Algebraic surface

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.

Related Topics:
Mathematics - Algebraic variety - Dimension - Complex number - Complex manifold - Non-singular - Smooth manifold

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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.

Related Topics:
Algebraic curve - Compact - Riemann surface - Surface - Italian school of algebraic geometry

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Examples of algebraic surfaces include:

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• the projective plane
• quadrics in P³
• cubic surfaces
• Veronese surface
• Del Pezzo surfaces
• ruled surfaces
• K3 surfaces
• abelian surfaces
• surfaces of general type.

The first three 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 functions in two indeterminates. The cartesian product of two curves also provides examples.

Related Topics:
Birationally equivalent - Function field - Projective plane - Rational function

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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).

Related Topics:
Birational geometry - Blowing up - Monoidal transformation - Projective line

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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 P3 lies in it, for example).

Related Topics:
Hodge index theorem - Classification of algebraic surface - Kodaira dimension

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There are essential three Hodge number invariants of a surface. Of those, h1,0 was classically called the irregularity and denoted by q; and h2,0 was called the geometric genus pg. The third, h1,1, is not a birational invariant, because blowing up can add whole curves, with classes in H1,1. It is known that Hodge cycles are algebraic, and that algebraic equivalence coincides with homological equivalence, so that h1,1 identifies with ρ, the rank of the Néron-Severi group. The arithmetic genus pa is the difference

Related Topics:
Hodge number - Birational invariant - Blowing up - Hodge cycle - Algebraic equivalence - Homological equivalence - Néron-Severi group - Arithmetic genus

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:geometric genus − irregularity.

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In fact this explains why the irregularity got its name, as a kind of 'error term'.

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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.

Related Topics:
Riemann-Roch theorem - Max Noether

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