General-type algebraic surface

algebraic surface of general type

A surface of one of the broadest classes of algebraic surfaces (cf. Algebraic surface) in the Enriques classification. Namely, a smooth projective surface $ X $ over an algebraically closed field $ k $ is called an algebraic surface of general type if

$$ \kappa ( x) = 2, $$

where $ \kappa $ is the Kodaira dimension. This condition is equivalent to the fact that for an integer $ n > 0 $ the linear system $ | nK | $, where $ K $ is the canonical divisor on $ X $, defines a birational mapping of $ X $ onto its image in $ P ^ {N} $ for a certain $ N $. Every algebraic surface of general type possesses a birational morphism onto its minimal model.

Minimal algebraic surfaces of general type are characterized (see [1], [3], [6]) by each of the following sets of properties:

a) $ K ^ { 2 } > 0 $ and $ KD \geq 0 $ for any effective divisor $ D $;

b) $ K ^ { 2 } > 0 $ and $ P _ {2} \geq 2 $, where $ P _ {2} = \mathop{\rm dim} | 2 K | + 1 $ is the second plurigenus of $ X $;

c) $ K ^ { 2 } > 0 $ and the surface $ X $ is not rational (cf. Rational surface);

d) there exists an integer $ n _ {0} $ such that, for any $ n \geq n _ {0} $, the mapping $ \phi _ {nK} $ defined by the system $ | n K | $ is a birational morphism of $ X $ onto its image in $ P ^ { \mathop{\rm dim} | n K | } $.

For algebraic surfaces of general type, relations (in the form of inequalities) exist between the numerical characteristics. Let $ p _ {g} $ be the geometric genus and let $ q $ be the irregularity of $ X $. Then for a minimal algebraic surface of general type the following inequalities hold:

1) $ q \leq p _ {g} $;

2) $ p _ {g} \leq K ^ { 2 } /2 + 2 $ if $ K ^ { 2 } $ is even, $ p _ {g} \leq ( K ^ { 2 } + 3)/2 $ if $ K ^ { 2 } $ is odd (these two inequalities are called Noether's inequalities);

3) $ K ^ { 2 } \leq 3C _ {2} $, where $ C _ {2} $ is the second Chern class of $ X $( or the topological Euler characteristic).

The most complete result on multi-canonical mappings $ \phi _ {nK} $ of algebraic surfaces of general type is the Bombieri–Kodaira theorem: Let $ X $ be a minimal algebraic surface of general type over an algebraically closed field of characteristic 0, then the mapping

$$ \phi _ {nK} : X \rightarrow P ^ { \mathop{\rm dim} | n K | } $$

is a birational morphism onto its own image for all $ n \geq 5 $. Algebraic surfaces of general type for which $ \phi _ {4K} $ does not possess this property exist (see [5], , [9]).

#### References[edit]

[1] |  I.R., et al. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001

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[2] |  F.A. Bogomolov, "Holomorphic tensors and vector bundles on projective varieties" Math. USSR-Izv. , 13 : 3 (1979) pp. 499–555 Izv. Akad. Nauk SSSR Ser. Mat. , 42 (1978) pp. 1227–1287 Zbl 0439.14002
[3] |  A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) MR0485887 Zbl 0394.14014
[4] |  E. Bombieri, "Canonical models of surfaces of general type" Publ. Math. IHES , 42 (1972) pp. 447–495 MR0318163 Zbl 0259.14005
[5] |  E. Bombieri, F. Catanese, "The tricanonical map of surfaces with , " K.G. Ramanathan (ed.) , C.P. Ramanujam, a tribute , Springer (1978) pp. 279–290 MR541028
[6] |  D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1974) pp. 329–420 MR0506292 Zbl 0326.14009
[7] |  E. Horikawa, "Algebraic surfaces of general type with small , I" Ann. of Math. , 104 (1976) pp. 357–387
[8a] |  E. Horikawa, "Algebraic surfaces of general type with small , II" Invent. Math. , 37 (1976) pp. 121–155
[8b] |  E. Horikawa, "Algebraic surfaces of general type with small , III" Invent. Math. , 47 (1978) pp. 209–248 MR501370
[8c] |  E. Horikawa, "Algebraic surfaces of general type with small , IV" Invent. Math. , 50 (1978–1979) pp. 103–128
[9] |  K. Kodaira, "Pluricanonical systems on algebraic surfaces of general type" J. Math. Soc. Japan , 20 (1968) pp. 170–192 MR0224613 Zbl 0157.27704
[10] |  Y. Miyaoka, "On the Chern numbers of surfaces of general type" Invent. Math. , 42 (1977) pp. 225–237 MR0460343 Zbl 0374.14007

#### Comments[edit]

Some of the above results have only been proved in characteristic zero; for instance, the inequality $ K ^ { 2 } \leq 3 C _ {2} $ only holds in characteristic zero.

For results on canonical models of surfaces of general type in positive characteristic see [a2].

#### References[edit]

[a1] |  A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023

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[a2] |  T. Ekedahl, "Canonical models of surfaces of general type in positive characteristic" Publ. Math. IHES , 67 (1988) pp. 97–144 MR0972344 Zbl 0674.14028