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

 [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 [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

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.