Clebsch diagonal surfaces

E262450

Fano surface algebraic surface cubic surface projective variety smooth cubic surface

Clebsch diagonal surfaces are classical 19th-century algebraic surfaces in projective three-space, famous as the first explicit smooth cubic surface with all 27 lines defined over the real numbers.

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All labels observed (2)

Label	Occurrences
Clebsch cubic surface	2
Clebsch diagonal surfaces canonical	1

Statements (46)

Predicate	Object
instanceOf	Fano surface ⓘ algebraic surface ⓘ cubic surface ⓘ projective variety ⓘ smooth cubic surface ⓘ
allLinesDefinedOver	real numbers ⓘ
appearsIn	classical classification of cubic surfaces ⓘ
canBeEmbeddedIn	P^3 via linear projection from P^4 ⓘ
canBeRealizedAs	intersection of a cubic hypersurface and a hyperplane in P^4 ⓘ
definedOver	complex numbers ⓘ real numbers ⓘ
degree	3 ⓘ
dimension	2 ⓘ
discoveredInCentury	19th century ⓘ
embeddedIn	P^3 ⓘ projective 3-space ⓘ
fieldOfStudy	algebraic geometry ⓘ classical algebraic geometry ⓘ
hasAnticanonicalEmbedding	into P^3 as a cubic surface ⓘ
hasAutomorphismGroup	symmetric group S5 ⓘ
hasBettiNumber	b2 = 7 ⓘ
hasCanonicalBundle	anti-ample ⓘ
hasEquationForm	sum x_i = 0 and sum x_i^3 = 0 in P^4 ⓘ
hasEulerCharacteristic	3 ⓘ
hasHodgeNumbers	h^{1,0} = 0 ⓘ h^{1,1} = 7 ⓘ h^{2,0} = 0 ⓘ
hasLinesConfiguration	27 lines in classical cubic surface configuration ⓘ
hasNumberOfLines	27 ⓘ
hasPicardNumber	7 ⓘ
hasProperty	all 27 lines are pairwise skew or intersect according to cubic surface incidence rules ⓘ all 27 lines are real ⓘ highly symmetric cubic surface ⓘ
hasRealForm	unique up to projective equivalence with all 27 lines real ⓘ
hasRealStructure	yes ⓘ
hasSymmetryGroup	S5 ⓘ
hasType	smooth projective rational surface ⓘ
isBirationalTo	blow-up of P^2 in six points ⓘ
isClassicalObjectIn	19th-century projective geometry ⓘ
isExampleOf	del Pezzo surface of degree 3 ⓘ
isOftenDefinedBy	homogeneous coordinates satisfying x0 + x1 + x2 + x3 + x4 = 0 and x0^3 + x1^3 + x2^3 + x3^3 + x4^3 = 0 ⓘ
isRationalSurface	yes ⓘ
namedAfter	Alfred Clebsch ⓘ
notableFor	being the first explicit smooth cubic surface with all 27 lines real ⓘ
relatedTo	Weyl group ⓘ surface form: Weyl group of type E6 via lines configuration
usedAs	standard example in the theory of cubic surfaces ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alfred Clebsch → notableWork → Clebsch diagonal surfaces ⓘ

this entity surface form: Clebsch cubic surface

Alfred Clebsch → notableConcept → Clebsch diagonal surfaces ⓘ

this entity surface form: Clebsch cubic surface