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. A first test case for this graded viewpoint is the situation where all the geometry is controlled by one equation. In [affine space](/page/Affine%20Space) any single polynomial cuts out a hypersurface, but in projective space the equation has to respect scalar equivalence of representatives. Because projective points are equivalence classes of nonzero vectors, such an equation must be homogeneous so that vanishing is independent of the chosen representative. This motivates isolating the projective analogue of a one-equation variety: the basic codimension-one projective objects are the zero loci of nonzero homogeneous forms. The definition packages two requirements that will be used repeatedly: the equation must have positive degree so it cuts out a genuine condition, and it must be homogeneous so the condition descends from affine coordinates to projective points. Naming these objects separately lets later arguments refer to projective curves, surfaces, and their degrees through a single standard class of examples. [definition: Projective Hypersurface] Let $F \in k[x_0, \ldots, x_n]$ be a nonzero homogeneous polynomial of positive degree. The projective hypersurface defined by $F$ is the projective variety \begin{align*} V_+(F) \subset \mathbb{P}^n_k. \end{align*} [/definition] Projective hypersurfaces include plane curves in $\mathbb{P}^2_k$, surfaces in $\mathbb{P}^3_k$, and higher-dimensional examples. For a reduced classical hypersurface, the degree is the degree of a squarefree homogeneous equation defining the same zero set over an [algebraic closure](/page/Algebraic%20Closure). If one keeps the equation $F$ with repeated factors, one is remembering scheme-theoretic or divisor multiplicity data beyond the reduced projective variety. ## Homogeneous Ideals and Vanishing Sets The definition uses equations, but the practical working object is usually the homogeneous ideal they generate. A list of equations is presentation-dependent: adding redundant equations, replacing generators, or separating a polynomial into homogeneous pieces should not change the geometry. Homogeneous ideals are the stable algebraic containers that remember exactly the degree-compatible equations available on projective space. The next theorem makes that replacement legitimate. [quotetheorem:9495] This theorem lets us write $V_+(I)$ without remembering a chosen generating set. It also turns geometric inclusion into algebraic reverse inclusion: more equations produce smaller zero loci. The next definition names the exceptional ideal forced on us by the fact that projective space excludes the origin. [definition: Irrelevant Ideal] The irrelevant ideal in $k[x_0, \ldots, x_n]$ is \begin{align*} (x_0, \ldots, x_n) \trianglelefteq k[x_0, \ldots, x_n]. \end{align*} [/definition] The irrelevant ideal vanishes only at the origin in affine space, and the origin is not a projective point. This is why projective algebraic geometry treats it as a special ideal rather than as the equation of a genuine projective subvariety. To compare ideals by their projective zero sets, one must ignore elements that differ only by behaviour supported at this missing origin. The algebraic operation that performs this removal is saturation. [definition: Saturation with Respect to the Irrelevant Ideal] Let $I \trianglelefteq k[x_0, \ldots, x_n]$ be a homogeneous ideal, and let $\mathfrak{m}=(x_0,\ldots,x_n)$ be the irrelevant ideal. The saturation of $I$ with respect to $\mathfrak{m}$ is \begin{align*} I^{\mathrm{sat}} = I : \mathfrak{m}^{\infty} = \{F \in k[x_0,\ldots,x_n] : \mathfrak{m}^rF \subset I \text{ for some } r \in \mathbb{N}\}. \end{align*} [/definition] Saturation is invisible on projective points: multiplying by sufficiently many homogeneous coordinates only changes behaviour at the affine cone vertex. Thus the naive question "which ideal has this projective zero set?" is not answered by the original ideal itself, because extra algebra supported only at the missing origin cannot be detected in projective space. Over an algebraically closed field, this is the source of the [projective Nullstellensatz](/theorems/2135) dictionary: before comparing homogeneous ideals by their projective zero loci, one first removes the part supported only at the irrelevant ideal. In the algebraically closed setting used throughout this discussion, the dictionary identifies projective zero sets only after this saturation step. The projective version therefore has a warning that the affine version does not: ideals are compared only after removing the irrelevant behaviour at $(x_0,\ldots,x_n)$. Powers of the irrelevant ideal define no projective points, and two homogeneous ideals can have the same projective zero set before saturation. For nonempty projective algebraic sets, the resulting correspondence is with saturated radical homogeneous ideals not containing the irrelevant ideal. The empty projective algebraic set is the boundary case: it occurs when the defining equations force every projective representative to disappear, equivalently when the irrelevant ideal lies in the radical of the defining homogeneous ideal. In that case the saturation becomes the whole ring, so the usual nonempty correspondence is best read with this exceptional case kept separate. ## Standard Examples The first example should feel almost empty: if no equations are imposed, every projective point is allowed. This matters because it sets the ambient geometry in which all other projective varieties live. [example: Projective Space as a Projective Variety] Let $k$ be a field and impose no homogeneous equations, so $S=\varnothing$. By the definition of projective zero locus, a point $[a_0:\cdots:a_n]\in \mathbb{P}^n_k$ lies in $V_+(\varnothing)$ exactly when every polynomial in $\varnothing$ vanishes at $(a_0,\ldots,a_n)$. Since there are no polynomials to check, every projective point satisfies the condition, and therefore \begin{align*} V_+(\varnothing)=\mathbb{P}^n_k. \end{align*} Thus $\mathbb{P}^n_k$ is a projective variety, because it is the projective zero locus of a set of homogeneous polynomials. Its defining ideal is the zero homogeneous ideal, so its homogeneous coordinate ring is \begin{align*} k[\mathbb{P}^n_k]=k[x_0,\ldots,x_n]/(0)=k[x_0,\ldots,x_n]. \end{align*} This example shows that projective varieties include the ambient projective spaces themselves, not only proper closed subsets inside them. [/example] After seeing that the whole ambient space is itself a projective variety, the next basic closed subsets are those defined by linear homogeneous equations. Linear equations are the projective analogue of linear subspaces through the origin, but a system of them may cut out only the zero vector in $k^{n+1}$, in which case it has no projective points at all. The definition therefore records the needed nonzero-solution condition and isolates the linear objects used for coordinates, hyperplanes, and embeddings. [definition: Projective Linear Subspace] A projective linear subspace of $\mathbb{P}^n_k$ is a projective variety of the form \begin{align*} V_+(L_1, \ldots, L_r) \end{align*} where $L_1, \ldots, L_r \in k[x_0, \ldots, x_n]$ are homogeneous linear polynomials and their common solution space in $k^{n+1}$ contains a nonzero vector. [/definition] A projective linear subspace is the projectivisation of a vector subspace of $k^{n+1}$. This relationship is one of the reasons linear algebra remains visible throughout projective geometry. [example: A Projective Line in the Projective Plane] In $\mathbb{P}^2_k$, let \begin{align*} L=V_+(x_0+x_1+x_2). \end{align*} The polynomial $x_0+x_1+x_2$ is homogeneous of degree $1$, so its vanishing condition is independent of the chosen homogeneous representative: for every $\lambda \in k^\times$, \begin{align*} (\lambda x_0)+(\lambda x_1)+(\lambda x_2)=\lambda(x_0+x_1+x_2). \end{align*} Thus $L$ is a well-defined projective linear subspace of $\mathbb{P}^2_k$. On the affine chart $x_2 \ne 0$, every point has a representative of the form $[u:v:1]$, obtained from $[a_0:a_1:a_2]$ by multiplying by $a_2^{-1}$. Substituting $x_0=u$, $x_1=v$, and $x_2=1$ into the homogeneous equation gives \begin{align*} u+v+1=0. \end{align*} So the part of $L$ inside this chart is the affine line $u+v+1=0$. The points of $L$ with $x_2=0$ satisfy \begin{align*} x_0+x_1+0=0. \end{align*} Hence $x_1=-x_0$. Since $[x_0:x_1:0]$ is a projective point, not both $x_0$ and $x_1$ are zero, so $x_0 \ne 0$. Multiplying the representative by $x_0^{-1}$ gives \begin{align*} [x_0:-x_0:0]=[1:-1:0]. \end{align*} Therefore the projective line is the affine line $u+v+1=0$ together with the single point at infinity $[1:-1:0]$, showing how projective closure records the limiting direction of an affine linear equation. [/example] Projective lines show how adding points at infinity completes affine linear equations. The next natural class asks what happens for the first nonlinear plane equations: homogeneous quadratics. These equations package the affine appearances called ellipses, parabolas, and hyperbolas into a single chart-independent projective object. [definition: Projective Plane Conic] A projective plane conic over a field $k$ is a projective hypersurface in $\mathbb{P}^2_k$ defined by a nonzero homogeneous polynomial of degree $2$. [/definition] The affine shape seen by a conic depends on the chart, while its projective equation is coordinate-homogeneous. This separation between intrinsic projective object and chosen affine view is a recurring theme. [example: Projective Closure of an Affine Parabola] Let $k$ be a field with $\operatorname{char}(k) \ne 2$. On the affine chart $x_2 \ne 0$ of $\mathbb{P}^2_k$, write \begin{align*} x=\frac{x_0}{x_2} \end{align*} and \begin{align*} y=\frac{x_1}{x_2}. \end{align*} The affine equation $y=x^2$ therefore becomes \begin{align*} \frac{x_1}{x_2}=\left(\frac{x_0}{x_2}\right)^2. \end{align*} Since $x_2 \ne 0$ on this chart, multiplying both sides by $x_2^2$ gives \begin{align*} x_1x_2=x_0^2. \end{align*} Equivalently, the homogeneous equation is \begin{align*} x_1x_2-x_0^2=0. \end{align*} Thus the projective closure of the affine parabola is \begin{align*} X=V_+(x_1x_2-x_0^2)\subset \mathbb{P}^2_k. \end{align*} On the line at infinity, we have $x_2=0$. Substituting $x_2=0$ into the defining equation gives \begin{align*} x_1\cdot 0-x_0^2=0. \end{align*} So \begin{align*} -x_0^2=0. \end{align*} Multiplying by $-1$ gives \begin{align*} x_0^2=0. \end{align*} Because $k$ is a field, this implies $x_0=0$. Hence any point of $X$ at infinity has the form $[0:x_1:0]$. A projective point cannot have all coordinates zero, so $x_1\ne 0$, and rescaling by $x_1^{-1}$ gives \begin{align*} [0:x_1:0]=[0:1:0]. \end{align*} Therefore the projective closure adds exactly one point at infinity, namely $[0:1:0]$, recording the single direction in which the affine parabola escapes to infinity. [/example] The homogeneity condition is not cosmetic. Without it, an equation may depend on the chosen representative of a projective point, so it cannot define a subset of projective space. [example: A Non-Homogeneous Equation Fails in Projective Space] Assume $2\ne 0$ in $k$, and consider the polynomial $f=x_0+1 \in k[x_0,x_1]$. The two nonzero vectors $(-1,1)$ and $(-2,2)$ represent the same point of $\mathbb{P}^1_k$, because \begin{align*} (-2,2)=2(-1,1) \end{align*} and $2\in k^\times$. Hence both representatives determine the projective point $[-1:1]$. Evaluating $f$ on the first representative gives \begin{align*} f(-1,1)=(-1)+1=0. \end{align*} Evaluating $f$ on the second representative gives \begin{align*} f(-2,2)=(-2)+1=-1. \end{align*} Since $-1\ne 0$ in a field, the same projective point has one representative on which $f$ vanishes and another representative on which $f$ does not vanish. Therefore the condition $x_0+1=0$ depends on the chosen representative and does not define a well-defined projective zero set; homogeneity is precisely the compatibility condition that prevents this representative-dependence. [/example] ## Properties Projective varieties form the closed sets of a natural topology on projective space. Ordinary metric or Euclidean closedness is not the right invariant notion for algebraic geometry, because it does not remember which sets are forced by homogeneous polynomial equations. The topology used here declares precisely those equation-defined projective zero loci to be closed. [definition: Projective Zariski Topology] The projective Zariski topology on $\mathbb{P}^n_k$ is the topology whose closed sets are the projective zero loci $V_+(S)$ for sets $S \subset k[x_0, \ldots, x_n]$ of homogeneous polynomials. [/definition] Under the set-theoretic convention used on this page, a projective variety in $\mathbb{P}^n_k$ is a subset that is closed in the projective Zariski topology. Equivalently, it is a set of the form $V_+(S)$ for some set $S \subset k[x_0,\ldots,x_n]$ of homogeneous polynomials. Authors who reserve variety for irreducible closed sets add the extra hypotheses that the set be nonempty and irreducible; this page states those hypotheses explicitly when they are needed. This property is more than terminology. It means topological operations such as closure, [irreducible decomposition](/theorems/2122), and continuity of regular maps can be studied through homogeneous equations. A projective point, however, represents an entire line of nonzero affine vectors, so passing between projective geometry and ordinary affine algebra requires keeping track of all representatives at once. The affine cone packages those representatives, together with the missing origin, into a single affine subset. [definition: Affine Cone] Let $X \subset \mathbb{P}^n_k$ be a projective variety. The affine cone over $X$ is \begin{align*} C(X)=\{a \in k^{n+1} : [a] \in X \text{ when } a \ne 0\}\cup\{0\}. \end{align*} [/definition] The cone includes the origin because every homogeneous polynomial vanishes at the origin. This creates a useful affine object, but it also introduces two features not present in projective space itself: the cone vertex and the many scalar representatives of each projective point. The key question is whether the cone construction loses information. The recovery statement below explains exactly how the original projective variety is obtained again: remove the vertex and identify nonzero vectors that differ by a scalar. [quotetheorem:9496] The affine cone viewpoint is especially useful when translating questions about projective varieties into questions about graded rings, singularities at the cone vertex, and homogeneous ideals. The next definition isolates projective varieties that behave like one algebraic piece rather than a union of smaller closed pieces. [definition: Irreducible Projective Variety] A projective variety $X \subset \mathbb{P}^n_k$ is irreducible if it is nonempty and cannot be written as \begin{align*} X = X_1 \cup X_2 \end{align*} with $X_1$ and $X_2$ proper projective subvarieties of $X$. [/definition] Irreducibility is the geometric counterpart of primality for homogeneous ideals, after the irrelevant ideal issue is excluded. The obstruction is the same projective one seen earlier: algebra supported only at the irrelevant ideal should not count as a genuine projective component. Once that exceptional behaviour is removed by using the correct homogeneous ideal, being one geometric piece is detected algebraically by primeness. [quotetheorem:9497] The criterion is useful because it turns the geometric test for being one piece into an algebraic test on the homogeneous vanishing ideal. For example, a union of two projective subvarieties has a vanishing ideal that reflects a decomposition, while an irreducible projective variety has no such decomposition and is detected by primality. The hypothesis that the ideal be the actual homogeneous vanishing ideal, rather than an arbitrary homogeneous ideal with the same projective zero set, is essential: components supported only at the irrelevant ideal do not appear as points of projective space. In practice, the theorem is used by computing or recognizing $I(X)$ and then applying the familiar prime-ideal language of quotient rings, dimension, and later function fields. ## Relationship to Other Concepts Projective varieties are closely related to affine varieties, but neither language replaces the other. Affine varieties are suited to coordinate rings of ordinary polynomial functions; projective varieties are suited to homogeneous coordinates, compactification, and global geometry. A basic passage from affine to projective geometry is homogenisation. This is not just a way to manufacture examples; it is the operation that records what an affine variety does at infinity. An affine equation in $k[x_1, \ldots, x_n]$ can often be converted into a homogeneous equation in $k[x_0, \ldots, x_n]$ by adding a new coordinate. The next definition names the smallest projective object containing the affine variety in a chosen projective chart. [definition: Projective Closure] Let $Y \subset \mathbb{A}^n_k$ be an affine variety, and embed $\mathbb{A}^n_k$ into the chart $x_0 \ne 0$ of $\mathbb{P}^n_k$ by \begin{align*} (a_1,\ldots,a_n) \mapsto [1:a_1:\cdots:a_n]. \end{align*} The projective closure of $Y$ is the closure $\overline{Y}^{\mathbb{P}}$ of its image in the projective Zariski topology on $\mathbb{P}^n_k$. [/definition] The projective closure is therefore the smallest projective variety containing the chosen affine copy of $Y$, but the topology is doing the work. Its added points at infinity often encode asymptotic directions, missing intersections, or limiting behaviour invisible in a chosen affine chart. The definitions so far are classical and reduced: they describe subsets of projective space cut out by homogeneous equations. Scheme theory keeps the same homogeneous equations but also remembers nilpotent structure, multiplicity, and behaviour after base change. The next definition explains how classical projective varieties pass into modern scheme language without identifying the two conventions. [definition: Proj of a Graded Ring] Let $R=\bigoplus_{d=0}^{\infty} R_d$ be a graded ring, and let \begin{align*} R_+=\bigoplus_{d=1}^{\infty}R_d. \end{align*} The set underlying $\operatorname{Proj}(R)$ consists of the homogeneous prime ideals $\mathfrak{p} \trianglelefteq R$ such that $R_+ \not\subset \mathfrak{p}$. [/definition] The full scheme $\operatorname{Proj}(R)$ also carries the Zariski topology and a structure sheaf built from degree-zero parts of localisations of $R$. For a projective variety $X \subset \mathbb{P}^n_k$, the associated projective scheme is built from $\operatorname{Proj}(k[X])$. This is the bridge from classical projective varieties to modern scheme language. The next definition records the standard way homogeneous forms define maps into projective space. [definition: Projective Morphism Given by Homogeneous Forms] Let $X \subset \mathbb{P}^n_k$ be a projective variety, and let $F_0, \ldots, F_m \in k[x_0, \ldots, x_n]$ be homogeneous polynomials of the same positive degree. If the $F_i$ do not vanish simultaneously at any point of $X$, then they define the map $\varphi: X \to \mathbb{P}^m_k$ by \begin{align*} \varphi([a_0:\cdots:a_n])=[F_0(a_0,\ldots,a_n):\cdots:F_m(a_0,\ldots,a_n)]. \end{align*} [/definition] The same-degree condition is the map-level analogue of homogeneity for equations. It ensures that rescaling the homogeneous coordinate representative $(a_0,\ldots,a_n)$ rescales every coordinate of the image by the same factor. [example: Veronese Embedding of the Projective Line] Over a field $k$, the quadratic forms $x_0^2$, $x_0x_1$, and $x_1^2$ have the same degree $2$. They do not vanish simultaneously on any projective point of $\mathbb{P}^1_k$: if $x_0^2=0$ and $x_1^2=0$, then $x_0=0$ and $x_1=0$, which is not allowed for a homogeneous representative. Also, for every $\lambda \in k^\times$, \begin{align*} ((\lambda x_0)^2,(\lambda x_0)(\lambda x_1),(\lambda x_1)^2)=\lambda^2(x_0^2,x_0x_1,x_1^2). \end{align*} Thus these forms define a projective map \begin{align*} \nu_2:\mathbb{P}^1_k\to \mathbb{P}^2_k,\qquad \nu_2([x_0:x_1])=[x_0^2:x_0x_1:x_1^2]. \end{align*} We show that the image is exactly the conic \begin{align*} V_+(y_0y_2-y_1^2)\subset \mathbb{P}^2_k. \end{align*} First let $[y_0:y_1:y_2]=\nu_2([x_0:x_1])$. Then $y_0=x_0^2$, $y_1=x_0x_1$, and $y_2=x_1^2$ up to a common nonzero scalar, so it is enough to check the displayed representative: \begin{align*} y_0y_2-y_1^2=(x_0^2)(x_1^2)-(x_0x_1)^2. \end{align*} Since $(x_0x_1)^2=x_0^2x_1^2$, this becomes \begin{align*} x_0^2x_1^2-x_0^2x_1^2=0. \end{align*} Hence every point in the image lies on the conic. Conversely, let $[y_0:y_1:y_2]\in \mathbb{P}^2_k$ satisfy $y_0y_2-y_1^2=0$. If $y_0\ne 0$, set $t=y_1/y_0$. The equation gives \begin{align*} y_0y_2=y_1^2. \end{align*} Dividing by $y_0^2$ gives \begin{align*} \frac{y_2}{y_0}=\left(\frac{y_1}{y_0}\right)^2=t^2. \end{align*} Therefore \begin{align*} [y_0:y_1:y_2]=[1:t:t^2]=\nu_2([1:t]). \end{align*} If $y_0=0$, the equation becomes \begin{align*} 0\cdot y_2-y_1^2=0. \end{align*} Thus $y_1^2=0$, so $y_1=0$ because $k$ is a field. Since $[y_0:y_1:y_2]$ is a projective point, $y_2\ne 0$, and hence \begin{align*} [y_0:y_1:y_2]=[0:0:y_2]=[0:0:1]=\nu_2([0:1]). \end{align*} So every point of the conic is in the image of $\nu_2$. Thus the Veronese map identifies $\mathbb{P}^1_k$ with the projective plane conic $V_+(y_0y_2-y_1^2)$, showing how homogeneous coordinate maps produce projective varieties as their images. [/example] Over $\mathbb{C}$, projective varieties also have an analytic life. A complex projective variety inherits a topology from complex projective space, and smooth complex projective varieties are compact complex manifolds. The next theorem records the compactness property that makes projective geometry a natural setting for global arguments. [quotetheorem:9498] This theorem is not part of the algebraic definition, but it explains much of the geometric force of projective varieties. Projective space behaves like a compact completion of affine space, and projective varieties inherit that global finiteness. ## Beyond and Connections Projective varieties are best read as the global counterpart to affine varieties. Passing from affine space to projective space adds points at infinity, but it also changes which equations are allowed: homogeneous equations replace arbitrary polynomial equations, and the projective Zariski topology records vanishing that is compatible with scaling. This is why many affine questions are first homogenized, studied projectively, and then returned to an affine chart. Several later topics refine the basic picture introduced here. Irreducible projective varieties lead to function fields, [dimension](/page/Dimension), and birational geometry. Smooth projective varieties form the main setting for intersection theory and many cohomological invariants. Projective morphisms generalize the compactness behavior of projective space and explain why projective varieties occupy such a central position in algebraic geometry. ## References Hartshorne, *Algebraic Geometry* (1977). Shafarevich, *Basic Algebraic Geometry 1* (1994). Harris, *Algebraic Geometry: A First Course* (1992). [Projective Space](/page/Projective%20Space).