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In mathematics, an algebraic surface is an algebraic variety of dimension two.
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The classification of algebraic surfaces was a bold and successful attempt to repeat the division of curves by their genus g.
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The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1).
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This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism.
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The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form.
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In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class.
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N In 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but Mumford (1968) showed that this is false for surfaces of positive geometric genus.
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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.
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At first this did not matter too much, as Enriques's intuition was so good that essentially all the results he claimed were in fact correct, and using this more informal style of argument allowed him to produce spectacular results about algebraic surfaces.
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In particular in the 1960s Kodaira and Shafarevich and his students rewrote the Enriques classification of algebraic surfaces in a more rigorous style, and also extended it to all compact complex surfaces, while in the 1970s Fulton and MacPherson put the classical calculations of intersection theory on rigorous foundations.
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Neither are algebraic curves or surfaces defined over fields other than the complex numbers.
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Algebraic geometry has traditionally meant the understanding of geometric objects, such as algebraic curves and surfaces, through the study of the algebraic equations for those objects.
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The most important are those relating to algebraic curves and surfaces, especially the short paper Allgemeine Eigenschaften algebraischer Curven.
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In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes.
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Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 Neue Theorie der Elektrodynamik and several papers on algebraic curves and surfaces, in the hope that these applications would lead others to take his theory seriously.
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Any complex nonsingular algebraic curve viewed as a complex manifold is a Riemann surface.
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It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
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This Togliatti surface is an algebraic surface of degree five.
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In fact, an algebraic surface is an algebraic variety of dimension two.
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The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology.
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Neither are algebraic curves defined over fields other than the complex numbers, nor are algebraic surfaces defined over fields other than the real numbers.
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The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.
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Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.
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The theory of compact Riemann surfaces can be shown to be equivalent to that of projective algebraic curves that are defined over the complex numbers and non-singular.
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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.
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