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For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]

For a finite separable morphism $f:X\to Y$ of smooth projective curves over an algebraically closed field, the ramification divisor is the zero divisor of the nonzero differential map $f^{\ast }{\Omega }_Y^1\to {\Omega }_X^1$. WriteThese coefficients are the local different exponents. The canonical relation and degree formula areUnder tame ramification, $d_P=e_P-1$. In wild ramification it can be larger, so using $\sum (e_P-1)P$ without the tame hypothesis loses part of the divisor.^[ref]

For example, in characteristic $p$, the map ${\mathbb{P}}_t^1\to {\mathbb{P}}_x^1$ with $x=t^p-t$ is separable of degree $p$ and is unramified at finite points. Infinity has index $p$. Since $dx=-dt$, the differential canonical relation givesso $R_f=(2p-2)\infty$, exceeding $(p-1)\infty$. This verifies the genus-zero Riemann-Hurwitz identity and exhibits the wild correction explicitly. A purely inseparable map has identically zero differential, so this construction requires separability.

A locally Noetherian scheme $X$ is regular in codimension one, or satisfies $(R_1)$, if every local ring ${\mathcal{O}}_{X,x}$ of Krull dimension at most one is a regular local ring. This includes both the generic points and the codimension-one points of its irreducible components. Together with the depth condition $(S_2)$, it is equivalent to normality by Serre criterion for normality.^[ref]

For an integral scheme, the local ring at its generic point is already its function field, so only the codimension-one points need checking; their local rings must be discrete valuation rings. For a general scheme, omitting the generic-point test changes the condition. The nonreduced double point has no codimension-one points at all, but its zero-dimensional local ring $k[\epsilon ]/({\epsilon }^2)$ is not a regular local ring. Thus it fails $(R_1)$ despite satisfying the empty list of exactly-codimension-one tests.

Let $X$ be a scheme and $\mathcal{A}$ a quasi-coherent sheaf of nonnegatively graded algebras with ${\mathcal{A}}_0={\mathcal{O}}_X$. Its relative Proj is obtained by gluing $\mathrm{Proj}\mathcal{A}(U)$ over affine open subschemes $U\subseteq X$. The standard affine opens of Proj and localization identify these constructions on overlaps. It is compatible with arbitrary base change:^[ref]

If $\mathcal{A}$ is generated in degree one and a finite locally free sheaf $\mathcal{E}$ surjects onto ${\mathcal{A}}_1$, the resulting surjective morphism of sheaves $\mathrm{Sym}\mathcal{E}\to \mathcal{A}$ gives a closed immersion into ${\mathrm{Proj}}_X(\mathrm{Sym}\mathcal{E})$, the projective bundle in the quotient-line convention. Local degree-one generators always give such embeddings over suitable affine open subschemes. A global generating vector bundle requires an additional existence hypothesis on ${\mathcal{A}}_1$; it does not follow merely from finite type over an arbitrary scheme.

The sheaf of total quotient rings, or sheaf of meromorphic functions, of a scheme $X$ is constructed by locally allowing denominators that are nonzerodivisors. For an open set $U$, let $S(U)$ consist of sections of ${\mathcal{O}}_X(U)$ whose germ is a nonzerodivisor at every point of $U$. Sheafifying the presheafgives ${\mathcal{K}}_X$. The stalkwise regular-denominator condition ensures that restriction to smaller open sets is defined. Merely taking arbitrary fraction rings of section rings is not a substitute for this construction.^[ref]

For a locally Noetherian scheme, ${\mathcal{K}}_{X,x}$ is the total quotient ring of ${\mathcal{O}}_{X,x}$, and on an affine open $\mathrm{Spec}A$ its sections are the total quotient ring of $A$. These identifications require the stated hypothesis.^[ref]

For an integral scheme this recovers its field of rational functions locally. For $A=k[\epsilon ]/({\epsilon }^2)$, every nonzerodivisor is already a unit, so its total quotient ring is $A$ itself and retains a nonzero nilpotent. Total quotient rings therefore need not be fields or remove infinitesimal structure.

For a variety $X$ over a perfect field, the singular locus is the closed complement of its smooth locus. It consists exactly of points whose local rings are not regular local rings.^[ref]

A normal variety in this setting has no singular points of codimension zero or one. Its generic local rings are fields, and its codimension-one local rings are one-dimensional Noetherian normal domains, hence discrete valuation rings and therefore regular. Any singular locus of a normal variety thus has codimension at least two.

Over an imperfect field, the nonsmooth locus and the nonregular locus can differ. For $k={\mathbb{F}}_p(s)$, the scheme $X=\mathrm{Spec}k[u]/(u^p-s)$ is the spectrum of a field and is regular. After base change adjoining a $p$th root of $s$, its defining equation becomes a $p$th power, so it is not smooth over $k$. The perfect-field qualifier is therefore essential when using normality to exclude nonsmooth points.

At a closed point $p$ of a variety over an algebraically closed field, singularity means failure of the regular local ring condition:For an irreducible variety of dimension $d$, the right side is $d$. Thus the usual test is $\dim T_{p}X>d$. The Zariski tangent space measures first-order directions, while the local dimension measures the actual local components; excess tangent directions detect singularity.^[ref]

For the ordinary cusp $y^2=x^3$, both first derivatives vanish at the origin, giving tangent dimension two although the curve has dimension one. On a variety with components of different dimensions, compare with $\dim {\mathcal{O}}_{X,p}$, not the largest dimension elsewhere. Over a perfect field, the nonregular and nonsmooth point notions agree; over an imperfect field they need not, as explained in singular locus.

A smooth algebraic curve over a field $k$ is an algebraic curve whose structural morphism to $\mathrm{Spec}k$ is smooth. Over a perfect field, this is equivalent to all its local rings being regular local rings. At each closed point, those rings have dimension one and are discrete valuation rings, so a local uniformizer measures orders of zeros and poles.

Over an imperfect field, regular local rings alone do not suffice. Let $k={\mathbb{F}}_p(s)$, let $K=k[u]/(u^p-s)$, and take $C=\mathrm{Spec}K[t]$. The polynomial ring $K[t]$ is regular. After the base change adjoining a $p$th root $v$ of $s$, the equation becomes $(u-v{)}^p=0$, giving a nonzero nilpotent element. The resulting curve is not geometrically reduced and cannot be smooth over $k$.

Thus smoothness tests regularity after extending the ground field, rather than only before that extension. The familiar nonsingular plane-curve Jacobian criterion over an algebraic closure is a geometric version of the same condition.^[ref]

The smooth locus of a variety over a field $k$ is the open subset where its structure morphism to $\mathrm{Spec}k$ is smooth. Over a perfect field, these are precisely the points with regular local rings, and for a reduced finite-type variety the locus is dense.^[ref]

At a closed point over an algebraically closed field, the test isThe left side is the embedding dimension, and equality is the regular local ring criterion. If $X$ is pure of dimension $d$, its closed-point local dimensions are all $d$, so the familiar test is $\dim T_{p}X=d$. On a mixed-dimensional variety the local dimension must be retained. A point on a disjoint smooth one-dimensional component remains smooth even when another component has dimension two.

Over an imperfect field, a regular local ring need not give a smooth point over that field; geometric regularity, rather than regularity alone, is the relevant condition.

For a proper morphism $f:X\to Y$ with $Y$ a locally Noetherian scheme, Stein factorization separates its connected-fiber part from its finite part:The morphism $g$ is proper with geometrically connected scheme-theoretic fibres, $h$ is a finite morphism, and $g_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Z$. The coherent sheaf $f_{\ast }{\mathcal{O}}_X$ makes the relative spectrum finite over $Y$; the connectedness assertion is the substantive part of the theorem.^[ref]

For a finite morphism, $Z=X$ and $g$ is the identity. For $X=Y\times {\mathbb{P}}^1$ with the projection to $Y$, the intermediate scheme is $Y$. Thus the construction records disconnected finite-sheet data without retaining the positive-dimensional geometry inside each connected fibre.

If $X$ is normal, so is $Z$. For a surjective proper morphism from a smooth algebraic surface to a smooth algebraic curve over an algebraically closed field, $Z$ is therefore a smooth algebraic curve. It is projective when the target curve is projective; properness relative to an affine curve does not itself imply projectivity over the ground field.

For a morphism of schemes $S^{\prime }\to S$ and an $S^{\prime }$-scheme $X$, the Weil restriction is the functor defined on $S$-schemes byWhen this functor is represented by a scheme, that representing object is also called the Weil restriction. Representability needs hypotheses; it is not part of a formal guarantee for every $X$ and every base map.^[ref]

For $\mathbb{C}/\mathbb{R}$, restricting the affine line gives ${\mathrm{Res}}_{\mathbb{C}/\mathbb{R}}{\mathbb{A}}_{\mathbb{C}}^1\cong {\mathbb{A}}_{\mathbb{R}}^2$. Indeed, a complex coordinate is expressed as $a+bi$, and for any real algebra $R$ the corresponding points are $R{\otimes }_{\mathbb{R}}\mathbb{C}\cong R^2$. The construction expresses geometry over an extension field using coordinates over the smaller field, while retaining its full point functor.

The Zariski closure of a subset $S$ of an algebraic variety or a scheme is the smallest Zariski-closed subset containing $S$. For $S\subseteq k^n$, define the ideal $I(S)$ of all polynomials vanishing on $S$. ThenThis description follows because an intersection of polynomial zero sets is again the common zero set of their equations.^[ref]

Every infinite subset of the complex affine line is Zariski dense: a nonzero one-variable polynomial has only finitely many roots of a polynomial. Thus $S=\{1,1/2,1/3,\dots \}$ has Zariski closure all of $\mathbb{C}$, although its closure in the ordinary Euclidean topology is only $S\cup \{0\}$. In algebraic geometry, taking a closure retains all polynomial consequences, rather than all ordinary sequential limits.