1. The Kontsevich moduli space of stable maps.1.1. Preliminaries1.2. Kontsevich's count of rational curves1.3. The quantum cohomology ring2. Divisor classes on the Kontsevich moduli space and enumerative geometry2.1. An algorithm for computing the genus zero characteristic numbers in projective space3. Counting genus zero curves in Pn: Vakil's algorithm4. The cones of ample and effective divisors on the Kontsevich moduli space4.1. The ample cone of the Kontsevich moduli space4.2. The effective cone of the Kontsevich moduli spaceReferencesTHE KONTSEVICH MODULI SPACES OF STABLE MAPS 1. The Kontsevich moduli space of stable maps. 1.1. Preliminaries. We will begin with a detailed study of the Kontsevich moduli spaces of stable maps to Pr . These spaces can be defined much more generally. However, we will have very little to say about the general situation. We will mostly concentrate on the case of genus zero maps to Pr . The best introduction to Kont-sevich moduli spaces is [FP] where you can find details about the construction of the space. Definition 1.1. Let X be a smooth projective variety. Let � ≤ H2(X, Z) denote the class of a curve. The Kontsevich moduli space Mg,n(X, �) of n-pointed, genus g stable maps to X in the class parameterizes isomorphism classes of the following data (1) (C, p1, . . . , pn, f) an at worst nodal curve C of arithmetic genus g with n distinct, smooth points p1, . . . , pn of C and a morphism f : C ∩ X such that f⊕[C] = �, (2) The map is required to be stable; that is if f is constant on any component of C, then that component is required to have at least 3 distinguished points. The distinguished points are either marked points, or points lying over nodes in the normalization of the curve. We have already encountered some examples of Kontsevich moduli spaces. Example 1.2. The moduli space of stable maps to a point coincides with the moduli space of curves: Mg,n(P0 , 0) ⊕=Mg,n. Example 1.3. The moduli space of degree zero stable maps, similarly, is easy to describe. Mg,n(X, 0) = Mg,n × X. Since a degree 0 map from a connected curve is determined by specifying a point on X, this identification is immediate. Example 1.4. The moduli space of degree one maps to Pr is isomorphic to the Grassmannian: M0,0(Pn , 1) = G(2, n + 1) = G(1, n). A generalization of this example is the moduli space of degree one maps to a smooth quadric hypersurface Q in Pn for n > 3. In that case the Kontsevich moduli space is isomorphic to the orthogonal Grassmannian. Example 1.5. The Kontsevich moduli space M0,0(P2 , 2) is isomorphic to the space of complete conics or alternatively it is isomorphic to the blow up of the Hilbert scheme of conics in P2 along the Veronese surface of double lines. 1Exercise 1.6. Prove the previous assertion by exhibiting a map (using the universal property of complete conics) from M0,0(P2 , 2) to the space of complete conics. Check that this is a bijection on points. The claim then follows from Zariski’s Main Theorem once we know that M 0,0(P2 , 2) is smooth. The main existence theorems for Kontsevich moduli spaces are the following. We refer you to [FP] for their proof. If XTheorem 1.7. is a complex, projective variety, then there exists a projective coarse moduli scheme Mg,n(X, �). Note that even when X is a nice, simple variety (such as P2), Mg,n(X, �) may have many components of different dimensions. Example 1.8. Consider the Kontsevich moduli space M1,0(P2 , 3) of genus one degree three stable maps to P2 . This space has three components: two of dimension 9 and one of dimension 10. Naively, we might expect an open subset of M1,0(P2 , 3) to parameterize smooth cubic curves in P2 . Indeed an open subset of one of the components does so. However, there is a second component whose general member is a map from a reducible curve with a genus zero component and a genus one component to P2 that contracts the genus one component and gives a degree three map on the genus zero component. Note that this component of M1,0(P2 , 3) has dimension 10. The dimension of rational cubics in P2 is 8, but the moduli of the contracted elliptic curve and the point of attachment add two more moduli. Similarly, one obtains a third component of dimension 9 by considering maps from elliptic curves with two rational tails which contract the elliptic curve and map the rational tails as a line and a conic. Example 1.9. Even if we restrict ourselves to genus zero stable maps the Kontse-vich moduli spaces may have many components of different dimensions. Consider degree two genus zero stable maps to a smooth degree seven hypersurface X in P7 . Assume that X contains a P3 . M0,0(X, 2) contains at least two components. One component covers X and has dimension 5. The conics in the P3 give a different component of dimension 8. In order to obtain an irreducible moduli space with mild singularities one needs to impose some conditions on X. One possibility is to require that X is convex. Recall that a variety X is convex if for every map f : P1 ∩ X, f⊕TX is generated by global sections. Since every vector bundle on P1 decomposes as a direct sum of line bundles, a variety is convex if for every map f : P1 ∩ X, the summands appearing in f ⊕TX are non-negative. If we consider genus zero stable maps to convex varieties, the Kontsevich moduli space has very nice properties. Theorem 1.10. Let X be a smooth, projective, convex variety. (1) M0,n(X, �) is a normal, projective variety of pure dimension dim(X) + c1(X) · � + n − 3. 2(2) M0,n(X, �) is locally the quotient of a non-singular variety by a finite group. The locus of automorphism free maps is a fine moduli space with a universal family and it is smooth. (3) The boundary is a normal crossings divisor. Observe that the previous theorem in particular applies to homogeneous varieties since homogeneous varieties are convex. In fact, if X is a homogeneous variety, then M0,n(X, �) is irreducible (see [KP]). Remark 1.11. Although when we do not restrict ourselves to the case of genus zero maps to homogeneous varieties Kontsevich moduli spaces may be reducible with components of different dimensions, Mg,n(X, �) possesses a virtual fundamental class of the expected dimension. The existence of the virtual fundamental class is