nLab
algebraic curve

Idea

An algebraic curve is an algebraic variety of dimension $1$. Typically one restricts considerations to either affine or projective algebraic curves. Most often one treats the plane algebraic curves, i.e. curves with an embedding into $A^2$ or $P^2$; they are the locus of solutions of a single algebraic equation.

An algebraic curve over a field $F$ is the locus of solutions of $(n-1)$-polynomials in $n$-variables of type $F$, provided the Krull dimension of the ring is $1$.

Properties

• Every projective algebraic curve is birationally equivalent to a plane algebraic curve
• Mordell conjecture: every algebraic curve of genus $g\ge 2$ defined over rationals has at least one point over rationals
• To a nonsingular curve $C$ over the field of complex numbers one associates an abelian variety, namely its Jacobian variety together with the period map or Abel-Jacobi map $C\to J(C)$.

References

Related $n$Lab entries include moduli space of curves, stable curve?, Jacobian variety, Mordell conjecture, Riemann surface, elliptic curve, Bezout's theorem?

• Wikipedia, Algebraic curve