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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 .
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
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 .
11 Algebraic surfaces. 12 Miscellaneous surfaces. 13 Fractals. Toggle Fractals subsection. 13.1 Random fractals. 14 Regular polytopes. Toggle Regular polytopes ...
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by Castelnuovo ( 1896 , 1897 ), after preliminary versions of it were found by Max Noether ( 1886 ) and Enriques ( 1894 ).
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).
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.
and can be expressed as an order-15 algebraic surface. [2] It can be viewed as an immersion of a punctured projective plane. [3] Up until 1981 it was the only known non-orientable minimal surface. [4] The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem. [5] [6]