projectivization

English

English Wikipedia has an article on:

projectivization

Wikipedia

Alternative forms

• projectivisation

Etymology

From projective +‎ -ization.

Noun

projectivization (plural projectivizations)

1. (mathematics, algebraic geometry, birational geometry) A process (more formally, a mapping) that, given a vector space, specifies an associated projective space; (loosely) the projective space so specified.
   • 1992, Maks A. Akivis, Alexander M. Shelekhov, translated by Vladislav V. Goldberg, Geometry and Algebra of Multidimensional Three-Webs, Springer, page 11:

     Let us consider the vector space ${\displaystyle T_{p}}$. Its projectivization ${\displaystyle {PT}_{p}}$, is a projective space, which is obtained from ${\displaystyle T_{p}}$, by factorization relative to the collinearity of vectors. The projectivization ${\displaystyle {PT}_{p}}$ is a set of straight lines, passing through the point ${\displaystyle p}$. Under projectivization, the cone ${\displaystyle C(2,r)}$ becomes a manifold ${\displaystyle S(1,r-1)}$ of dimension ${\displaystyle r}$.

   • 1997, M. E. Alferieff (translator), Alexei Kostrikin, Yuri Manin, Linear Algebra and Geometry, Gordon and Breach Science Publishers, Paperback, page 233,

     Therefore, the projectivization ${\displaystyle P(f)}$ is determined only on the complement ${\displaystyle U_{f}=P(L)\backslash P(\ker f)}$.

   • 2004, Alexey Glutsyuk, “Confluence of singular points and Stokes phenomena”, in Yulij Ilyashenko, Christiane Rousseau, Gert Sabidussi, editors, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Kluwer Academic, page 290:

     The projectivization of a two-dimensional irregular equation (1.1) is a holomorphic vector field on ${\displaystyle \mathbb {P} ^{1}\times \left\{\vert t\vert <1\right\}}$ having a pair of singularities on the fiber ${\displaystyle \mathbb {P} ^{1}\times 0}$ (which correspond to the eigenlines of the matrix ${\displaystyle A(0)}$, the coordinate lines in our case). […] Now the preceding corollary applied to the family of projectivizations says that the horizontal separatrices converge to the sectorial separatrices of the projectivized nonperturbed equation.

Related terms

• projectivize
• projectivized

Translations

specification of a projective space given a vector space

See also

• blowup

Further reading

• Blowing up on Wikipedia.Wikipedia
• Projective space on Wikipedia.Wikipedia
• Algebraic geometry of projective spaces on Wikipedia.Wikipedia