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