Projectivization

1997, M. E. Alferieff (translator), Alexei Kostrikin, Yuri Manin, Linear Algebra and Geometry, Gordon and Breach Science Publishers, Paperback, page 233, Therefore, the projectivization P(f) is determined only on the complement U_f=P(L)P( ker f).

Let us consider the vector space T#95;p. Its projectivization #123;PT#125;#95;p, is a projective space, which is obtained from T#95;p, by factorization relative to the collinearity of vectors. The projectivization #123;PT#125;#95;p is a set of straight lines, passing through the point p. Under projectivization, the cone C(2,r) becomes a manifold S(1,r-1) of dimension r.

The projectivization of a two-dimensional irregular equation (1.1) is a holomorphic vector field on #92;mathbbP¹#92;times#92;left#92;#123;#92;vertt#92;vertlt;1#92;right#92;#125; having a pair of singularities on the fiber #92;mathbbP¹#92;times 0 (which correspond to the eigenlines of the matrix 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.