Petri’s theorem
E898504
Petri’s theorem is a fundamental result in algebraic geometry that characterizes the ideal of a canonically embedded algebraic curve by describing it as being generated by quadrics under suitable conditions.
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
result in the theory of algebraic curves
ⓘ
theorem in algebraic geometry ⓘ |
| appliesTo |
canonically embedded curves
ⓘ
non-hyperelliptic curves ⓘ smooth projective curves ⓘ |
| asserts | for a non-hyperelliptic smooth projective curve of genus g ≥ 3, the canonical ideal is generated by quadrics except for certain special cases ⓘ |
| characterizes | canonical ideal of a curve in terms of quadrics ⓘ |
| concerns |
canonical embeddings of algebraic curves
ⓘ
canonical models of curves ⓘ homogeneous ideal of a canonically embedded curve ⓘ |
| describes | generators of the canonical ideal of a non-hyperelliptic curve ⓘ |
| field |
algebraic geometry
ⓘ
theory of algebraic curves ⓘ |
| hasCondition |
curve must be non-hyperelliptic
ⓘ
curve must be projective ⓘ curve must be smooth ⓘ genus at least 3 ⓘ |
| hasException |
certain special linear series on curves
ⓘ
plane quintic curves ⓘ trigonal curves ⓘ |
| hasGeneralization |
Green–Lazarsfeld results on syzygies
NERFINISHED
ⓘ
results on higher syzygies of canonical curves ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies | projective normality of the canonical embedding under its hypotheses ⓘ |
| namedAfter | Karl Petri NERFINISHED ⓘ |
| relatedTo |
Brill–Noether theory
NERFINISHED
ⓘ
Green’s conjecture on syzygies of canonical curves NERFINISHED ⓘ canonical linear system of a curve ⓘ projective normality of canonical curves ⓘ syzygies of canonical curves ⓘ |
| standardReference |
classical textbooks on algebraic curves
ⓘ
modern treatments of canonical curves and syzygies ⓘ |
| states | the canonical ideal is generated by quadrics for a general non-hyperelliptic curve of genus g ≥ 3 ⓘ |
| usedIn |
analysis of generators and relations of canonical rings
ⓘ
classification of algebraic curves by their canonical models ⓘ study of special linear series on curves ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.