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Canonical static concept record

Smooth projective quadric

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Summary

For a nondegenerate quadratic form $q$ of dimension at least three over a field of characteristic not two, the projective zero locus $q=0$ is a smooth projective quadric and a geometrically integral variety.

Record metadata

Canonical name: Smooth projective quadric
Concept ID: smooth_projective_quadric
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Number theory, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

vector space over a field
nondegenerate quadratic form

Data

projective hypersurface $q=0$

Axioms / constraints

dimension at least three
characteristic not two
nondegeneracy

Canonically induces

A rational-point problem equivalent to isotropy of $q$

Notes

The dimension and nondegeneracy conditions avoid reducible or nonintegral low-dimensional zero loci. Over number fields the characteristic condition is automatic.

Concept sources

Wikipedia — Quadric (algebraic geometry) — Quadric (algebraic geometry) · encyclopedic reference · source ID wp-smooth-projective-quadric
Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory
Stacks Project — Varieties — Varieties · reference · source ID stacks-varieties

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Algebraic variety Smooth projective quadric

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_algebraic_variety_to_smooth_projective_quadric_impose_axiom
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: specialize to a smooth projective degree-two hypersurface

Authored explanation

A smooth projective quadric is an algebraic variety cut out by one nondegenerate homogeneous quadratic equation.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties
Stacks Project — Varieties — Varieties · reference · source ID stacks-varieties

Quadratic form Smooth projective quadric

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_quadratic_form_to_smooth_projective_quadric_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: take the projective zero locus under regularity hypotheses

Authored explanation

For a nondegenerate quadratic form of dimension at least three over a field of characteristic not two, the equation $q=0$ defines a smooth projective quadric.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

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Each relation below starts at this concept.