Let
π
:
S
→
B
\pi :S \to B
be an elliptic surface with a section
σ
:
B
→
S
\sigma :B \to S
. Let
L
−
1
→
B
{L^{ - 1}} \to B
be the normal bundle of
σ
(
B
)
\sigma (B)
in S, and let
W
=
P
(
L
⊗
2
⊕
L
⊗
3
⊕
1
)
W = P({L^{ \otimes 2}} \oplus {L^{ \otimes 3}} \oplus 1)
be a
P
2
{{\mathbf {P}}^2}
-bundle over B. Let
S
∗
{S^\ast }
be the surface obtained from S by contracting those components of fibres of S which do not intersect
σ
(
B
)
\sigma (B)
. Then
S
∗
{S^\ast }
may be imbedded in W and defined by a “Weierstrass equation":
\[
y
2
z
=
x
3
−
g
2
x
z
2
−
g
3
z
3
{y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}
\]
where
g
2
∈
H
0
(
B
,
O
(
L
⊗
4
)
)
{g_2} \in {H^0}(B,\mathcal {O}({L^{ \otimes 4}}))
and
g
3
∈
H
0
(
B
,
O
(
L
⊗
6
)
)
{g_3} \in {H^0}(B,\mathcal {O}({L^{ \otimes 6}}))
. The only singularities (if any) of
S
∗
{S^\ast }
are rational double points. The triples
(
L
,
g
2
,
g
3
)
(L,{g_2},{g_3})
form a set of invariants for elliptic surfaces with sections, and a complete set of invariants is given by
{
(
L
,
g
2
,
g
3
)
}
/
G
\{ (L,{g_2},{g_3})\} /G
where
G
≅
C
∗
×
Aut
(
B
)
G \cong {{\mathbf {C}}^\ast } \times {\operatorname {Aut}}\;(B)
.