|
Schubert Cycle
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linear algebra, inside a fixed vector subspace ''W''. Here ''W'' may be a vector space over an arbitrary field, though most commonly over the complex numbers. A typical example is the set ''X'' whose points correspond to those 2-dimensional subspaces ''V'' of a 4-dimensional vector space ''W'', such that ''V'' non-trivially intersects a fixed (reference) 2-dimensional subspace ''W''2: :X \ =\ \. Over the real number field, this can be pictured in usual ''xyz''-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of \mathbb(W), we obtain an open subset ''X''° ⊂ ''X''. This is isomorphic to the set of all lines ''L'' (not necessarily through the origin) which ... [...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] |
