Invertible Sheaves or Line Bundles
The Tautological and Hyperplane Bundles
Consider the line bundle
where acts via inverse scalar multiplication on both and , i.e. if and , then each variable has weight one with respect to the -action.
is a space that maps to by projecting onto the first factor, and the fibre of this projection over any point is a line. More generally, we can consider the line bundle
where now acts on with weight , i.e. for any and with .
::: proposition Tautological and Hyperplane Bundles
The set that consists of all pairs with forms in a natural way a holomorphic line bundle over , called the tautological bundle.
Its dual bundle is called the hyperplane or anti-tautological bundle.