projectivization

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projectivization

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• projectivisation

From projective +‎ -ization.

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.

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