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The theory of algebraic surfaces is much more complicated than that of Algebraic Curve s (including the Compact Riemann Surface s, which are genuine Surface s 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 ):
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 Function s in two indeterminates. The cartesian product of two curves also provides examples. The Birational Geometry of algebraic surfaces is rich, because of Blowing Up (also known as a Monoidal Transformation ); under which a point is replaced by the ''curve'' of all limiting tangent directions coming into it (a Projective Line ). Certain curves may also be blown ''down'', but there is a restriction (self-intersection number must be −1). 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). There are essential three Hodge Number invariants of a surface. Of those, ''h''1,0 was classically called the irregularity and denoted by ''q''; and ''h''2,0 was called the '''geometric genus''' ''p''''g''. The third, ''h''1,1, is not a Birational Invariant , because Blowing Up can add whole curves, with classes in ''H''1,1. It is known that Hodge Cycle s are algebraic, and that Algebraic Equivalence coincides with Homological Equivalence , so that ''h''1,1 is an upper bound for ρ, the rank of the Néron-Severi Group . The Arithmetic Genus ''p''''a'' is the difference :geometric genus − irregularity. In fact this explains why the irregularity got its name, as a kind of 'error term'. The Riemann-Roch Theorem for surfaces was first formulated by Max Noether . The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry. EXTERNAL LINKS
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