special fiber

Tags: #AG Refs: degeneration

special fiber

  • For \(R\in \mathsf{Loc}\mathsf{Ring}\) a local ring, \(\operatorname{Spec}R\) has
    • A generic point \(\left\langle{0}\right\rangle\) and
    • A special point (or closed point) \({\mathfrak{m}}_R\).
  • Easiest example: for \(R\in \mathsf{DVR}\), note \(\operatorname{Spec}R = \left\{{\left\langle{0}\right\rangle, {\mathfrak{m}}_R}\right\}\) has a single generic point and a single closed point.
  • For \(R\in \mathsf{DVR}\) or \(\mathsf{Loc}\mathsf{Ring}\) and a morphism \(f: X\to \operatorname{Spec}R\),
    • The generic fiber is the fiber over the generic point \((0)\), so \(f^{-1}(\left\langle{0}\right\rangle)\)
    • The special fiber is the fiber above \({\mathfrak{m}}\), so \(f^{-1}({\mathfrak{m}})\).
  • degeneration is the theory of passing from the generic fiber to the special fiber.

Closed points

attachments/Pasted%20image%2020220214093029.png

attachments/Pasted%20image%2020220424122108.png

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