LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 199of degree six. Find the limit stable map and prove that Un Gromov converges to
your limit. Hint: The limit is a stable map, modelled over a tree with six vertices,
two each of degree 0, 1, and 2. D
4.3. Deligne-Mumford compactification
We begin by repeating the formal definition of a stable map in the simplified context
where the target space is a point.
Definition 4.14. A stable Riemann surface of genus zero with n marked
points, modelled over a tree (T, E), is a tuple z = ( {za,B}aE,B, {ai, zih<i<n)
consisting of points Za,B E 82 for a, (3 ET with aE(J and pairs (ai, zi) ET x S^2 for
i = 1, ... , n such that the following holds.
( i) If aE (3, a.E"(, and (3 =f. 'Y, then Zap =f. Zai · If ai = a j with i =f. j then
Zi =f. Zj. If CY.i = a and a.E (3 then Zi =f. Za,B.
(ii) For each a E T the setZa = Za(z) = {za,B : (3 ET, aE(J} U {zi 1:::; i:::; n, CY.i =a}
contains at least three elements.Definition 4.15. Two stable Riemann surfaces
of genus zero with n marked points are called equivalent if there exists a tree
isomorphism f : T -+ T and a collection of Mobius transformations <p = { <pa}aET
such that the following holds.(i) If a.,(3 ET with a.E(J then ZJ(a)f(,B) = <pa(Za,B)·
(ii) For i = 1, ... ,n, ai = f(ai) and Zi = <pa; (zi)·
Definition 4.16. A sequence zv = ({z~.e}aEv,B, {ai,zi}1~i~n) of stable Riemann
surfaces of genus zero with n marked points is said to DM-converge to a stable
Riemann surface z = ({za,B}aE,B,{a.i,zih~i~n) if, for 1.1 sufficiently large, thereexists a surjective tree homomorphism JV : T -+ rv and a collection of Mobius
transformations { <p~}aET such that the following holds.(i) Let a, (3 ET with a.E(J. If r1 (a) =f. r1 ((3) for some subsequence llj then
Za,B = }~1!(<p;{)-^1 (z;[1(aJr1(,B)).If r1 (a) = r1 ((3) for some subsequence vj then (<pd )-^1 o <p~^1 converges to
Za,B, uniformly on compact subsets of 82 - {z.ea}·(ii) For i = 1, ... 'n, ai = r(a.i) and Zi = limv->oo(<p~J-^1 (zi).
Consider the moduli space Mo,n of equivalence classes [z] of stable Riemann
surfaces of genus zero with n marked points under the equivalence relation of Defini-
tion 4.15. This quotient space inherits a topology from Definition 4.16 as described
in the previous section. The goal of this section is to explain how Mo,n naturally
admits the structure of a compact smooth manifold. To begin with let us denote
by
82 x···x8^2 -6..
Mo,n = G
the space of equivalence classes of ordered n-tuples of distinct points in 82 under
the diagonal action of the conformal group G = PSL(2, q. This quotient is the