12.3 Plumbing Fixture
261
where I is the inversion (the superscript on the coordinates z a and z b has been
removed to indicate that they are now seen as coordinates on the same surface
g,n ).
The Riemann surface g,n is a point of M g,n . By varying the moduli parameters
of g 1 ,n 1 and g 2 ,n 2 , one obtains other surfaces in M g,n . But the number of
parameters furnished by g 1 ,n 1 and g 2 ,n 2 does not match the dimension (11.53)
of M g,n :
M g 1 ,n 1 + M g 2 ,n 2 = 6g 1 − 6 + 2n 1 + 6g 2 − 6 + 2n 2 = M g,n − 2.
(12.36)
This means that the subspace of M g,n obtained by gluing all the possible surfaces
in M g 1 ,n 1 and M g 2 ,n 2 is of codimension 2. The missing complex parameter is q: in
writing the plumbing fixture, it was taken to be fixed, but it can be varied to generate
a 2-parameter family of Riemann surfaces in M g,n , with the moduli of the original
surfaces held fixed.
The surface g,n is equipped with local coordinates inherited from the original
surfaces g 1 ,n 1 and g 2 ,n 2 . Hence, the plumbing fixture of points in P g 1 ,n 1 and
P g 2 ,n 2 automatically leads to a point of P g,n . The fact that the local coordinates
are inherited from lower-order surfaces is called gluing compatibility. It is also not
necessary to add parameters to describe the fibre direction.
12.3.2 Non-separating Case
In the previous section, the plumbing fixture was used to glue punctures on two
different surfaces. In fact, one can also glue two punctures on the same surface to
get a new surface with an additional handle:
g,n = # g 1 ,n 1 ,
g = g 1 + 1,
n = n 1 − 2,
(12.37)
defining # as a unary operator. This gluing is called non-separating because there is
a single surface before the identification of the disks.
In terms of the local coordinates, the gluing relation reads
w
(1)
n 1 −1 w
(1)
n 1
= q,
(12.38)
where we consider the last two punctures for definiteness.
The dimensions of both moduli spaces are related by
M g 1 ,n 1 = M g,n − 2.
(12.39)
261
where I is the inversion (the superscript on the coordinates z a and z b has been
removed to indicate that they are now seen as coordinates on the same surface
g,n ).
The Riemann surface g,n is a point of M g,n . By varying the moduli parameters
of g 1 ,n 1 and g 2 ,n 2 , one obtains other surfaces in M g,n . But the number of
parameters furnished by g 1 ,n 1 and g 2 ,n 2 does not match the dimension (11.53)
of M g,n :
M g 1 ,n 1 + M g 2 ,n 2 = 6g 1 − 6 + 2n 1 + 6g 2 − 6 + 2n 2 = M g,n − 2.
(12.36)
This means that the subspace of M g,n obtained by gluing all the possible surfaces
in M g 1 ,n 1 and M g 2 ,n 2 is of codimension 2. The missing complex parameter is q: in
writing the plumbing fixture, it was taken to be fixed, but it can be varied to generate
a 2-parameter family of Riemann surfaces in M g,n , with the moduli of the original
surfaces held fixed.
The surface g,n is equipped with local coordinates inherited from the original
surfaces g 1 ,n 1 and g 2 ,n 2 . Hence, the plumbing fixture of points in P g 1 ,n 1 and
P g 2 ,n 2 automatically leads to a point of P g,n . The fact that the local coordinates
are inherited from lower-order surfaces is called gluing compatibility. It is also not
necessary to add parameters to describe the fibre direction.
12.3.2 Non-separating Case
In the previous section, the plumbing fixture was used to glue punctures on two
different surfaces. In fact, one can also glue two punctures on the same surface to
get a new surface with an additional handle:
g,n = # g 1 ,n 1 ,
g = g 1 + 1,
n = n 1 − 2,
(12.37)
defining # as a unary operator. This gluing is called non-separating because there is
a single surface before the identification of the disks.
In terms of the local coordinates, the gluing relation reads
w
(1)
n 1 −1 w
(1)
n 1
= q,
(12.38)
where we consider the last two punctures for definiteness.
The dimensions of both moduli spaces are related by
M g 1 ,n 1 = M g,n − 2.
(12.39)