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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
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
The surface is a quotient of a K3 surface by the group scheme Z/2Z. Supersingular: dim(H 1 (O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K = 0, and Pic τ is α 2. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α 2. All Enriques surfaces are elliptic or quasi elliptic.
Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.
An investigation of the relative positions of the branches of real algebraic curves of degree n (and similarly for algebraic surfaces). The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions. The first problem is yet unsolved ...
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V.It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such one-dimensional ...
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 ...