Some Geometry of Curves in Projective Space

Abstract

We begin with some general remarks. If M is a complex manifold of dimension n and A ⊂ M is a submanifold of dimension n −, 1 (codimension 1), A defines a holomorphic line bundle as on a Riemann surface: if {U _i} is an open covering of $M,f_i \in \mathcal{O}(U_i )$ is such that U _i ∩ A = {x ∈ U _i f _i (x) = 0, df _i ≠ 0 at any pomit of U _i }, then g _ij = f _i /f _j is holomorphic, nowhere zero on U _i ∩ U _j and form the transition functions for a line bundle L(A). The family {f _i } define the standard section s _A of L(A) (whose divisor is A).