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Canonical class of the rational elliptic surface
(
K
E
(
1
)
=
−
F
)
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@codex,
0
)
...
Integrable almost complex structure
Complex manifold
Blowup of a complex manifold at a point
Pencil of plane cubics
Rational elliptic surface
Elliptic fibration of the rational elliptic surface
2026-09-24
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Writing
H
for the
pullback
of
a
line and
E
1
,
…
,
E
9
for the exceptional classes,
a
fiber has class
F
=
3
H
−
∑
i
E
i
, while the blowup
formula
gives
K
E
(
1
)
=
−
3
H
+
∑
i
E
i
=
−
F
.
(1)
Table of contents
Genus formula for a multisection of the rational elliptic surface
Canonical class of the rational elliptic surface
Genus formula for a multisection of the rational elliptic surface
0
0
0
Canonical class of the rational elliptic surface
If
a
smooth connected complex
curve
C
⊂
E
(
1
)
satisfies
C
2
=
k
and has degree
d
=
C
⋅
F
over the base, the
Adjunction formula
gives
2
g
(
C
)
−
2
=
C
2
+
K
E
(
1
)
⋅
C
=
k
−
d
,
g
(
C
)
=
1
+
2
k
−
d
.
(1)
Ancestors
(13)
Elliptic fibration of the rational elliptic surface
Rational elliptic surface
Pencil of plane cubics
Blowup of a complex manifold at a point
Complex manifold
Integrable almost complex structure
Almost complex manifold
Complex structure
Complex geometry
Geometry and topology
Area of mathematics
Mathematics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2024
/
iii
/
Paper 146
/
4
/
c
/
Solution
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