6.4 Modular Operads
137
Example 6.25 Property (6.59) is the abstraction of the stability of complex curves.
A stable curve with marked points is a connected complex projective curve P whose
only singularities are ordinary double points (nodal singularities), together with a
“marking” given by an embedding of a set S into the set of smooth points of P . The
stability means that there are no infinitesimal automorphisms of P fixing the marked
and double points. Equivalently, each smooth component of P isomorphic to the
complex projective space CP
1 has at least three special points and each smooth
component isomorphic to the torus has at least one special point, where a special
point is either a double point or a marked point.
The dual graph Δ = Δ(P ) of a stable curve P is a labeled graph whose vertices
are the components of P , edges are the nodes and its legs are the elements of S.
An edge e y corresponding to a nodal point y joins the vertices corresponding to the
components intersecting at y. The vertex v K corresponding to a branch K is labeled
by the genus of the normalization of K. The construction of Δ(P ) from a curve P
is visualized in Fig. 6.5 taken from [12].
Let us denote by M (S; g) the coarse moduli space [5, p. 347] of marked curves
P whose dual graph Δ(P ) has genus g. Obviously,
M =
M (S; g) | (S, g) ∈ Cor × N
is a modular module in the category of projective varieties. Since there are no stable
curves of genus g if 2(g − 1) + card(S) ≤ 0, M automatically satisfies the stability
condition (6.59).
For stable curves P 1 ∈ M (S 1 {a}; g 1 ) and P 2 ∈ M (S 2 {b}; g 2 ) we define
P 1 a ◦ b P 2 ∈ M (S 1 S 2 ; g 1 + g 2 )
to be the curve obtained by the identification of the point a ∈ P 1 with b ∈ P 2
introducing a nodal singularity. The contraction ◦ uv (P ) ∈ M (S; g + 1) of a curve
P ∈ M (S {u, v}; g) is defined similarly. With these operations, M is a modular
operad in the category of complex projective varieties, with A = N and s = 1.
x 0
x 2
x 1
Dual graph Δ (P):
•
a 4
a 3
a 2
a 1
a 5
•
•
•
•
x 2
x 1
x 0
Curve P:
A 5
A 4
A 3
A 2
A 1
•
•
•
•
•
•
•
•
•
Fig. 6.5 A stable curve and its dual graph. The curve P on the left has five components, A 1 , A 2 ,
A 3 , A 4 , and A 5 , and points marked by S = {x 0 , x 1 , x 2 ). The dual graph Δ(P ) on the right has five
vertices a 1 , a 2 , a 3 , a 4 , and a 5 corresponding to the components of the curve and three legs labeled
by the marked points
137
Example 6.25 Property (6.59) is the abstraction of the stability of complex curves.
A stable curve with marked points is a connected complex projective curve P whose
only singularities are ordinary double points (nodal singularities), together with a
“marking” given by an embedding of a set S into the set of smooth points of P . The
stability means that there are no infinitesimal automorphisms of P fixing the marked
and double points. Equivalently, each smooth component of P isomorphic to the
complex projective space CP
1 has at least three special points and each smooth
component isomorphic to the torus has at least one special point, where a special
point is either a double point or a marked point.
The dual graph Δ = Δ(P ) of a stable curve P is a labeled graph whose vertices
are the components of P , edges are the nodes and its legs are the elements of S.
An edge e y corresponding to a nodal point y joins the vertices corresponding to the
components intersecting at y. The vertex v K corresponding to a branch K is labeled
by the genus of the normalization of K. The construction of Δ(P ) from a curve P
is visualized in Fig. 6.5 taken from [12].
Let us denote by M (S; g) the coarse moduli space [5, p. 347] of marked curves
P whose dual graph Δ(P ) has genus g. Obviously,
M =
M (S; g) | (S, g) ∈ Cor × N
is a modular module in the category of projective varieties. Since there are no stable
curves of genus g if 2(g − 1) + card(S) ≤ 0, M automatically satisfies the stability
condition (6.59).
For stable curves P 1 ∈ M (S 1 {a}; g 1 ) and P 2 ∈ M (S 2 {b}; g 2 ) we define
P 1 a ◦ b P 2 ∈ M (S 1 S 2 ; g 1 + g 2 )
to be the curve obtained by the identification of the point a ∈ P 1 with b ∈ P 2
introducing a nodal singularity. The contraction ◦ uv (P ) ∈ M (S; g + 1) of a curve
P ∈ M (S {u, v}; g) is defined similarly. With these operations, M is a modular
operad in the category of complex projective varieties, with A = N and s = 1.
x 0
x 2
x 1
Dual graph Δ (P):
•
a 4
a 3
a 2
a 1
a 5
•
•
•
•
x 2
x 1
x 0
Curve P:
A 5
A 4
A 3
A 2
A 1
•
•
•
•
•
•
•
•
•
Fig. 6.5 A stable curve and its dual graph. The curve P on the left has five components, A 1 , A 2 ,
A 3 , A 4 , and A 5 , and points marked by S = {x 0 , x 1 , x 2 ). The dual graph Δ(P ) on the right has five
vertices a 1 , a 2 , a 3 , a 4 , and a 5 corresponding to the components of the curve and three legs labeled
by the marked points