In
mathematics, an algebraic surface is an
algebraic variety of
dimension two. In the case of geometry over the
complex number field, an algebraic surface is therefore of 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.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 surfacesκ= 0 :
K3 surfaces,
abelian surfaces,
Enriques surfaces,
hyperelliptic surfacesκ= 1:
Elliptic surfacesκ= 2:
surfaces of general type.
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