1. Preliminaries2. The canonical bundle of the moduli space of curves3. Ample divisors on the moduli space of curves4. The moduli space is of general type5. The compuatation of the classes of Brill-Noether Divisors 5.1. The Brill-Noether Theorem5.2. Limit linear series5.3. Calculating the classes of the Brill-Noether divisors6. The ample and effective cones of the moduli space of curvesReferencesTHE 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 defined to be the non-negative powers of L that have a non-zero section: N(X, L) := { m ∼ 0 : H0(X, L�m) > 0 }. Given m ⊕ N(X, L) we can consider the rational map πm associated to L�m . 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 image of πm for m ⊕ N (X, L) provided N(X, L) ∈= 0. If N(X, L) = 0, then the Iitaka dimension of L is defined to be −→. When X is smooth, the Kodaira dimension of X is defined to be the Iitaka dimension of its canonical bundle KX . If X is singular, the Kodaira dimension of X is defined to be the Kodaira dimension of any desingularization of X. Remark 1.2. Note that by definition the Iitaka dimension of a line bundle L on X is an integer between 0 and dim(X) or it is −→. Definition 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 definitions can be made for Cartier (or even Q-Cartier) divisors instead of line bundles. Below we will use the language of Cartier divisors and line bundles interchangably. An alternative definition 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) ∼ Cm n for all sufficiently large m ⊕ N(X, L). Kodaira’s Lemma allows us to obtain other useful characterizations of big line bundles. 1Lemma 1.6 (Kodaira’s Lemma). Let D be a big Cartier divisor and E be an arbitrary effective Cartier divisor on a normal, projective variety X. Then H0(X, OX (mD − E)) ∈= 0 for all sufficiently 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 effective 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 effective divisor E such that mD is linearly equivalent to A + E. (3) There exists an ample divisor A, an integer m > 0 and an effective divisor E such that mD is linearly equivalent to A + E. (4) There exists an ample divisor A, an integer m > 0 and an effective 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 effective. By Kodaira’s Lemma there is a positive integer m such that mD − (r + 1)A is effective, say linearly equivalent to an effective 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 K= 13� − 2ν − ν1.M g 2Proof. The cotangent bundle of Mg at a smooth, automorphism-free curve is given by the space of quadratic differentials. More generally, over the automorphism-free locus the canonical bundle will be the first chern class of ≤ � ).α�(ΓMg,1 /Mg Mg,1 /Mg We can easily calculate this class in the Picard group of the moduli functor: � ⎞2 2c1(Γ ≤ �) c1(Γ) c1(Γ) + c2(Γ)α� (1 + c1(Γ ≤ �) + − c2(Γ ≤ �))(1 − + )2 2 12 Expanding (and using the relations we proved in the last unit) we see that this expression equals �⎞2c1(�) + [Sing]2 2α� 2c1(�) − [Sing] − c1(�) + = 13� − 2ν. 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 effect 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 ν1 does not descend to the coarse moduli scheme because every curve in the boundary locus has an automorphism of order 2. However, ν2 descends to the coarse moduli scheme. 1 Accordingly we defined the