# 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.