Differential forms in algebraic geometry

Abstract

Before considering more general spaces we shall first discuss (1) the r-dimensional projective space Π r . In this space we shall consider a homogeneous coordinate system (Z0, Z1, ... , Z r ). Let U α be that part of Π r in which Z α ≠ 0. In U α we may then introduce non-homogeneous coordinates zαi = Zι/Zα (ι≠α). Any two distinct sets U α and U β will overlap and in U α ∩ U β we have the transformation law $$ z_\alpha ^i = \frac{z_{\beta ^i} } {z_{\beta} ^\alpha }\left( {i \ne \alpha ,\beta } \right)\,\,;\,z_\alpha ^\beta = \frac{1} {{z_{\beta} ^\alpha }}\,\,. $$ (1.1)