266
12 Geometry of Moduli Spaces and Riemann Surfaces
Next, the subspace of M 0,4 obtained from the plumbing fixture is
F 0,4 = V 0,3 #V 0,3 ,
(12.46)
and V 0,4 is characterized as the remaining region. Then, one has
F 0,5 = M 0,4 #M 0,3
= F 0,4 #V 0,3 + V 0,4 #V 0,3 = V 0,3 #V 0,3 #V 0,3 + V 0,4 #V 0,3 ,
(12.47)
and V 0,5 is what remains of M 0,5 . The pattern continues for g = 0. The same story
holds for g ≥ 1: the first such space is
F 1,1 = #V 0,3 ,
(12.48)
and V 1,1 = M 1,1 − F 1,1 . The gluing of a 3-punctured sphere and the addition of a
handle are the two most elementary operations.
To keep track of which moduli spaces can contribute, it is useful to find a function
of g,n , called the index, which increases by 1 for each of the two elementary
operations:
r(( g 1 ,n 1 # 0,3 ) = r(( g 1 ,n 1 ) + 1,
r(# g 1 ,n 1 ) = r(( g 1 ,n 1 ) + 1.
(12.49)
An appropriate function is
r(( g,n ) = 3g + n − 2 ∈ N
∗ ,
(12.50)
which is normalized such that
r(( 0,3 ) = 1.
(12.51)
For a generic separating plumbing fixture, we find
r(( g 1 ,n 1 # g 2 ,n 2 ) = r(( g 1 ,n 1 ) + r(( g 2 ,n 2 ).
(12.52)
Since the index increases, surfaces with a given r can be obtained by considering all
the gluings of surfaces with r < r.
12.3.4 Stubs
To conclude this chapter, we introduce the concept of stubs. Previously in (12.31),
the range of the parameter s was the complete line of positive numbers, s ∈ R + .
This means that tubes of all lengths were considered to glue surfaces. But, we could
12 Geometry of Moduli Spaces and Riemann Surfaces
Next, the subspace of M 0,4 obtained from the plumbing fixture is
F 0,4 = V 0,3 #V 0,3 ,
(12.46)
and V 0,4 is characterized as the remaining region. Then, one has
F 0,5 = M 0,4 #M 0,3
= F 0,4 #V 0,3 + V 0,4 #V 0,3 = V 0,3 #V 0,3 #V 0,3 + V 0,4 #V 0,3 ,
(12.47)
and V 0,5 is what remains of M 0,5 . The pattern continues for g = 0. The same story
holds for g ≥ 1: the first such space is
F 1,1 = #V 0,3 ,
(12.48)
and V 1,1 = M 1,1 − F 1,1 . The gluing of a 3-punctured sphere and the addition of a
handle are the two most elementary operations.
To keep track of which moduli spaces can contribute, it is useful to find a function
of g,n , called the index, which increases by 1 for each of the two elementary
operations:
r(( g 1 ,n 1 # 0,3 ) = r(( g 1 ,n 1 ) + 1,
r(# g 1 ,n 1 ) = r(( g 1 ,n 1 ) + 1.
(12.49)
An appropriate function is
r(( g,n ) = 3g + n − 2 ∈ N
∗ ,
(12.50)
which is normalized such that
r(( 0,3 ) = 1.
(12.51)
For a generic separating plumbing fixture, we find
r(( g 1 ,n 1 # g 2 ,n 2 ) = r(( g 1 ,n 1 ) + r(( g 2 ,n 2 ).
(12.52)
Since the index increases, surfaces with a given r can be obtained by considering all
the gluings of surfaces with r < r.
12.3.4 Stubs
To conclude this chapter, we introduce the concept of stubs. Previously in (12.31),
the range of the parameter s was the complete line of positive numbers, s ∈ R + .
This means that tubes of all lengths were considered to glue surfaces. But, we could