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.
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| notation | F_n ⓘ |
| PicardNumber | 2 ⓘ |
| specialCase |
F_0 is isomorphic to P^1 × P^1
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.