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A divisor is a finite formal integer combination of closed points. For a divisor $D$,

This is the one-dimensional case of normalization of an algebraic variety, constructed from integral closures of coordinate rings.

The normalization of an irreducible algebraic curve is a normal, hence regular, curve with a finite birational morphism to the original curve. Over a perfect field this regular curve is smooth.

Irreducible varieties are birational exactly when they have isomorphic function fields.

This equivalence is the relation studied by birational geometry.

A determinantal variety is cut out by minors imposing an upper bound on matrix rank.

A degree-$d$ projective hypersurface $X_d\subseteq {\mathbb{P}}_k^n$ is the closed subscheme cut out by one nonzero homogeneous polynomial of degree $d$.

This is a homogeneous-polynomial hypersurface in projective space.

The zero set of a nonconstant polynomial in $n$ affine variables has dimension $n-1$. Passing to the square-free part gives its radical principal ideal.

This is a polynomial-defined hypersurface in affine space.

The Krull dimension of an affine variety $X$ is the supremum of lengths of strict chains of nonempty irreducible closed subsets. It equals the Krull dimension of its coordinate ring, and is the maximum of the dimensions of its irreducible components.

For an irreducible affine variety over an algebraically closed field, it also equals the minimum closed-point Zariski tangent space dimension, by dimension from minimum tangent dimension. This second description uses irreducibility or a common component dimension. For a union of a plane and a line not contained in it, the global dimension is two but the minimum tangent dimension is one along the isolated part of the line. The chain definition remains valid without that extra hypothesis.

For $P\in X\subseteq {\mathbb{A}}^n$, the Zariski tangent space is the common kernel at $P$ of the differentials of all polynomials in $I(X)$.

A twisted cubic and a smooth plane cubic embedded in ${\mathbb{P}}^3$ both have degree three, but their genera are zero and one. Thus equal projective degree does not imply isomorphism.

The twisted cubic is the image ofin ${\mathbb{P}}^3$. Its ideal is generated by the three $2\times 2$ minors ofand it has degree three.

If the Hilbert polynomial of the homogeneous coordinate ring of a projective curve isthen $\mathrm{deg}C=d$. Equivalently, a sufficiently general hyperplane section has scheme-theoretic length $d$.

The graph $\{(x,e^x):x\in \mathbb{C}\}$ is Zariski dense in ${\mathbb{A}}_{\mathbb{C}}^2$. If ${\sum }_j{p}_j(x)e^{jx}=0$, repeated application of $D-m$ removes the largest exponential term while acting injectively on the others, proving inductively that every $p_j$ vanishes.

For an algebraically closed field $k$ and an ideal $I\subseteq k[x_1,\dots ,x_n]$, the strong Hilbert Nullstellensatz isIts weak form says that an ideal with empty affine zero set is the unit ideal. Equivalently, if finitely many polynomials have no common zero, a polynomial combination of them equals one.

Every finitely generated algebra $A$ over a field has algebraically independent elements $y_1,\dots ,y_d$ such that $A$ is finite as a module over the polynomial subalgebra $k[y_1,\dots ,y_d]$. The integer $d$ is the Krull dimension of $A$.

A Zariski-open set is the complement of a Zariski-closed set. A distinguished affine open has the form $D(f)=\{x:f(x)\ne 0\}$.

A Zariski-closed set is a common zero set of polynomials. Arbitrary intersections and finite unions of such sets are again Zariski closed.

A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.

An algebraic variety over a field $k$ is, in a common modern convention, a reduced scheme that is a separated scheme of finite type over $k$. Some authors also require it to be irreducible; this wiki states that extra condition when reducible spaces matter. A classical convention over an algebraically closed field describes the same geometry using polynomial zero sets and their regular functions.

An affine variety has a coordinate ring $A$, a finitely generated reduced $k$-algebra, and its scheme model is $\mathrm{Spec}A$. General algebraic varieties glue finitely many such affine charts along open subsets. The regular functions are part of the structure: their local rings record which functions can be evaluated near each point.

For example, $x^2+y^2=1$ defines a real algebraic curve, while $y^2=x^3$ defines a cuspidal cubic. Both are one-dimensional, but the cusp has a singular local ring at the origin. The scheme defined by $x^2=0$ additionally remembers a nonzero nilpotent element; it is not a reduced scheme and so is outside this variety convention. This illustrates why an underlying zero set alone cannot describe every scheme.

On an integral scheme $X$, a Cartier divisor is represented on an open cover $U_i$ by nonzero rational functions $f_i$, with $f_i/f_j$ a regular unit on every overlap. Changing the local equation by a regular unit leaves the divisor unchanged. A principal Cartier divisor uses one global nonzero rational function on all the charts.

The transition ratios glue the local rank-one modules $f_i^{-1}{\mathcal{O}}_{U_i}$ into the divisor line bundle ${\mathcal{O}}_X(D)$. If each $f_i$ is regular, it cuts out an effective Cartier divisor. Allowing poles gives signed divisor data.

On the projective line, the coordinate $t$ defines the principal Cartier divisor $[0]-[\infty ]$: it has one simple zero at zero and one simple pole at infinity. For a general scheme, the analogous definition uses invertible elements of the sheaf of total quotient rings, so nonzero elements must be distinguished from non-zero-divisors.^[ref]

A Weil divisor on a Noetherian integral scheme is a finite integer combination of integral codimension-one closed subschemes.