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An imbedding of an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201601.png" /> into a projective space using a multiple of the [[Canonical class|canonical class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201602.png" /> (see [[Linear system|Linear system]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201603.png" /> be a non-singular projective curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201604.png" />; a mapping defined by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201605.png" /> is an imbedding for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201606.png" /> provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201607.png" />. Here one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201608.png" /> for non-hyper-elliptic curves, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c0201609.png" /> for hyper-elliptic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016011.png" /> for curves of genus 2. These results have been used for the classification of algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016012.png" /> (see [[Canonical curve|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016013.png" /> is played by surfaces for which some multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016015.png" /> already determines a regular mapping into a projective space which is a birational mapping. For example, a non-singular surface of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016017.png" /> is a surface of general type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016018.png" />. In this case the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016019.png" /> itself gives a birational mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016021.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016022.png" /> is the self-intersection index and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016023.png" /> is the geometric genus), then one can replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016025.png" />. Surfaces for which no multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016026.png" /> gives an imbedding are divided into the following five families: rational surfaces, ruled surfaces, Abelian varieties, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016027.png" />-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 [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Severi, "Vorlesungen über algebraische Geometrie" , Teubner (1921) (Translated from Italian) {{MR|0245574}} {{ZBL|48.0687.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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 {{MR|0506292}} {{ZBL|0326.14009}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Ueno, "Introduction to classification theory of algebraic varieties and compact complex spaces" , ''Lect. notes in math.'' , '''412''' , Springer (1974) pp. 288–332 {{MR|0361174}} {{ZBL|0299.14006}} </TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016028.png" /> be the line bundle, the canonical bundle, defined by a divisor representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016029.png" /> (cf. [[Divisor|Divisor]]). The mapping defined by its global sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016030.png" /> is called the canonical mapping. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016031.png" /> are a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016032.png" /> and it is assumed that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016033.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016035.png" />, cf. [[Linear system|Linear system]].) Correspondingly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016036.png" /> is used instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020160/c02016037.png" />, one speaks of a multi-canonical mapping and, if it is an imbedding, of a multi-canonical imbedding.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Ueno, "Classification theory of algebraic varieties and compact complex spaces" , Springer (1975) {{MR|0506253}} {{ZBL|0299.14007}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) {{MR|}} {{ZBL|0491.14006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , '''1''' , Springer (1985) {{MR|0770932}} {{ZBL|0559.14017}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>

[[Category:Algebraic geometry]]