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  • valuation
    • Archimedean valuation
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algebraic curve

Notes

• Encode points on a smooth projective curve as valuations measuring orders of poles/zeros.
  • Bounded valuations: points on the variety
  • Unbounded: points “at infinity”, like puncture points on a Riemann surface
• Compactifications are not unique. Example:
  • \({\mathbf{A}}^2 \subseteq {\mathbf{P}}^2\)
  • \({\mathbf{A}}^2 \subseteq ({\mathbf{P}}^1)^{\times 2}\)
  • But \({\mathbf{P}}^2 \neq ({\mathbf{P}}^1)^{\times 2}\)!

The algebraic analogues of a compact proper complex algebraic curve of genus \(g\).

Prym varieties