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



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THE KODAIRA DIMENSION OF THE MODULI SPACE OF CURVES

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Pages:
18
School:
Massachusetts Institute of Technology
Course:
18 727 - Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces
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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 other useful characterizations of big line bundles 1 Lemma 1 6 Kodaira s Lemma Let D be a big Cartier divisor and E be an arbitrary e ective Cartier divisor on a normal projective variety X Then H 0 X OX mD E 0 for all su ciently large m N X D Proof Consider the exact sequence 0 OX mD E OX mD OE mD 0 Since D is big by assumption the dimension of global sections of OX mD grows like mdim X On the other hand dim E dim X hence the dimension of global sections of OE mD grows at most like mdim X 1 It follows that h0 X OX mD h0 E OE mD for large enough m N X D The lemma follows by the long exact sequence of cohomology associated to the exact sequence of sheaves A corollary of Kodaira s Lemma is the characterization of big divisors as those divisors that are numerically equivalent to the sum of an ample and an e ective divisor We will use this characterization in determining the Kodaira dimension of the moduli space of curves Proposition 1 7 Let D be a divisor on a normal irreducible projective variety X Then the following are equivalent 1 D is big 2 For any ample divisor A there exists an integer m 0 and an e ective divisor E such that mD is linearly equivalent to A E 3 There exists an ample divisor A an integer m 0 and an e ective divisor E such that mD is linearly equivalent to A E 4 There exists an ample divisor A an integer m 0 and an e ective divisor E such that mD is numerically equivalent to A E Proof To prove that 1 implies 2 given any ample divisor A take a large enough positive number r such that both rA and r 1 A are e ective By Kodaira s Lemma there is a positive integer m such that mD r 1 A is e ective say linearly equivalent to an e ective divisor E We thus get that mD is linearly equivalent to A rA E proving 2 Clearly 2 implies 3 and 3 implies 4 To see that 4 implies 1 since mD is numerically equivalent to A E mD E is numerically equivalent to an ample divisor Since ampleness is numerical mD E is ample Since ample divisors are big and h0 X mD h0 X mD E D is big 2 The canonical bundle of the moduli space of curves We can calculate the canonical class of the moduli space of curves using the Grothendieck Riemann Roch formula Theorem 2 1 The canonical class of the coarse moduli scheme M g is given by KM g 13 2 1 2 Proof The cotangent bundle of Mg at a smooth automorphism free curve is given by the space of quadratic di erentials More generally over the automorphism free locus the canonical bundle will be the rst chern class of Mg 1 Mg Mg 1 Mg We can easily calculate this class in the Picard group of the moduli functor c21 c1 c21 c2 c2 1 1 c1 2 2 12 Expanding and using the relations we proved in the last unit we see that this expression equals c2 Sing 2c21 Sing c21 1 13 2 12 We need to adjust this formula to take into account that every element of the locus of curves with an elliptic tail have an automorphism given by the hyperelliptic involution on the elliptic tail The e ect of this can be calculated in local coordinates to see that it introduces a simple zero along that locus Remark 2 2 One word of caution is in order Recall that 1 does not descend to the coarse moduli scheme because every curve in the boundary locus has an automorphism of order 2 However 12 descends to the coarse moduli scheme Accordingly we de ned the class 1 as half of the class of the boundary locus 1 In terms of the class of the loci of reducible curves the canonical class is 1 13 2 1 2 3 Ample divisors on the moduli space of curves In order to show that the moduli space is of general type we need to show that the canonical bundle is big on a desingularization In view of the discussion in the rst section we can try to express the canonical bundle as a sum of an ample and an e ective divisor The G I T construction gives us a large collection of ample divisors For our purposes we need only the following …


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