7.0 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Elliptic surface | 2/2 | https://en.wikipedia.org/wiki/Elliptic_surface | reference | science, encyclopedia | 2026-05-05T09:08:24.692093+00:00 | kb-cron |
Here the multiple fibers of f (if any) are written as
f
∗
(
p
i
)
=
m
i
D
i
{\displaystyle f^{*}(p_{i})=m_{i}D_{i}}
, for an integer mi at least 2 and a divisor Di whose coefficients have greatest common divisor equal to 1, and L is some line bundle on the smooth curve S. If S is projective (or equivalently, compact), then the degree of L is determined by the holomorphic Euler characteristics of X and S: deg(L) = χ(X,OX) − 2χ(S,OS). The canonical bundle formula implies that KX is Q-linearly equivalent to the pullback of some Q-divisor on S; it is essential here that the elliptic surface X → S is minimal. Building on work of Kenji Ueno, Takao Fujita (1986) gave a useful variant of the canonical bundle formula, showing how KX depends on the variation of the smooth fibers. Namely, there is a Q-linear equivalence
K
X
∼
Q
f
∗
(
K
S
+
B
S
+
M
S
)
,
{\displaystyle K_{X}\sim _{\bf {Q}}f^{*}(K_{S}+B_{S}+M_{S}),}
where the discriminant divisor BS is an explicit effective Q-divisor on S associated to the singular fibers of f, and the moduli divisor MS is
(
1
/
12
)
j
∗
O
(
1
)
{\displaystyle (1/12)j^{*}O(1)}
, where j: S → P1 is the function giving the j-invariant of the smooth fibers. (Thus MS is a Q-linear equivalence class of Q-divisors, using the identification between the divisor class group Cl(S) and the Picard group Pic(S).) In particular, for S projective, the moduli divisor MS has nonnegative degree, and it has degree zero if and only if the elliptic surface is isotrivial, meaning that all the smooth fibers are isomorphic. The discriminant divisor in Fujita's formula is defined by
B
S
=
∑
p
∈
S
(
1
−
c
(
p
)
)
[
p
]
{\displaystyle B_{S}=\sum _{p\in S}(1-c(p))[p]}
, where c(p) is the log canonical threshold
lct
(
X
,
f
∗
(
p
)
)
{\displaystyle {\text{lct}}(X,f^{*}(p))}
. This is an explicit rational number between 0 and 1, depending on the type of singular fiber. Explicitly, the lct is 1 for a smooth fiber or type
I
ν
{\displaystyle I_{\nu }}
, and it is 1/m for a multiple fiber
m
I
ν
{\displaystyle {}_{m}I_{\nu }}
, 1/2 for
I
ν
∗
{\displaystyle I_{\nu }^{*}}
, 5/6 for II, 3/4 for III, 2/3 for IV, 1/3 for IV*, 1/4 for III*, and 1/6 for II*. The canonical bundle formula (in Fujita's form) has been generalized by Yujiro Kawamata and others to families of Calabi–Yau varieties of any dimension.
== Logarithmic transformations ==
A logarithmic transformation (of order m with center p) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a point p of the base space into a fiber of multiplicity m. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers. Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces. Example: Let L be the lattice Z+iZ of C, and let E be the elliptic curve C/L. Then the projection map from E×C to C is an elliptic fibration. We will show how to replace the fiber over 0 with a fiber of multiplicity 2. There is an automorphism of E×C of order 2 that maps (c,s) to (c+1/2, −s). We let X be the quotient of E×C by this group action. We make X into a fiber space over C by mapping (c,s) to s2. We construct an isomorphism from X minus the fiber over 0 to E×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s2). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration X is certainly not isomorphic to the fibration E×C over all of C.) Then the fibration X has a fiber of multiplicity 2 over 0, and otherwise looks like E×C. We say that X is obtained by applying a logarithmic transformation of order 2 to E×C with center 0.
== See also == Enriques–Kodaira classification Néron minimal model
== Notes ==
== References == Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004) [1984], Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 4, Springer, doi:10.1007/978-3-642-57739-0, ISBN 978-3-540-00832-3, MR 2030225 Cossec, François; Dolgachev, Igor (1989). Enriques Surfaces. Boston: Birkhäuser. doi:10.1007/978-1-4612-3696-2. ISBN 3-7643-3417-7. MR 0986969. Kodaira, Kunihiko (1963). "On compact analytic surfaces. II". Ann. of Math. 77 (3): 563–626. doi:10.2307/1970131. JSTOR 1970131. MR 0184257. Zbl 0118.15802. Kodaira, Kunihiko (1964). "On the structure of compact complex analytic surfaces. I". Am. J. Math. 86 (4): 751–798. doi:10.2307/2373157. JSTOR 2373157. MR 0187255. Zbl 0137.17501. Kollár, János (2007), "Kodaira's canonical bundle formula and adjunction", Flips for 3-folds and 4-folds, Oxford University Press, pp. 134–162, doi:10.1093/acprof:oso/9780198570615.003.0008, ISBN 978-0-19-857061-5, MR 2359346 Néron, André (1964). "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux". Publications Mathématiques de l'IHÉS (in French). 21: 5–128. doi:10.1007/BF02684271. MR 0179172. Zbl 0132.41403.