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Intersection theory assigns algebraic cycles and intersection multiplicities to intersections of algebraic varieties, and organizes these calculations modulo rational equivalence. On a smooth projective variety $X$, the intersection product makes its Chow groups into a graded ring:When representatives meet properly, one counts their common components with the appropriate intersection multiplicities. A moving argument extends this construction to arbitrary classes and makes it independent of the representatives.^[ref]

For projective plane curves of degrees $d$ and $e$ without a common component, Bézout theorem gives total intersection number $de$. The value counts a tangency more than once; it is not just the number of distinct points. The same formalism defines a self-intersection number even though a curve does not meet itself properly. For a line $L$ in ${\mathbb{P}}^2$, $L^2=1$ because another line represents the same Chow class and meets it once. By contrast, the exceptional divisor of the blowup of a smooth algebraic surface has square $-1$, measured by its normal bundle. These calculations connect divisors, Chern classes and birational geometry.

Toric geometry studies algebraic varieties with an action of an algebraic torus having a dense open orbit isomorphic to that algebraic torus. For normal algebraic varieties with a split algebraic torus, toric fans encode the geometry using integral toric cones and their intersections.

The simplest chart is the affine line: the multiplicative algebraic torus acts by scaling, with one open orbit and the origin as a fixed point. The projective line adds another fixed point at infinity. Its toric fan has the zero toric cone and the positive and negative half-lines in the real extension of the rank-one toric lattice.

The dictionary turns geometric questions into finite arithmetic data: smoothness criterion for a toric variety is checked by primitive toric ray generators extending to a lattice basis, properness by a complete fan, and line bundles by compatible integral linear functions on toric cones. The normal variety and split algebraic torus conventions matter; an arbitrary algebraic variety with a dense algebraic torus need not be described by the same ordinary toric fan construction.^[ref]

After coordinates place $p=[0:0:1]$, projection away from $p$ is $[x:y:z]\mapsto [x:y]$. Its restriction to a plane curve avoiding $p$ is a morphism to ${\mathbb{P}}^1$ whose degree equals the degree of the curve.

If $C$ is a smooth projective curve of genus at least three, then the smooth projective surface $C\times C$ contains no curve of geometric genus below three. On the normalization of any curve in the product, at least one coordinate projection to $C$ is nonconstant, and the Riemann-Hurwitz formula cannot decrease genus.

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 plane curve is a curve contained in a plane. In algebraic geometry, one studies plane curves defined by polynomial equations. General plane curves, such as an Archimedean spiral, need not be algebraic curves.

A plane cubic is nonsingular when its homogeneous equation and all first partial derivatives have no common projective zero.

A rational map is a morphism on a dense open subset, considered up to agreement on a smaller dense open subset.

It is the projective-source-and-target case of a rational map of algebraic varieties.

A hyperelliptic curve of genus at least two admits a degree-two map to ${\mathbb{P}}^1$ and can be represented in characteristic other than two by an equation $y^2=f(x)$ with $f$ square-free.

The gonality of a smooth projective curve is the least degree of a nonconstant morphism from the curve to the projective line.

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.