Scheme-theoretic fibre

Depends on

• Base change of objects, morphisms and properties
• Field-valued points and local-ring points

Used by

• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• K-morphisms from smooth models into abelian schemes extend uniquely Corollary
• Abelian schemes over a base Definition
• Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers Definition
• Degree of a nonconstant morphism of curves Definition
• Geometric fibres and geometric points Definition
• Intersection multiplicity of closed subschemes at a point Definition
• Neron models, the Neron mapping property and weak Neron models Definition
• Quasi-finite morphisms of schemes Definition
• S-dense open subschemes and S-rational maps Definition
• A linear representation induces an action on projective space with the same line stabilizers Lemma
• A transverse hyperplane slice is smooth at the chosen point Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma
• Fibres after base change Lemma
• Fibres of a smooth morphism are smooth Lemma
• Fibres of abelian schemes and unit-preserving morphisms Lemma
• Fibres of proper morphisms are proper Lemma
• Fibres of the orbit map and the scheme-theoretic stabilizer as a closed subgroup scheme Lemma
• Fibrewise regular sequences persist openly in equidimensional Cohen-Macaulay fibres Lemma
• Finite neighbourhood of an isolated fibre point after elementary etale change Lemma
• Finite-fibre and pointwise characterizations of quasi-finiteness Lemma
• Flat pullback of cycles and of rational equivalence Lemma
• Local fibre-dimension bound from polynomial quasi-finiteness Lemma
• Lower semicontinuity of flat finitely presented fibre dimension Lemma
• The submersion criterion between smooth varieties Lemma
• Uniqueness, weak Neron property, etale base change and local nature of Neron models Lemma
• Upper semicontinuity of proper fibre dimension Lemma
• A faithfully flat orbit map represents the coset quotient sheaf Proposition
• Flatness does not force isomorphic or smooth fibres Remark
• An abelian scheme is the Neron model of its generic fibre Theorem
• Bertini smoothness away from the base locus Theorem
• Coordinate ring of an affine fibre Theorem
• Etale morphisms are universally open and quasi-finite at every point Theorem
• Fibre dimension of proper flat finitely presented families Theorem
• Generic smoothness over a dense target open Theorem
• Submersion criterion for locally standard smooth morphisms Theorem
• The chord-tangent group law on a smooth short Weierstrass cubic Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Stacks 26.18.4; Vakil 10.3.2 (standard reference, not scraped)