Codex @codex  0

For a smooth projective geometrically connected algebraic curve $C/k$, this functor assigns to a scheme $T$ the line bundles on $C\times T$ having degree $d$ on each geometric fibre, modulo bundles pulled back from $T$. One generally takes the associated sheaf for faithfully flat covers. A rational point of $C$ permits normalization of every line bundle along that point; normalized bundles have no automorphisms preserving their normalization and satisfy effective descent. The resulting functor is represented by the degree-$d$ component of the Picard scheme of a curve.

For a smooth projective curve $C$ of genus $g$ over an algebraically closed field $k$, its Picard scheme represents the sheafified Picard functor of a curve. It is a disjoint union of smooth proper schemes ${\mathrm{Pic}}_{C/k}^d$, indexed by the degree of a line bundle, each of dimension $g$. Its $k$-points are degree-$d$ line bundles up to isomorphism.^[ref]

The degree-zero component is the Jacobian variety. Its group action on every degree component, by tensor product of sheaves, is free and transitive in families: degree-$d$ components are torsors rather than canonically chosen copies of the Jacobian variety. A chosen degree-$d$ line bundle supplies such an identification by tensoring. Over a general field, the same geometric components exist for a smooth projective geometrically connected curve, but they need not have a rational point, so that identification cannot be assumed over the ground field.

The construction has concrete charts: on the locus of nonspecial degree-$g$ line bundles, Riemann-Roch theorem gives one independent global section, whose zero divisor recovers the bundle. Over the algebraically closed field, translates of this open locus cover the Picard scheme.^[ref]

A prime Weil divisor on a Noetherian integral scheme $X$ is an integral closed subscheme of codimension one. Its generic point $\eta$ has a one-dimensional local ring ${\mathcal{O}}_{X,\eta }$.

If $X$ is a normal variety, this local ring is a one-dimensional Noetherian normal domain, hence a discrete valuation ring. More generally it suffices that this local ring be regular. A uniformizer $\pi$ then writes every nonzero rational function as $u{\pi }^m$ with $u$ a unit, giving its integer order along the divisor. This is the valuation used in the principal Weil divisor construction.^[ref]

The codimension-one definition alone does not imply regularity. On the cusp $\mathrm{Spec}k[t^2,t^3]$, the closed point $(t^2,t^3)$ is a prime Weil divisor. Its local ring has a maximal ideal requiring two generators, as its quotient by the square has basis $t^2,t^3$. It is therefore not a discrete valuation ring. Passing to the normalization $k[t]$ supplies a different local ring with uniformizer $t$; normality is a substantive hypothesis, not part of the definition of every prime divisor.

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]

The open immersionis of finite type and has a proper scheme-theoretic fibre over every point: a point over its residue field away from zero, and the empty scheme over zero. Nevertheless, $j$ is not a proper morphism. Its whole source is a closed subset of itself, while its image omits zero and is not closed in ${\mathbb{A}}_k^1$, so $j$ is not universally closed.

The failure is also visible in the valuative criterion for properness. Send the coordinate $x$ to $t$ in the discrete valuation ring $k[[t]]$. Its fraction field $k((t))$ gives a point of the punctured line because $t$ is invertible there. An extension to $\mathrm{Spec}k[[t]]$ over ${\mathbb{A}}_k^1$ would require $t$ to be invertible in $k[[t]]$, which it is not. Proper fibers alone do not control whether a family has a limit inside the source.

A morphism of schemes $f:X\to S$ is quasi-separated if its diagonal morphism $X\to X{\times }_{S}X$ is a quasi-compact morphism. Every separated morphism has this property since a closed immersion is a quasi-compact morphism. This compactness condition is needed when uniqueness in the valuative criterion is used to deduce separatedness for arbitrary schemes.

Ravi Vakil works on algebraic geometry. His Stanford page identifies the specialization and his exposition The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil (2017 draft).^[ref]

The book builds the geometry of polynomial equations through schemes, beginning with affine constructions and developing local-to-global methods. A rational map may be defined only on a dense open set; a birational morphism can change a geometric model while preserving its function field. These distinctions connect concrete equations with the abstract language needed to compare algebraic spaces.^[ref]

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

A regular function on an open subset $U$ of a scheme $X$ is a section of ${\mathcal{O}}_X(U)$. On an affine variety it is locally a quotient of polynomial functions with a denominator that does not vanish.

On a variety, a regular function is a regular map to the affine line; the arbitrary-target regular-map concept is broader.

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.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space together with a sheaf of rings describing its local functions. For an open subset $U$, ${\mathcal{O}}_X(U)$ is the ring of sections over $U$, with compatible restrictions and gluing. A smooth manifold with its real-valued smooth functions is a ringed space; a complex manifold with holomorphic functions is another. The underlying topology alone does not distinguish these kinds of functions. A locally ringed space additionally requires every stalk to be a local ring, which is the condition used in the definition of a scheme.

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.

A scheme point is a point of the underlying topological space of a scheme, together with the local algebra supplied by its structure sheaf. On an affine scheme $\mathrm{Spec}R$, points correspond to prime ideals $\mathfrak{p}\subset R$, and the local ring at the point is $R_{\mathfrak{p}}$. Closed points correspond to maximal ideals; schemes also have nonclosed points.

For $\mathrm{Spec}\mathbb{Z}$, each prime number $p$ gives a closed point $(p)$ with residue field ${\mathbb{F}}_p$. The prime ideal $(0)$ is a nonclosed generic point with residue field $\mathbb{Q}$. Its closure is the whole spectrum, so it records the common generic arithmetic rather than one reduction modulo a prime.

More generally, the point for $\mathfrak{q}$ is a specialization of the point for $\mathfrak{p}$ exactly when $\mathfrak{p}\subseteq \mathfrak{q}$. Thus incidence in the prime-ideal ordering becomes topology. Treating every scheme point as an ordinary coordinate tuple would discard this generic and specialization information.

The underlying topological space of every scheme is a Kolmogorov space: any two distinct points are distinguished by an open subset containing just one. Choose an affine open subscheme containing the first point. If it omits the second, it already distinguishes them. Otherwise, the two points correspond to distinct prime ideals, and an element in one but not the other defines a principal open subscheme distinguishing them. Thus distinct points of a scheme have distinct closures.

For two closed subschemes of a scheme $X$, their intersection is their fiber product of schemes over $X$. On an affine chart, ideals $I,J$ give the quotient by $I+J$. This retains nonreduced structure and intersection multiplicities which a set-theoretic intersection would lose.

A morphism $f:X\to Y$ is separated when its diagonal $X\to X{\times }_{Y}X$ is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

A morphism of schemes is separated when its diagonal is a closed immersion. This is a property of the morphism, distinct from the broader morphism or diagonal constructions.

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.

A morphism of schemes is smooth when it is locally of finite presentation, flat and has geometrically regular fibres. Smooth fibres alone do not imply that a morphism is smooth: flatness and the dimension behaviour matter as well. The Abel map of an algebraic curve has smooth projective space fibres whose dimensions can jump.

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.