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

Previewing pages 1, 2, 3, 4, 5, 6 of 18 page document View the full content.
View Full Document

THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES



Previewing pages 1, 2, 3, 4, 5, 6 of actual document.

View the full content.
View Full Document
View Full Document

THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES

33 views


Pages:
18
School:
Massachusetts Institute of Technology
Course:
18 727 - Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces
Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces Documents

Unformatted text preview:

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



View Full Document

Access the best Study Guides, Lecture Notes and Practice Exams

Loading Unlocking...
Login

Join to view THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?