nLab
algebraic curve

Contents

Idea

An algebraic curve is an algebraic variety of dimension 11. 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\mathbf{A}^2 or P 2\mathbf{P}^2; they are the locus of solutions of a single algebraic equation.

An algebraic curve over a field FF is the locus of solutions of (n1)(n-1)-polynomials in nn-variables of type FF, provided the Krull dimension of the ring is 11.

Properties

  • Every projective algebraic curve is birationally equivalent to a plane algebraic curve
  • Mordell conjecture: every algebraic curve of genus g2g\geq 2 defined over rationals has at least one point over rationals
  • To a nonsingular curve CC over the field of complex numbers one associates an abelian variety, namely its Jacobian variety together with the period map or Abel-Jacobi map CJ(C)C\to J(C).

References

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

Revised on November 2, 2012 21:23:36 by Zoran Škoda (31.45.202.129)