# MIT 18 727 - THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES (18 pages)

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THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES 1 Preliminaries 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 de ned to be the non negative powers of L that have a non zero section N X L m 0 H 0 X L m 0 Given m N X L we can consider the rational map m associated to L m De nition 1 1 Let L be a line bundle on a normal irreducible projective variety Then the Iitaka dimension of L is de ned to be the maximum dimension of the image of m for m N X L provided N X L 0 If N X L 0 then the Iitaka dimension of L is de ned to be When X is smooth the Kodaira dimension of X is de ned to be the Iitaka dimension of its canonical bundle KX If X is singular the Kodaira dimension of X is de ned to be the Kodaira dimension of any desingularization of X Remark 1 2 Note that by de nition the Iitaka dimension of a line bundle L on X is an integer between 0 and dim X or it is De nition 1 3 A line bundle L on a normal projective variety is called big if its Iitaka dimension is equal to the dimension of X A smooth projective variety is called of general type if its canonical bundle is big A singular variety is called of general type if a desingularization is of general type Remark 1 4 Of course the same de nitions can be made for Cartier or even QCartier divisors instead of line bundles Below we will use the language of Cartier divisors and line bundles interchangably An alternative de nition of big line bundles in terms of cohomology is given by the following well known lemma Lemma 1 5 A line bundle L on a normal projective variety X of dimension n is big if and only if there exists a positive constant C such that h0 X L m Cmn for all su ciently large m N X L Kodaira s Lemma allows us to obtain

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