# Canonical imbedding

Jump to: navigation, search

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 .

How to Cite This Entry:
Canonical imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_imbedding&oldid=46195
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article