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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.

For $a>0$, this algebraic curve can be constructed using a circle of radius $a$ centred at $(a,0)$ and its tangent line $x=2a$. On a ray from the origin, subtract the distance to its nonzero circle intersection from the distance to the line. The resulting radial distance is $r=2a{\sin }^2\theta /\cos \theta$, for $-\pi /2<\theta <\pi /2$. Thus a parametric curve for it isEliminating $t$ gives the displayed cubic equation. It has a cusp at the origin, horizontal tangent direction there, and vertical asymptote $x=2a$. Its name honours Diocles.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

A unibranch singular point of an algebraic variety on a plane algebraic curve. The ordinary cusp has local equation $y^2=x^3$ and parametrization $(x,y)=(t^2,t^3)$; its branch has a repeated tangent and a vanishing first derivative at the singular point. Higher-order cusps may have different local equations. A cuspidal cubic gives the basic projective example.

The projective algebraic curve $D=\{y^{2}z=x^3\}\subset {\mathbb{P}}^2$ is an irreducible cubic with a cusp at $[0:0:1]$. In the chart $z=1$, its equation is $y^2=x^3$, with parametrization $(x,y)=(t^2,t^3)$. Its affine version $y^2=z^3$, used below, differs only by renaming a coordinate. At the point $[0:1:0]$, the chart $y=1$ gives $z=x^3$, which is smooth and exhibits its unique smooth flex of a cuspidal cubic.

Choose an effective Cartier divisor $E$ of degree at least $g$ on a smooth projective genus-$g$ algebraic curve. For a family $M$ of degree-$g-1$ line bundles, the evaluation map from ${\pi }_{\ast }M(E)$ to ${\pi }_{\ast }(M(E){|}_E)$ is between vector bundles of the same rank. Its determinant vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero Cartier divisor is the theta divisor. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1({\mathcal{O}}_C)\to \mathrm{Hom}(H^0(M),H^1(M))$. By Serre duality it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this Cartier divisor is generically reduced.

For a smooth irreducible algebraic curve, send a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$ to the line bundle associated to a divisor defined locally by ${\mathcal{O}}_X(D{)}_P=t_P^{-n_P}{\mathcal{O}}_{X,P}$. A nonzero rational section of a line bundle gives its inverse construction, and changing that section changes its divisor by a principal divisor. Thus the divisor class group is the Picard group. The quotient of the sheaf of nonzero rational functions on an irreducible variety by the sheaf of units of the structure sheaf has stalk $\mathbb{Z}$ at each closed point, via the discrete valuation. Its global sections are finite sums of points. The corresponding long exact sequence in sheaf cohomology, and the vanishing for the rational-function sheaf, identify both groups with $H^1(X,{\mathcal{O}}_X^{\ast })$.

Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

For nonzero real $a$, the folium of Descartes is the algebraic curve $x^3+y^3=3axy$, with rational parametrization of an algebraic curveIt is symmetric about $y=x$ and has a node at the origin, with the coordinate axes as tangents. For $a>0$, $0\le t<\infty$ traces a loop in the first quadrant, returning to the origin as $t\to \infty$. Its enclosed area, by Green's theorem, isThe other branches approach the asymptote $x+y+a=0$ as $t\to -1$. Negative $a$ rotates the positive-$a$ picture through $\pi$.

For a projective scheme $X$ and a fixed Hilbert polynomial $P$, the Hilbert scheme parametrizes flat families of closed subschemes of $X$ with that polynomial. This turns a family of embedded algebraic curves into an algebraic parameter space.

Igor Dolgachev studies algebraic geometry, including the classical geometry of plane curves and their symmetries. With Michela Artebani, he wrote a study of the Hesse pencil and its associated configurations.^[ref]

The pencil combines plane cubic algebraic curves sharing nine base points. Those points encode the flex geometry of a smooth cubic, while special members split into lines. The paper connects this classical configuration with elliptic surfaces and K3 surfaces, illustrating how a one-parameter family of equations organizes several geometric objects at once.^[ref]

An algebraic curve which is projective, reduced and irreducible. Over an algebraically closed field, its only global regular functions are constants and its arithmetic genus is $h^1({\mathcal{O}}_C)\ge 0$.

The Jacobian variety of a smooth projective algebraic curve parametrizes degree-zero line bundles. Its dimension is the genus of the algebraic curve. An effective degree-$d$ Cartier divisor defines a point in a degree-$d$ torsor through the Abel map of an algebraic curve. The theta divisor induces an isomorphism with the dual abelian variety.

A linear system of divisors is a projective family of effective Weil divisors linearly equivalent to a fixed divisor. A complete linear system of a divisor contains every effective representative of that class; a projective subspace gives a subsystem. For a Cartier divisor, the family is obtained from global sections of its associated line bundle. The algebraic curve case recovers the usual moving family of effective divisors.

Let $C$ be a smooth integral algebraic curve and let $p$ be a closed point. Its local ring ${\mathcal{O}}_{C,p}$ is a one-dimensional regular local ring, hence a discrete valuation ring. If $\pi$ is a uniformizer, every nonzero rational function has a unique expressionwith $u$ a unit in ${\mathcal{O}}_{C,p}$. The integer $m$ is its order of vanishing at $p$: positive for a zero, negative for a pole, and zero for a local unit.^[ref]

On the affine line at $t=0$, the local ring is $k[t{]}_{(t)}$, the element $t$ is a uniformizer, and $t^3/(1+t)$ has order of vanishing three. At the generic point of $C$, the local ring is the function field $k(C)$, which has dimension zero; the closed-point hypothesis distinguishes these cases.

Michela Artebani works on algebraic geometry. Her joint paper with Igor Dolgachev examines the Hesse pencil of plane cubic algebraic curves, its flex configuration and associated K3 surfaces.^[ref]

A useful feature of the example is that a smooth cubic and its Hessian belong to a tightly organized family. Incidences among the nine common points and twelve distinguished lines expose symmetries that are harder to see in an arbitrary cubic equation. Their treatment connects explicit coordinates with the global geometry of surfaces.^[ref]

A morphism of algebraic varieties whose source and target are algebraic curves. A nonconstant morphism between smooth projective curves is finite and surjective; its degree is the degree of the induced extension of function fields.

A negative curve is an irreducible algebraic curve on a smooth projective surface whose self-intersection number is negative. The negativity prevents it from moving in a nontrivial effective linear system and constrains how many curves can represent a fixed numerical class.

A nodal crossing locally consists of two complex branches meeting transversely and is not a one-dimensional complex manifold at the intersection.

For a plane algebraic curve this crossing is an ordinary double point, with two distinct tangent directions.

For an affine plane algebraic curve $F(x,y)=0$, a point over an algebraic closure is nonsingular when the displayed condition holds. Its Zariski tangent space has dimension one, and a local coordinate can be chosen in the direction of a nonzero partial derivative. A smooth algebraic curve has all points nonsingular. Over a complete local field, the Hensel lemma lifts a nonsingular residue point by solving for a coordinate with unit partial derivative.