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• Tags:
  • #AG/moduli-spaces
• Refs:
  • #todo/add-references
• Links:
  • Abel-Jacobi theorem
  • Torelli

Jacobian

For \(C\) a nonsingular algebraic curve, \(\operatorname{Jac}(C)\) is the connected component of the identity in the Picard group \(\operatorname{Pic}(C)\), i.e. the moduli space of degree 0 line bundles on \(C\). Over \({\mathbf{C}}\), can be realized as \(\operatorname{Jac}(C) \cong H^0(X; \Omega^1_{C}) {}^{ \vee }/ H^1(X; {\mathcal{O}}_X)\), where the embedding \(H^1\hookrightarrow H^0\) uses theta functions.