Projective geometry begins with a simple refusal: points differing only by a nonzero scalar should be regarded as the same. This convention turns lines through the origin into points and allows polynomial equations to describe objects without privileging an affine chart. The price is that only equations compatible with rescaling coordinates make sense. A projective variety is the resulting algebraic object: a subset of [projective space](/page/Projective%20Space) cut out by [homogeneous polynomial](/page/Homogeneous%20Polynomial) equations.

Projective varieties are central because many geometric constructions naturally produce them. Plane conics close up affine parabolas by adding points at infinity; smooth projective curves provide the natural home for divisors and meromorphic functions; projective embeddings organise algebraic varieties through [homogeneous coordinates](/page/Homogeneous%20Coordinates). The language also connects geometry to graded commutative algebra through homogeneous ideals in a [polynomial ring](/page/Polynomial%20Ring).

## Definition

A projective point is not a tuple of coordinates but a proportionality class of tuples. Therefore the first question is not merely which equations vanish, but which equations have a vanishing condition that survives rescaling. A projective variety is the geometric object obtained when this compatibility is built into the definition from the start.

[definition: Projective Variety] Let $k$ be a field. In the classical set-theoretic convention used on this page, a projective variety over $k$ is a subset $X \subset \mathbb{P}^n_k(k)$ of the form \begin{align*} X = V_+(S)(k) \end{align*} for some $n \in \mathbb{N}$ and some set $S \subset k[x_0, \ldots, x_n]$ of homogeneous polynomials, where $V_+(S)$ denotes their common projective zero locus. [/definition]

The phrase "over $k$" can hide a convention. Here $\mathbb{P}^n_k(k)$ means the set of $k$-rational projective points, and the equations are evaluated on homogeneous coordinate representatives in $k^{n+1}\setminus\{0\}$. When $k$ is algebraically closed, this is the usual classical picture. Over a non-[algebraically closed field](/page/Algebraically%20Closed%20Field), the same equations also determine geometric points after extending scalars to $\overline{k}$, and in modern algebraic geometry they determine a closed subscheme of $\mathbb{P}^n_k$. Those refinements carry more information than the set of $k$-rational points, so later statements that use the Nullstellensatz will explicitly assume $k$ algebraically closed.

## Homogeneous Equations and Coordinate Rings

The definition deliberately puts the homogeneous equations, rather than affine coordinates, in charge. This is necessary because a projective point has many representatives: $(a_0,\ldots,a_n)$ and $(\lambda a_0,\ldots,\lambda a_n)$ describe the same point whenever $\lambda \in k^\times$. An ordinary polynomial equation may change its value under this replacement, so it may fail to define a condition on projective points at all. The equations that survive this test are the homogeneous ones, and the next definition isolates that compatibility.

[definition: Homogeneous Polynomial] A polynomial $F \in k[x_0, \ldots, x_n]$ is homogeneous of degree $d \in \mathbb{N}$ if every monomial of $F$ has total degree $d$. [/definition]

For a homogeneous polynomial $F$ of degree $d$, rescaling a representative gives

\begin{align*} F(\lambda a_0, \ldots, \lambda a_n) = \lambda^d F(a_0, \ldots, a_n). \end{align*}

Thus the condition $F(a_0, \ldots, a_n)=0$ is independent of the representative of the projective point.

A single equation is rarely enough for the geometry one wants to study. Intersections of equations, such as a conic meeting a line or a curve lying on a surface, require simultaneous vanishing. The next definition records this common vanishing set directly in projective space.

[definition: Projective Zero Locus] Let $k$ be a field, let $n \in \mathbb{N}$, and let $S \subset k[x_0, \ldots, x_n]$ be a set of homogeneous polynomials. The projective zero locus of $S$ is \begin{align*} V_+(S)(k) = \{[a_0 : \cdots : a_n] \in \mathbb{P}^n_k(k) : F(a_0, \ldots, a_n)=0 \text{ for every } F \in S\}. \end{align*} [/definition]

The subscript $+$ reminds the reader that homogeneous equations are being used. When the base field is fixed, authors often write $V_+(S)$ for the set $V_+(S)(k)$. When the distinction matters, especially over non-algebraically closed fields, the notation $V_+(S)(K)$ records the $K$-rational points after extending scalars from $k$ to a field $K \supset k$. The construction is analogous to the affine zero locus $V(I)$, but the ambient space is projective and the equations must respect the scaling relation.

Some authors reserve the word variety for irreducible closed sets and call the more general objects projective algebraic sets. On this page, projective variety means a projective algebraic set unless irreducibility is explicitly imposed. Since different lists of equations may define the same set, the next definition packages equations into the algebraic object that controls them.

[definition: Homogeneous Ideal] An ideal $I \trianglelefteq k[x_0, \ldots, x_n]$ is homogeneous if it is generated by homogeneous polynomials. [/definition]

Equivalently, a polynomial $F \in I$ belongs to a homogeneous ideal $I$ precisely when each homogeneous component of $F$ belongs to $I$. This reformulation is a structural fact about graded rings, and it is what allows projective algebraic geometry to use commutative algebra without losing track of degree. The next definition attaches such algebra to a projective variety itself.

[definition: Homogeneous Coordinate Ring] Let $X \subset \mathbb{P}^n_k(k)$ be a projective variety in the classical set-theoretic convention of this page. The homogeneous vanishing ideal of $X$ is \begin{align*} I_+(X)=\{F \in k[x_0, \ldots, x_n] : \text{each homogeneous component of } F \text{ vanishes on every point of } X\}. \end{align*} The homogeneous coordinate ring of $X$ is the graded [quotient ring](/page/Quotient%20Ring) \begin{align*} k[X] = k[x_0, \ldots, x_n]/I_+(X). \end{align*} [/definition]

The grading on $k[X]$ is part of the data. It remembers which functions arise from linear forms, quadratic forms, and higher-degree forms on the chosen projective embedding.