The background is that codimension-1 subvarieties are understood much better than higher-codimension subvarieties. This happens in both global and local ways. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-"r" subvariety need not be definable by only "r" equations when "r" is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles.