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White Surface
In algebraic geometry, a White surface is one of the rational surfaces in ''P''''n'' studied by , generalizing cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...s and Bordiga surfaces, which are the cases ''n'' = 3 or 4. A White surface in ''P''''n'' is given by the embedding of ''P''2 blown up in ''n''(''n'' + 1)/2 points by the linear system of degree ''n'' curves through these points. References *{{citation, first=F. P. , last=White, title=On certain nets of plane curves, journal=Proceedings of the Cambridge Philosophical Society, volume=22, year=1923, pages=1–10, doi=10.1017/S0305004100000037 Complex surfaces 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] |
Rational Surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birational geometry, birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to be investigated. Structure Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σ''r'' for ''r'' = 0 or ''r'' ≥ 2. Invariants: The plurigenera are all 0 and the fundamental group is trivial. Homological mirror symmetry#Hodge diamond, Hodge diamond: where ''n'' is 0 for the projective plane, and 1 for Hirzebruch surfaces and greater than 1 for other rational surfaces. The Picard group is the odd unimodular lattice I1,''n'', except for the Hirzebruch surfaces Σ2''m'' when it is the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 minus a lower-dimensional subset and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bordiga Surface
In algebraic geometry, a Bordiga surface is a certain sort of rational surface of degree 6 in ''P''4, introduced by Giovanni Bordiga. A Bordiga surface is isomorphic to the projective plane blown up in 10 points, the embedding into ''P''4 is given by the 5-dimensional space of quartics passing through the 10 points. White surface In algebraic geometry, a White surface is one of the rational surfaces in ''P'n'' studied by , generalizing cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic s ...s are the generalizations using more points. References * Complex surfaces Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blowing Up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups. Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by bl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Surfaces
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
