Jacobian variety

Jacobian varieties are the most important class of abelian varieties.

To every nonsingular algebraic curve CC (over the complex numbers) of genus gg one associates the Jacobian variety or simply Jacobian J(C)J(C) either via differential 1-forms or equivalently via line bundles: Jacobian is the the moduli space of degree 00 line bundles over CC, i.e. the connected component of the identity of the Picard group of CC.

The Abel-Jacobi map CJ(C)C\to J(C) is defined with help of periods.

(one should also cover Abel-Jacobi theorem)

  • wikipedia Jacobian variety, Abel-Jacobi map
  • P. Griffiths, J. Harris, Principles of algebraic geometry
  • A. Beauville, Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.
Revised on November 10, 2012 05:42:49 by Zoran Škoda (