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In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old. == Classification by the Kodaira dimension == (詳細はtopological genus, but dimension two, the difference between the arithmetic genus and the geometric genus turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. Let's summarize the results. (in detail, for each kind of surfaces refer to each redirections) Examples of algebraic surfaces include (κ is the Kodaira dimension): * κ=−∞: the projective plane, quadrics in P3, cubic surfaces, Veronese surface, del Pezzo surfaces, ruled surfaces * κ=0 : K3 surfaces, abelian surfaces, Enriques surfaces, hyperelliptic surfaces * κ=1: Elliptic surfaces * κ=2: surfaces of general type. For more examples see the list of algebraic surfaces. The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rational functions in two indeterminates. The cartesian product of two curves also provides examples. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old.== Classification by the Kodaira dimension ==(詳細はEnriques–Kodaira classificationを参照)In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. Let's summarize the results. (in detail, for each kind of surfaces refer to each redirections)Examples of algebraic surfaces include (κ is the Kodaira dimension):* κ=−∞: the projective plane, quadrics in P3, cubic surfaces, Veronese surface, del Pezzo surfaces, ruled surfaces* κ=0 : K3 surfaces, abelian surfaces, Enriques surfaces, hyperelliptic surfaces* κ=1: Elliptic surfaces* κ=2: surfaces of general type.For more examples see the list of algebraic surfaces.The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rational functions in two indeterminates. The cartesian product of two curves also provides examples.」の詳細全文を読む スポンサード リンク
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