No abstract

Representation theory: a first eourse / William Fulton and Joe Harris. p. em. -(Graduate texts in mathematies) IncJudes bibliographieal referenees and index. 1. Representations of groups. 2. Representations of Algebras. 3. Lie Groups. 4. Lie algebras. 1. Harris, Joe. II. Tîtle.

No abstract

A great reference for background about linear systems, big and ample line bun dles and Kodaira dimensions is [L]. Here we will only develop a few basics that will be necessary for our discussion of the Kodaira dimension of the moduli space of curves.Let L be a line bundle on a normal, irreducible, projective variety X. The semi-group N (X, L) of L is defined to be the non-negative powers of L that have a non-zero section:Definition 1.1. Let L be a line bundle on a normal, irreducible, projective variety. Then the Iitaka dimension of L is defined to be the maximum dimension of the imageWhen X is smooth, the Kodaira dimension of X is defined to be the Iitaka dimension of its canonical bundle K X . If X is singular, the Kodaira dimension of X is defined to be the Kodaira dimension of any desingularization of X. Kodaira's Lemma allows us to obtain other useful characterizations of big line bundles.