E. Moduli spaces of stable maps 77 nonsingular variety X convex if, for every mapµ : P1✲X, we have H1
P1, µ∗(TX)
= {0}. This follows ifTX is generated by global sec- tions; hence, any variety admitting a transitive group action is con- vex. For convex X, the space M0,n(X, γ) has properties very much like those of moduli spaces of stable curves. In particular,
• M0,n(X, γ)is locally normal of pure dimension dim(X)+
γc1(TX)+n−3 ;
• M0,n(X, γ)is locally a quotient of a smooth variety by a finite group;
• The locus of maps without automorphisms inM0,n(X, γ) is a fine moduli space for such maps; and,
• The boundary ofM0,n(X, γ)— that is, the locus of maps with reducible domainC — is a normal crossing divisor.
The next exercise makes this more concrete by analyzing the simple examples.
Exercise (2.76) 1) Show that an open set ofM0,0(P2,2)with domain a smooth rational curve parameterizes nonsingular conics. Maps with this domain also give all double covers of a line (which are deter- mined, up to isomorphism, by the line and the two branch points on it). Next, maps with domain the union of two rational curves meeting at a point parameterize those singular conics that are the union of two distinct lines as well as double lines with a distinguished point (the image of the point where the two rational curves meet). Conclude that M0,0(P2,2)is isomorphic to the classical space ofcomplete conics(see for example Vainsencher [146]).
2) Show that the same classification of stable maps extends to M0,0(Pn,2)whenn≥3. However, what we obtain isnot the classical space of complete conics (classifying a plane plus a complete conic in that plane): when the map has image a line, the locus of planes containing the line is blown down to a point inM0,0(Pn,2).
The boundary of M0,n(X, γ) can, once again, be broken up into subloci indexed by the ways in which a stable map can have a re- ducible domain. Now, however, it’s necessary to keep track not only of the decomposition of the curve, but also of the set of marked points and the classγ. For each partition of{1, . . . , n}into disjoint subsets AandBand each decompositionγ=α+β, we let∆(A, B;α, β)be the closure of the locus of stable maps for whichC=CA∪CB, the points indexed byAlie onCAand those indexed byBonCB, and the restric- tions ofµtoCAandCBrepresentαandβ, respectively. (Note that, if
α= 0, then the stability ofµ forces #A ≥2 and that∆(A, B;α, β)is empty unless the classesαandβare represented by stable maps of genus 0. Also, the irreducibility of∆(A, B;α, β)is only known whenX is a projective space.)
If we define
∆({i, j}|{k, l})=
{i,j}⊂A {k,l}⊂B α+β=γ
∆(A, B;α, β) ,
then we have the fundamental linear equivalence (2.77) ∆({i, j}|{k, l})∼∆({i, l}|{j, k}) .
In the case of the moduli spaceM0,4 P1, there are three such di- visors corresponding to the three points 0, 1 and ∞ in this space parameterizing reducible curves — see Exercise (2.19) — and the linear equivalence is simply that of points on P1. In general, the equivalence follows by pulling back this special case under the map M0,n(X, γ)✲M0,4 which forgets the map γ and the points not in- dexed by{i, j, k, l}.
When we takeXto be a projective space (certain other homogeneous spaces can be used as well), equation (2.77) can be used to obtain re- cursions for solutions to a wide range of enumerative questions about rational curves. The most direct approach is to write down a suitable curveY inM0,n(Pn, d)and to interpret its intersection numbers with various boundary divisors as enumerative quantities. Applying these interpretations to the two sides of (2.77) then produces relations be- tween the enumerative quantities.
Rather than even attempt to write down general results of this type, we sketch the now classic use ofM0,3d(P2, d)to calculate the number Ndof rational plane curves of degreedthrough 3d−1 general points.
In this case, the “right” curveY consists of those stable maps which send the first two marked points to points lying on two fixed but general lines and the other 3d−2 marked points to fixed general points. The only maps
µ:C=C{1,2}∪C{3,...,3d}✲P2
at which Y can meet ∆({1,2},{3, . . . ,3d}; 0, d) are those which col- lapse C{1,2} to the point of intersection of the two fixed lines. Since the map must also take the points of C{3,...,3d} to the 3d−2 fixed points, #
Y∩∆({1,2},{3, . . . ,3d}; 0, d)
=Nd.
Since the images of the marked points are general, Y is disjoint from every other ∆(A, B; 0, d) for which A contains {1,2} and, for 0< e < d, it can meet∆(A, B;e, d−e)only when #A=3e+1. If so, we can count as follows: there are3e−4
3e−1
partitions for which{1,2} ∈A
E. Moduli spaces of stable maps 79 and{3,4} ∈B;Nechoices for the image ofCAandNd−efor the image ofCB;echoices for the image of each the pointsp1andp2which must map to points of intersection ofµ(CA)with the corresponding fixed line; and e(d−e) choices for the image of the intersection CA∩CB
which must lie in the intersectionµ(CA)∩µ(CB). Thus,
#
Y ∩∆({1,2}|{3,4})
=Nd+
0<e<d
NeNd−ee3(d−e)
3d−4 3e−1
Exercise (2.78) Suppose now that{1,4} ∈A and{2,3} ∈ B. Show that #
Y∩∆(A, B;e, d−e)
=0 ife=0 ore=d. Otherwise, show that Y meets∆(A, B;e, d−e)
only if #A =3eand that, in each of these 3d−4
3e−2
cases, we have
#
Y ∩∆(A, B;e, d−e)
=NeNd−ee2(d−e)2.
Finally, use ∆({1,2}|{3,4}) = ∆({1,4}|{2,3})(cf. (2.77)) to deduce the recursion
Nd=
0<e<d
NeNd−e
e3(d−e)
3d−4 3e−1
−e2(d−e)2
3d−4 3e−2
. Of course, this argument depends coming up with the right curveY. This can be avoided by rephrasing (2.77) in terms of the formalism of Gromov-Witten invariants and quantum cohomology. Even defining these terms carefully would take us too far afield, so we give only the barest sketch and refer to [54] for all details.
Gromov-Witten invariants are numerical invariants of suitable col- lections of cohomology classes onX obtained by: pulling the classes back to M0,n(X, γ) using the evaluation maps which send a stable map to its value at a marked point; cupping together the pullbacks;
and finally integrating them over the fundamental class ofM0,n(X, γ).
WhenX is a homogeneous space, they have enumerative interpreta- tions. If, in addition, all classes on X which represent a stable map are expressible as nonnegative linear combinations of a finite num- ber γ1, . . . , γm, the Gromov-Witten invariants can be used to define an extension of the ring structure on the Chow ring ofA∗(X)to its tensor product with the formal power series ringQ[[γ1, . . . , γm]]: the extended ring is called aquantum cohomology ring. Equation (2.77) amounts to the associativity of this extended product. Once this is set up, enumerative results can be obtained by simply writing down the associativity equations and applying the enumerative interpretations of the Gromov-Witten invariants.
These ideas are currently being pursued in a number of different directions. The basic program is laid out in [103] and [104]. On the one hand, there is a symplectic approach to quantum cohomology
(cf. [135]) which has been used to prove associativity in some cases not yet handled by the kinds of methods discussed here. On the other, the genus 0 results have motivated work by Ran, Caporaso-Harris, Pandharipande, Vakil, Getzler and others on enumerative geometry of curves of higher genus (cf. [133], [17], [18], [19], [129], [128], [147], [55] and [56]) to which the quantum cohomological formalism doesn’t seem to extend directly.