Line Bundles on Projective Spaces Preliminary Draft

0.1. Notations and Conventions. During this note, we will fix a base field k (e.g. the complex numbers C). The multiplicative group of the field k will be denoted by Gm. All the vector spaces considered will be finite dimensional vector spaces over the field k. For a given vector space V of dimension≥ 1, we will denote the dual vector space, the projective space parameterizing 1-dimensional subspace in V and the projective space parameterizing subspaces of codimension 1 in order by V, P(V) and P(V) = P(V). By the projectivization of the vector space V , we will mean the projective space P(V). The projective space P(V) is called the dual projective space. Notice that the dimensions of the projective space P(V) and P(V) are one less than that of the vector space V . More generally, one can fix an integer l ≤ dim(V) and consider the variety parameterizing l-planes in the vector space V . The resulting variety is called the Grassmannian of l-planes in V and is denoted by Grass(l, V). If L is a line in the vector space V , we will denote the corresponding point in the projective space by [L ⊂ V], or [L] for short. More generally, if S is an l-dimensional plane in V , the corresponding point in the Grassmannian Grass(l, V) will be denoted by [S ⊂ V], or [S] for short. Notice that by definition, the equalities Grass(1, V) = P(V) and Grass(dim(V) − 1, V) = P(V) hold. The isomorphism classes of line bundles on a smooth variety X is denoted by Pic(X). Under the tensor product, the set Pic(X) becomes a an Abelian group, where the trivial element of the group is the trivial bundle,OX and the inverse of a line bundle is determined by the equation L⊗L = OX. Therefore, for a given line bundle L, the dual bundle is called the inverse of L or L inverse , and it is denoted by L. For arbitrary integer n, we define the nth multiple of the line bundle L by the formula: