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 over an algebraically closed field
is called an algebraic surface of general type if
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where is the Kodaira dimension. This condition is equivalent to the fact that for an integer
the linear system
, where
is the canonical divisor on
, defines a birational mapping of
onto its image in
for a certain
. 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) and
for any effective divisor
;
b) and
, where
is the second plurigenus of
;
c) and the surface
is not rational (cf. Rational surface);
d) there exists an integer such that, for any
, the mapping
defined by the system
is a birational morphism of
onto its image in
.
For algebraic surfaces of general type, relations (in the form of inequalities) exist between the numerical characteristics. Let be the geometric genus and let
be the irregularity of
. Then for a minimal algebraic surface of general type the following inequalities hold:
1) ;
2) if
is even,
if
is odd (these two inequalities are called Noether's inequalities);
3) , where
is the second Chern class of
(or the topological Euler characteristic).
The most complete result on multi-canonical mappings of algebraic surfaces of general type is the Bombieri–Kodaira theorem: Let
be a minimal algebraic surface of general type over an algebraically closed field of characteristic 0, then the mapping
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is a birational morphism onto its own image for all . Algebraic surfaces of general type for which
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 ![]() ![]() |
[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 ![]() |
[8a] | E. Horikawa, "Algebraic surfaces of general type with small ![]() |
[8b] | E. Horikawa, "Algebraic surfaces of general type with small ![]() |
[8c] | E. Horikawa, "Algebraic surfaces of general type with small ![]() |
[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
Some of the above results have only been proved in characteristic zero; for instance, the inequality only holds in characteristic zero.
For results on canonical models of surfaces of general type in positive characteristic see [a2].
References
[a1] | A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023 |
[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 |
General-type algebraic surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=General-type_algebraic_surface&oldid=42128