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An open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.

For a classical algebraic variety $X$ and affine variety $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the coordinate ring generators also prove uniqueness.

For a smooth algebraic variety of dimension $r$ over a field, the sheaf of Kähler differentials over a field is locally free of rank $r$ and defines its algebraic cotangent bundle. Its dual is the tangent sheaf of a scheme. For a smooth hypersurface of degree $d$ in ${\mathbb{P}}^n$, the Conormal exact sequence for Kähler differentials is the short exact sequence $0\to {\mathcal{O}}_X(-d)\to {\Omega }_{{\mathbb{P}}^n}^1{|}_X\to {\Omega }_X^1\to 0$.

An algebraic curve is a one-dimensional algebraic variety. On a smooth algebraic curve, the local ring at every closed point is a discrete valuation ring, which allows orders of zeros and poles to be recorded as a divisor on an algebraic curve.

A group variety is an algebraic variety whose multiplication and inversion are morphisms.

An algebraic set is a zero locus of polynomial equations, allowing reducible sets. An affine algebraic set uses ordinary polynomial equations; a projective algebraic set uses homogeneous equations in projective coordinates. On affine or projective space, closed sets in the Zariski topology are algebraic sets, and an irreducible algebraic set is an algebraic variety in the classical convention.

An algebraic surface is a two-dimensional algebraic variety. The birational classification of smooth projective surfaces uses their curves, divisor intersections and pluricanonical maps.

For a vector bundle $E$ of rank $r$ on a smooth algebraic variety, algebraic Chern classes have $c_0(E)=1$, vanish for $i>r$, commute with pullback, and obey the Whitney sum formula for Chern classes for every short exact sequence of vector bundles. A line bundle $\mathcal{O}(D)$ has total class $1+[D]$. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular, $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$ and $c_1(L\otimes M)=c_1(L)+c_1(M)$ in the ordinary Chow ring.

A closed subvariety is an integral closed subscheme of an algebraic variety. It supplies the lower-dimensional loci on which positivity is tested by the Nakai–Moishezon criterion.

For a smooth complex projective algebraic variety, the cycle class map sends an algebraic cycle of codimension $r$ to the Poincare dual of its homology class. Rational equivalence gives the same cohomology class, so the map descends to the Chow group. It respects intersection products, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of Chow classes.

The degree of a projective algebraic variety of dimension $r$ is the length, with intersection multiplicities, of its intersection with a sufficiently general linear subspace of complementary codimension. Equivalently, its Hilbert polynomial has leading coefficient $\mathrm{deg}X/r!$. For a projective curve, a general hyperplane section measures this degree.

This is the supremum of lengths of strict chains of nonempty irreducible closed subsets. Closedness is relative to the given space. This is the relevant dimension for locally closed subsets of algebraic varieties, rather than a general Hausdorff-space notion of dimension.

An algebraic variety whose irreducible components all have the same dimension of an algebraic set. The components of a generic fibre in the fiber dimension theorem have this property even when that fibre is reducible.

Étale cohomology is sheaf cohomology on the site whose objects are étale morphisms $Y\to X$ and whose coverings are jointly surjective families of such morphisms. It provides cohomological invariants for algebraic varieties over fields of positive characteristic. In Deligne-Lusztig theory, compactly supported groups with $\ell$-adic coefficients, for $\ell$ different from the field characteristic, give finite-dimensional representations of the finite groups acting on the varieties.

The function field $k(X)$ of an irreducible algebraic variety consists of its rational functions.

The Hasse–Weil zeta function of an algebraic variety over a number field is assembled from local point-counting factors at finite places. At a good place with residue field of size $q$, the factor is $Z(V_q,q^{-s})$, where $Z(V_q,T)=\exp ({\sum }_{n\ge 1}\#V_q({\mathbb{F}}_{q^n})T^n/n)$. Defining all bad-place factors requires a specified integral model or convention. The zeta function of an elliptic curve over a finite field is a local factor, rather than this global product.

For closed irreducible subvarieties $Y,Z$ of a smooth $d$-dimensional algebraic variety $X$, every irreducible component of their intersection has dimension at least $\dim Y+\dim Z-d$. Locally the diagonal of $X\times X$ is cut out by $d$ parameters. Intersecting it with $Y\times Z$ and applying the Krull height theorem gives the bound. The three-dimensional affine quadric cone shows that the same lower bound can fail when $X$ is singular.

An algebraic variety whose underlying topological space with its Zariski topology is an irreducible topological space: it is nonempty and cannot be a union of two proper closed subsets. For a reduced affine variety, this is equivalent to its coordinate ring being an integral domain.

Let $f:X\to Y$ be a proper morphism of irreducible algebraic varieties with equal dimension. If some fibre has an isolated point, its corresponding irreducible component has dimension zero. Apply the fiber dimension theorem to $X\to f(X)$: $0\ge \dim X-\dim f(X)$. Thus the closed image has dimension $\dim Y$ and equals $Y$. The incidence proof that a cubic surface contains a line is an application; the entire witness fibre need not be finite.

The normalization of an integral algebraic variety is obtained by taking the integral closure of its affine coordinate rings in its function field. Its natural morphism is finite and birational, and its source is normal. For Cartier divisors, bigness under finite normalization compares section growth on the two varieties.