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Elliptic Singularity
In algebraic geometry, an elliptic singularity of a surface, introduced by , is a surface singularity such that the arithmetic genus of its local ring is 1. See also *Rational singularity In mathematics, more particularly in the field of algebraic geometry, a scheme (mathematics), scheme X has rational singularities, if it is normal scheme, normal, of finite type over a field of characteristic of a ring, characteristic zero, and the ... References * Algebraic surfaces Singularity theory {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Genus
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' p_a of ''X'' is defined asp_a(X)=(-1)^r (\chi(\mathcal_X)-1).Here \chi(\mathcal_X) is the Euler characteristic of the structure sheaf \mathcal_X. Complex projective manifolds The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely :p_a=\sum_^ (-1)^j h^. When ''n=1'', the formula becomes p_a=h^. According to the Hodge theorem, h^=h^. Consequently h^=h^1(X)/2=g, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When ''X'' is a compact Kähler manifold, applying ''h''''p'',''q'' = ''h''''q'',''p'' recovers the earlier definition for projective varieties. Kähler manifolds By u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non-units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x'' is any element of ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Singularity
In mathematics, more particularly in the field of algebraic geometry, a scheme (mathematics), scheme X has rational singularities, if it is normal scheme, normal, of finite type over a field of characteristic of a ring, characteristic zero, and there exists a proper morphism, proper birational map :f \colon Y \rightarrow X from a Glossary of scheme theory#Properties of schemes, regular scheme Y such that the higher direct images of f_* applied to \mathcal_Y are trivial. That is, :R^i f_* \mathcal_Y = 0 for i > 0. If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third. For surfaces, rational singularities were defined by . Formulations Alternately, one can say that X has rational singularities if and only if the natural map in the derived category :\mathcal_X \rightarrow R f_* \mathcal_Y is a quasi-isomorphism. Notice that this includes the statement that \mathcal_X \s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surfaces
In mathematics, an algebraic surface is an algebraic variety of dimension two. 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. 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. Classification by the Kodaira dimension In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. A summary of the results (in det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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