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Plücker Surface
In algebraic geometry, a Plücker surface, studied by , is a quartic surface in 3-dimensional projective space with a double line and 8 nodes. Construction For any quadric line complex In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian ''G''(2, 4) (embedded in projective space ''P''5 by Plücker coordinates) with a hypersurface. It is called a line complex because points of ''G''( ..., the lines of the complex in a plane envelop a quadric in the plane. A Plücker surface depends on the choice of a quadric line complex and a line, and consists of points of the quadrics associated to the planes through the chosen line. References * * * {{DEFAULTSORT:Plucker Surface Algebraic surfaces ... [...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] |
Quartic Surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surface is the solution set of an equation of the form :f(x,y,z)=0\ where is a polynomial of degree 4, such as . This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space of the same form, but now is a ''homogeneous'' polynomial of 4 variables of degree 4, so for example . If the base field is or the surface is said to be ''real'' or ''complex'' respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a ''real'' surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an ''arithmetic quartic surfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadric Line Complex
In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian ''G''(2, 4) (embedded in projective space ''P''5 by Plücker coordinates) with a hypersurface. It is called a line complex because points of ''G''(2, 4) correspond to lines in ''P''3, so a line complex can be thought of as a 3-dimensional family of lines in ''P''3. The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2; they are both rational varieties In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), t .... References * * * Algebraic varieties 3-folds {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
