Canonical imbedding

An imbedding of an algebraic variety $ X $ into a projective space using a multiple of the canonical class $ K _ {X} $( see Linear system). Let $ X $ be a non-singular projective curve of genus $ g $; a mapping defined by the class $ nK _ {X} $ is an imbedding for some $ n \geq 3 $ provided that $ g > 1 $. Here one can take $ n \geq 1 $ for non-hyper-elliptic curves, $ n \geq 2 $ for hyper-elliptic curves of genus $ g > 2 $ and $ n \geq 3 $ for curves of genus 2. These results have been used for the classification of algebraic curves of genus $ g > 1 $( see Canonical curve).

Similar questions have been considered for varieties of dimension greater than one, mainly surfaces. In this connection, the role of curves of genus $ g > 1 $ is played by surfaces for which some multiple $ nK _ {X} $ of the canonical class gives a birational mapping of the surface onto its image in projective space. They are called surfaces of general type; the main result concerning these surfaces is the fact that for them, the class $ 5K _ {X} $ already determines a regular mapping into a projective space which is a birational mapping. For example, a non-singular surface of degree $ m $ in $ P _ {3} $ is a surface of general type if $ m > 4 $. In this case the canonical class $ K _ {X} $ itself gives a birational mapping. If $ K _ {X} K _ {X} > 2 $ and $ p _ {g} (X) > 1 $( here $ K _ {X} K _ {X} $ is the self-intersection index and $ p _ {g} (X) $ is the geometric genus), then one can replace $ 5K _ {X} $ by $ 3K _ {X} $. Surfaces for which no multiple $ nK _ {X} $ gives an imbedding are divided into the following five families: rational surfaces, ruled surfaces, Abelian varieties, $ K3 $- surfaces, and surfaces with a pencil of elliptic curves. In this connection, the rational and ruled surfaces are analogues of rational curves, while the remaining three families are analogues of elliptic curves.

The first generalizations of these results to higher-dimensional varieties appeared in [5].

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	F. Severi, "Vorlesungen über algebraische Geometrie" , Teubner (1921) (Translated from Italian) MR0245574 Zbl 48.0687.01
[3]	"Algebraic surfaces" Trudy Mat. Inst. Steklov. , 75 (1965) (In Russian) Zbl 0154.33002 Zbl 0154.21001
[4]	D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982) pp. 329–420 MR0506292 Zbl 0326.14009
[5]	K. Ueno, "Introduction to classification theory of algebraic varieties and compact complex spaces" , Lect. notes in math. , 412 , Springer (1974) pp. 288–332 MR0361174 Zbl 0299.14006

Comments

Let $ {\mathcal O} _ {X} (K _ {X} ) $ be the line bundle, the canonical bundle, defined by a divisor representing $ K _ {X} $( cf. Divisor). The mapping defined by its global sections $ x \mapsto (s _ {1} (x) : \dots : s _ {g} (x)) \in \mathbf P ^ {g - 1 } $ is called the canonical mapping. (Here $ s _ {1} \dots s _ {g} $ are a basis of $ \Gamma (X; L) $ and it is assumed that for all $ x $ there is an $ i $ with $ s _ {i} (x) \neq 0 $, cf. Linear system.) Correspondingly, if $ nK _ {X} $ is used instead of $ K _ {X} $, one speaks of a multi-canonical mapping and, if it is an imbedding, of a multi-canonical imbedding.

References

[a1]	K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) MR0506253 Zbl 0299.14007
[a2]	S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) Zbl 0491.14006
[a3]	A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023
[a4]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) MR0770932 Zbl 0559.14017
[a5]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001