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The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor).
Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P 3 lies in it, for example).
Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces; Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane
An algebraic surface is an algebraic variety of dimension two. The Enriques-Kodaira classification gives an overview of the possibilities. Over the complex numbers, a non-singular algebraic surface is an example of a 4-manifold
In 1906, he obtained a theorem of existence of algebraic curves drawn on certain types of surfaces, thus beginning the search for the classification of rational surfaces. [2] Mobilized during World War I, Severi enlisted in the artillery. In 1921, he obtained the chair of algebraic geometry at La Sapienza University in Rome.
Volume one. Algebraic exterior calculus and local differential properties], Istituto Nazionale di Alta Matematica (in Italian), Roma: Docet edizioni universitarie, p. 520, MR 0049646, Zbl 0045.19702. [8] Segre, Beniamino (1951b), Arithmetical Questions on Algebraic Varieties, London: The Athlone Press, pp. V+55, MR 0043498, Zbl 0042.15204. [9]
More precisely, let be a smooth projective surface over and a (−1)-curve on (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from to another smooth projective surface such that the curve has been contracted to one point , and moreover this morphism is an isomorphism outside (i.e., is ...
The surface with w = 1 (real points, bounded by a sphere with radius=6). 3D model of same surface as above (w = 1) bounded by the cube [-10, 10] 3. In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by Eugenio G. Togliatti .
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