of homogeneous coordinates, capturing zeros of homogeneous ideals.

Demonstration

Example: a projective plane curve given by a single homogeneous polynomial F(x:y:z)=0, such as a nonsingular cubic in P^2, exhibits projective invariance and global properties like degree and genus.

Misapplication

Treating the affine zero set of nonhomogeneous polynomials in affine space as a projective variety without homogenization or adding points at infinity misapplies the projective setting.

Consequence

Correct use yields global invariants (degree, dimension, Hilbert polynomial), well-defined notions of intersection multiplicity, and compactness in the Zariski topology that facilitate algebraic-geometric methods.

Reversal

The reversal contrasts with affine varieties: affine sets are defined in affine space by arbitrary polynomials and do not enforce homogeneity or projective scaling symmetry, lacking canonical points at infinity.

Boundary

Scope: typically over algebraically closed fields for classical theory; excludes arbitrary analytic sets, nonhomogeneous loci in projective coordinates, and spaces lacking an underlying homogeneous coordinate ring.

Semantic Tension

Tension exists between 'projective variety' and 'projective scheme' or 'projective algebraic set': varieties often imply irreducibility and reducedness, whereas schemes allow nilpotents and more general gluing.

Synthesis

A projective variety is the homogeneous-zero locus in projective space determined by a homogeneous ideal; it is globally defined up to projective scaling, carries algebraic invariants like degree and dimension, and serves as a compact algebraic-geometric object for intersection theory and moduli questions.