LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 203point { w I ( z)} I Ein) as smooth functions of the given n - 3 cross ratios. The choice
of the n-3 coordinates is geometric. If there exists a marked point Zm such that the
corresponding vertex am has at least four special points, then there is a cross ratio
wim.Jm.km.m which completely determines the position of the point Zm and hence all
other cross ratios involving the point Zm. Now remove the point Zm and proceed
by induction until there is no marked point left which lies on a sphere with at least
four special points. Next choose a marked point Zm lying on an endpoint am =a
of the tree. Then there is precisely one other marked point Zkm. with akm. = a and
precisely one vertex /3 with aE/3. The marked points Zi"' and Zj"' should be chosen
such that Zf3im. =/:-Zf3jm. (see Figure 17). Then the cross ratio Wim.Jm.km.m uniquely
determines the position of Zm in a neighbourhood of the given stable Riemann
surface. Now proceed again by induction to find the required n - 3 coordinates.
For more details see [20]. D
Figure 17. Coordinates for Mo,nExercise 4.22. Prove that M 0 , 4 ~ CP^1 and M 0 , 5 ~ CP^2 #4CP
2
~ (S^2 x
S^2 )#3CP
2. Examine the natural projection Mo,n ----> Mo,n-I and describe it
with the above identifications in the case n = 5. Hint 1: The fiber of the projec-
tion is the curve corresponding to the point in Mo,n-I · In the case n = 5 there are
three exceptional fibers, corresponding to the three special points in Mo,4. Think
of the fibration as a family of quadrics passing through four generic points in the
(complex projective) plane. The three singular fibers correspond to the three pairs
of lines passing through these points. Blow up the four points to obtain Mo,5 (see
Figure 18). Hint 2: Fix three points at zo = 0, z1 = 1, z2 = oo. Show that Mo,5
consists of pairs ( z 3 , z 4 ) which are not equal to the three pairs ( 0, 0), ( 1, 1), ( oo, oo),
together with three 2-spheres representing the possible co nfigurations which arise
from a collision of both z 3 and z 4 with Zi for i = 0, 1, 2. Examine neighbourhoods
of these 2-spheres to obtain the product S^2 x S^2 with three points on the diagonal
blown up. D
Figure 18. Three pairs of lines determined by four generic points