Fermat surface
E530319
A Fermat surface is an algebraic surface in projective space defined by a homogeneous equation where each variable appears with the same exponent, generalizing the notion of Fermat curves to higher dimensions.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic surface
ⓘ
complex surface ⓘ hypersurface ⓘ projective variety ⓘ smooth surface ⓘ |
| appearsIn | classification of algebraic surfaces ⓘ |
| definedIn | projective 3-space ⓘ |
| definedOver |
algebraically closed field
ⓘ
complex numbers ⓘ |
| embeddedIn | projective space P^3 ⓘ |
| generalizes | Fermat curve NERFINISHED ⓘ |
| hasAutomorphismGroupContaining |
(μ_n)^4 / μ_n
ⓘ
symmetric group S_4 NERFINISHED ⓘ |
| hasBettiNumber | b_2 depending explicitly on n ⓘ |
| hasCanonicalBundle | O(n−4) ⓘ |
| hasDefiningEquation | x^n + y^n + z^n + w^n = 0 ⓘ |
| hasDegree | n ⓘ |
| hasDimension | 2 ⓘ |
| hasEulerCharacteristic | topological Euler characteristic depending polynomially on n ⓘ |
| hasHodgeStructure | pure Hodge structure of weight 2 on H^2 ⓘ |
| hasKodairaDimension |
0 for n = 4 (K3 case)
ⓘ
2 for n ≥ 5 ⓘ −∞ for n = 3 (cubic surface) ⓘ |
| hasLFunction | expressible in terms of Jacobi sums over finite fields ⓘ |
| hasModuli | discrete for fixed n up to projective equivalence ⓘ |
| hasNeronSeveriGroup | generated by explicit algebraic cycles for many n ⓘ |
| hasParameter | degree n ≥ 3 ⓘ |
| hasPicardNumber | often large compared to generic surface of same degree ⓘ |
| hasProperty |
Kähler surface
ⓘ
minimal surface of general type for n ≥ 5 ⓘ simply connected (over C) ⓘ |
| hasSpecialCase |
Fermat cubic surface (n = 3)
NERFINISHED
ⓘ
Fermat quartic surface (n = 4) NERFINISHED ⓘ Fermat quintic surface (n = 5) NERFINISHED ⓘ |
| hasSymmetryGroup |
(μ_n)^4 / μ_n
ⓘ
S_4 ⓘ |
| is | two-dimensional projective variety ⓘ |
| isSingularIf | characteristic of base field divides n ⓘ |
| isSmoothIf | characteristic of base field does not divide n ⓘ |
| isSpecialCaseOf | Fermat hypersurface NERFINISHED ⓘ |
| liesIn | projective 3-space P^3(k) ⓘ |
| namedAfter | Pierre de Fermat NERFINISHED ⓘ |
| relatedTo |
Jacobi sums
ⓘ
cyclotomic fields ⓘ |
| studiedIn |
Hodge theory
NERFINISHED
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| usedToStudy | zeta functions of varieties over finite fields ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.