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Encyclopedia > 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

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. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... An open surface with X-, Y-, and Z-contours shown. ... In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. ...

Examples of algebraic surfaces include (κ is the Kodaira dimension): In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is a graded commutative ring that is made up of the sections of powers of the canonical bundle K. More precisely, it is the graded ring R such that for n...

For more examples see the list of algebraic surfaces Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... A cubic surface is a projective variety studied in algebraic geometry. ... In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space. ... In mathematics, a del Pezzo surface is a complex two-dimensional Fano variety, i. ... In geometry, a surface is ruled if through every point of there is a straight line that lies on . ... A K3 manifold is a hyperkähler manifold of real dimension 4, i. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, an Enriques surface is an algebraic surface such that the irregularity q = 1 and the canonical line bundle is non-trivial but has trivial square. ... In mathematics, the Enriques-Kodaira classification is a classification of compact complex surfaces. ... In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected smooth morphism to an algebraic curve, almost all of whose fibers are elliptic curves. ... In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. ... This is a list of named (classes 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 functions in two indeterminates. The cartesian product of two curves also provides examples. In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

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). In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In mathematics, blowing up is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. ... In mathematics, blowing up or blowup is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. ... In mathematics, a projective line is a one-dimensional projective space. ...

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). 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... In mathematics, the Enriques-Kodaira classification is a classification of compact complex surfaces. ... In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is a graded commutative ring that is made up of the sections of powers of the canonical bundle K. More precisely, it is the graded ring R such that for n...

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 is an upper bound for ρ, the rank of the Néron-Severi group. The arithmetic genus pa is the difference In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension... In mathematics, a birational invariant in algebraic geometry is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties. ... In mathematics, blowing up is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. ... In mathematics, a Hodge cycle is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. ... In algebraic geometry, the NÃ©ronâ€“Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. ... In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalisations of the genus of an algebraic curve. ...

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. In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... Max Noether (September 24, 1844 - December 13, 1921) was a German mathematician. ... Results from FactBites:

 Algebraic surface - Wikipedia, the free encyclopedia (396 words) 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). 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).
 Algebraic number - encyclopedia article about Algebraic number. (2503 words) The algebraically closed field in which these numbers lie can be the complex numbers In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.
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