geometric fiber

geometric fiber

Idea: a point of \(X\) with values in an algebraically closed field.

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Geometric Fiber

For $X\in {\mathsf{Sch}}_{/ {k}} $ , a geometric point is a morphism \(x: {\operatorname{pt}}\to X\), where \({\operatorname{pt}}\coloneqq\operatorname{Spec}k\) . The geometric fiber of \(x\) is the pullback \(E_x \coloneqq\lim({\operatorname{pt}}\to X \leftarrow E) = E \underset{\scriptscriptstyle {X} }{\times} {\operatorname{pt}}\).

Idea: for any \(U \subseteq Y\) and \(f:X\to Y\), the fiber product \(U \underset{\scriptscriptstyle {Y} }{\times} X\) should be thought of as \(f^{-1}(U)\).

Note that if \(x\) is a finite etale cover , then \({\sharp}E_x < \infty\).

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  • unresolved links output
  • scheme
  • etale fundamental group

    Idea: \({ \mathsf{Gal}} (k^{ {}^{ \operatorname{sep} } }_{/ {k}} ) \cong \pi_1^\text{ét}(\operatorname{Spec}k, \mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu)\) for \(\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu\) a geometric point?

  • curves
    geometric point : a section to the structure map, \(s: \operatorname{Spec}k \to X\) so that \(\operatorname{Spec}k \xrightarrow{s} X \xrightarrow{S} \operatorname{Spec}k\) is the identity on \(\operatorname{Spec}k\)
  • Torelli

    Idea: a non-singular projective algebraic curve $C_{/ {k}} $ for any \(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) is determined by its Jacobian variety \(\operatorname{Jac}(C) \in {\mathcal{A}_g}\). Equivalently, there is a period map \(P: {\mathcal{M}_g}\to {\mathcal{A}_g}\) which is injective on geometric points.

  • Artin stack
    geometric points do not usually admit universal deformation rings.
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