Hirzebruch surfaces
E586791
Hirzebruch surfaces are a family of complex algebraic surfaces that serve as fundamental examples in algebraic geometry and the classification of complex surfaces.
All labels observed (2)
| Label | Occurrences |
|---|---|
| F_0 is isomorphic to P^1 × P^1 | 1 |
| Hirzebruch surfaces canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6337355 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hirzebruch surfaces Context triple: [Friedrich Hirzebruch, knownFor, Hirzebruch surfaces]
-
A.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
-
B.
Clebsch diagonal surfaces
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.
-
C.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
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D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
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E.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hirzebruch surfaces Target entity description: Hirzebruch surfaces are a family of complex algebraic surfaces that serve as fundamental examples in algebraic geometry and the classification of complex surfaces.
-
A.
Kummer surfaces
Kummer surfaces are special quartic algebraic surfaces in projective three-space characterized by having 16 ordinary double points, extensively studied in the context of complex geometry and abelian varieties.
-
B.
Clebsch diagonal surfaces
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.
-
C.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
D.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
E.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
complex algebraic surface
ⓘ
family of algebraic surfaces ⓘ |
| are |
Fano surfaces for n = 0,1
ⓘ
examples in Mori theory ⓘ examples of non-isomorphic ruled surfaces over P^1 ⓘ examples of projective bundles over P^1 ⓘ geometrically ruled over the projective line ⓘ minimal rational surfaces for n ≠ 1 ⓘ non-isomorphic for different n ⓘ not Fano for n ≥ 2 ⓘ rational surfaces ⓘ rationally connected ⓘ ruled surfaces ⓘ simply connected ⓘ smooth projective surfaces ⓘ toric surfaces ⓘ toric varieties ⓘ |
| base | complex projective line P^1 ⓘ |
| belongTo | Enriques–Kodaira classification of complex surfaces ⓘ |
| canBeRealizedAs | P(O ⊕ O(n)) over P^1 ⓘ |
| canonicalDivisor | K = −2C_0 − (n+2)f (up to linear equivalence) ⓘ |
| dimension | 2 ⓘ |
| fiber | complex projective line P^1 ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ topology ⓘ |
| have |
Kodaira dimension −∞
ⓘ
Zariski decomposition properties studied in surface theory ⓘ automorphism groups depending on n ⓘ effective cone generated by fiber and negative section ⓘ geometric genus p_g = 0 ⓘ irregularity q = 0 ⓘ nef cone generated by fiber and another section ⓘ ruling by lines over P^1 ⓘ section of self-intersection −n ⓘ |
| haveParameter | nonnegative integer n ⓘ |
| namedAfter | Friedrich Hirzebruch ⓘ |
| notation | F_n ⓘ |
| PicardNumber | 2 ⓘ |
| specialCase |
F_0 is isomorphic to P^1 × P^1
ⓘ
linked to:
Hirzebruch surfaces
F_1 is the blow-up of P^2 at one point ⓘ |
| usedAs |
basic examples of ruled surfaces
ⓘ
examples in classification of minimal models ⓘ examples in intersection theory ⓘ examples in toric geometry ⓘ test cases for vanishing theorems ⓘ |
| usedIn |
classification of algebraic surfaces
ⓘ
classification of complex surfaces ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hirzebruch surfaces Description of subject: Hirzebruch surfaces are a family of complex algebraic surfaces that serve as fundamental examples in algebraic geometry and the classification of complex surfaces.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hirzebruch surfaces