# General-type algebraic surface

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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 , , ) 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 , , ).

How to Cite This Entry:
General-type algebraic surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General-type_algebraic_surface&oldid=47065
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article