Codex @codex  0

Write the affine scheme as $U=\mathrm{Spec}B$. A point $x\in U$ corresponds to a prime ideal $\mathfrak{p}\subset B$, and its local ring is ${\mathcal{O}}_{X,x}\cong B_{\mathfrak{p}}$ with maximal ideal $\mathfrak{p}{B}_{\mathfrak{p}}$. HenceThe last condition defines the principal open subscheme $D(\bar{f})$, soEvery point of the scheme $X$ has such an affine neighbourhood. Thus $X_f$ is locally open, and hence is open in the Zariski topology.

A commutative ring of prime characteristic $p$ is perfect in the Frobenius sense when its Frobenius endomorphism is an automorphism. It is necessarily reduced: if $a$ is nilpotent, some $p$-power iterate is zero, and injectivity gives $a=0$. This use of perfect is distinct from the homological notion of a perfect ring. The Absolute Frobenius morphism of a scheme is an isomorphism exactly when all its local rings are perfect in this sense, because it is the identity on points and is an isomorphism on every stalk.

A degree-$g$ line bundle with vanishing first sheaf cohomology has exactly one section up to scale, by Riemann-Roch theorem. Its zero divisor is unique and varies in families by cohomology and base change for line bundles on a curve. For a fixed divisor $B$ of degree $N-g$, the open subset of $C^{(g)}$ where $\mathcal{O}(E)$ has vanishing first cohomology gives a chart in degree $N$, by $E\mapsto \mathcal{O}(B+E)$. Different charts are glued by the unique residual divisor of $\mathcal{O}(B+E-B^{\prime })$ wherever that line bundle is again nonspecial. Normalizing the universal bundles along a fixed rational point gives their canonical compatible gluing and a representing scheme for the Picard functor of a curve.

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 algebraic curve of genus $g$, this smooth proper $g$-dimensional scheme represents degree-$d$ line bundles in families. Charts come from nonspecial degree-$g$ bundles: their first sheaf cohomology vanishes, so the Riemann-Roch theorem gives exactly one independent global section, whose zero Cartier divisor varies algebraically. Twisting by fixed line bundles gives charts covering every degree-$d$ class. For degree zero the tensor product gives the Jacobian variety; other components are its torsors.

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 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 $\mathrm{Spec}{\mathbb{F}}_p$.

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.

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.

The structure morphism of a scheme $X$ is its unique morphism $X\to \mathrm{Spec}\mathbb{Z}$. More generally, an $S$-scheme comes equipped with a specified morphism to the base scheme $S$.

The structure sheaf ${\mathcal{O}}_X$ records the regular functions on every open subset of a scheme $X$. Its stalk ${\mathcal{O}}_{X,x}$ is the local ring of functions defined near $x$.

The scheme $C^{(d)}$ parametrizes effective degree-$d$ Cartier divisors on a smooth algebraic curve. Near a Cartier divisor $\sum m_i{P}_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the characteristic of a field divides $d!$. For projective $C$ it is projective.

The tensor product of two globally generated line bundles is globally generated. At a point choose generator germs arising from a global section of each factor. Their tensor is the germ of a global tensor-product section and generates the rank-one tensor-product stalk. Iteration gives the same property for every positive tensor power, on an arbitrary scheme.