THE 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 de ned 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 nd details about the construction of the space De nition 1 1 Let X be a smooth projective variety Let H2 X Z denote the class of a curve The Kontsevich moduli space M g 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 M g n X 0 M g n X Since a degree 0 map from a connected curve is determined by specifying a point on X this identi cation is immediate Example 1 4 The moduli space of degree one maps to Pr is isomorphic to the Grassmannian M 0 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 M 0 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 1 Exercise 1 6 Prove the previous assertion by exhibiting a map using the universal property of complete conics from M 0 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 Theorem 1 7 If X 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 di erent 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 di erent dimensions Consider degree two genus zero stable maps to a smooth degree seven hypersurface X in P 7 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 di erent 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 nite group The locus of automorphism free maps is a ne 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 di erent dimensions Mg n X possesses a virtual fundamental class of the expected dimension The existence of the virtual fundamental class is the key to Gromov Witten Theory Requiring a variety to be convex is a strong requirement on uniruled varieties For instance the blow up of a convex variety ceasses to be convex In fact I do not know any examples of rationally connected projective convex varieties that are not homogeneous Problem 1 12 Is every rationally connected convex projective variety a homo geneous space Either prove that it is or give counterexamples 1 2 Kontsevich s count of rational curves The Kontsevich moduli space is endowed with n evaluation morphisms evi M g n X X where evi sends the point C p1 pn f to f pi X From now on we will assume that X is a homogeneous variety and we will always restrict ourselves to the case of genus zero curves Given the classes 1 n of algebraic subvarieties of …