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A NOTE ON MIRANDA-MORRISON THEORY

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A NOTE ON MIRANDA-MORRISON THEORY

ICHIRO SHIMADA

Let (X, f, s) be a complex elliptic K3 surface; that is, X is a complex K3 surface, f : X → P

1

is a fibration whose general fiber is a curve of genus 1, and s : P

1

→ X a section of f . As in [2], we consider the following objects:

• S

X

is the N´ eron-Severi lattice of X , embedded primitively into the even unimodular lattice H

2

(X, Z ) with the cup product.

• U

f

is the sublattice of S

X

generated by the class of a fiber of f and the class of the zero section s( P

1

). Thus U

f

is a hyperbolic plane.

• Φ

f

is the set of classes of smooth rational curves on X that are mapped by f to a point and disjoint from s( P

1

). It is well-known that Φ

f

is a fundamental system of roots of type ADE.

• L(Φ

f

) is the sublattice of S

X

generated by Φ

f

.

• M

f

is the primitive closure of L(Φ

f

) in S

X

. It is obvious that L(Φ

f

) is orthogonal to U

f

in S

X

, and hence the orthogonal direct sum U

f

⊕ M

f

is embedded primitively into S

X

.

• A

f

is the finite abelian group M

f

/L(Φ

f

). It is well-known that A

f

is isomorphic to the torsion part of the Mordell-Weil group of (X, f, s).

• T

f

is the orthogonal complement of U

f

⊕ M

f

in H

2

(X, Z ). Then we have an isomorphism q

Tf

∼ = − q

Mf

.

• O(T

f

) → O(q

Tf

) is the natural homomorphism from the orthogonal group of T

f

to the automorphism group of the discriminant form q

Tf

of T

f

.

• G

Tf

is the genus of lattices containing the isomorphism class of T

f

. Suppose that rank T

f

≥ 3; that is, rank L(Φ

f

) ≤ 17. Miranda-Morrison theory [1]

enables us to put a structure of the abelian group on G

Tf

, and to calculate a group M

Tf

that fits in the exact sequence

0 → Coker(O(T

f

) → O(q

Tf

)) → M

Tf

→ G

Tf

→ 0.

We have calculated this group M

Tf

for all elliptic K3 surface (X, f, s) by means of the computational tools developed in [2].

Theorem 1. Suppose that rank T

f

≥ 3. Then the group G

Tf

is trivial.

Theorem 2. The list in the following page is the list of all the cases where rank T

f

≥ 3 and M

Tf

is non-trivial.

Remark 3. In order to calculate the connected components of the moduli of elliptic K3 surfaces in [2], we have to take into account the positive sign structures of T

f

, and the action of automorphisms of q

Tf

coming from the automorphisms of the diagram Φ

f

via the isomorphism q

Tf

∼ = − q

Mf

. For this purpose, we presented a refinement of the Miranda-Morrison theory in Section 4.4 of [2]. The purpose of this note is to present a simple part of the calculation, for which we need not use the refinement of Miranda-Morrison theory.

1

(2)

2 ICHIRO SHIMADA

The contents of the following table are

rank L(Φ

f

) the ADE type of Φ

f

A

f

|M

Tf

| .

16 [ "D4", "D4", "A2", "A2", "A2", "A2" ] [ 1 ] 2 16 [ "A4", "A4", "A4", "A4" ] [ 1 ] 2

16 [ "A4", "A4", "A4", "A3", "A1" ] [ 1 ] 2 16 [ "A4", "A4", "A4", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "E8", "A4", "A4", "A1" ] [ 1 ] 2

17 [ "E7", "A4", "A4", "A1", "A1" ] [ 1 ] 2 17 [ "E6", "A7", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "E6", "A4", "A4", "A3" ] [ 1 ] 2 17 [ "E6", "A4", "A4", "A2", "A1" ] [ 1 ] 2 17 [ "D8", "A4", "A4", "A1" ] [ 1 ] 2 17 [ "D8", "A3", "A2", "A2", "A2" ] [ 1 ] 2 17 [ "D7", "D4", "A2", "A2", "A2" ] [ 1 ] 2 17 [ "D7", "A4", "A4", "A2" ] [ 1 ] 2 17 [ "D7", "A3", "A3", "A2", "A2" ] [ 1 ] 2 17 [ "D6", "A7", "A2", "A2" ] [ 1 ] 2 17 [ "D5", "D4", "A4", "A4" ] [ 1 ] 2 17 [ "D5", "A4", "A4", "A4" ] [ 1 ] 2 17 [ "D5", "A4", "A4", "A3", "A1" ] [ 1 ] 2 17 [ "D5", "A4", "A4", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "D5", "A3", "A3", "A2", "A2", "A2" ] [ 1 ] 4 17 [ "D4", "A7", "A2", "A2", "A1", "A1" ] [ 2 ] 2 17 [ "D4", "A4", "A4", "A2", "A2", "A1" ] [ 1 ] 2 17 [ "D4", "A3", "A3", "A3", "A2", "A2" ] [ 2 ] 2 17 [ "A9", "A4", "A4" ] [ 1 ] 2

17 [ "A9", "A4", "A3", "A1" ] [ 2 ] 2 17 [ "A9", "A4", "A3", "A1" ] [ 1 ] 2 17 [ "A9", "A4", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "A8", "A4", "A4", "A1" ] [ 1 ] 2 17 [ "A7", "A5", "A5" ] [ 1 ] 2

17 [ "A7", "A5", "A2", "A1", "A1", "A1" ] [ 2 ] 2 17 [ "A7", "A4", "A4", "A1", "A1" ] [ 1 ] 2 17 [ "A7", "A4", "A2", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "A7", "A3", "A3", "A2", "A2" ] [ 1 ] 2 17 [ "A6", "A6", "A2", "A2", "A1" ] [ 1 ] 2 17 [ "A6", "A4", "A4", "A3" ] [ 1 ] 2 17 [ "A6", "A4", "A4", "A2", "A1" ] [ 1 ] 2 17 [ "A5", "A5", "A3", "A2", "A1", "A1" ] [ 2 ] 2 17 [ "A5", "A4", "A4", "A4" ] [ 1 ] 2

17 [ "A5", "A4", "A4", "A3", "A1" ] [ 1 ] 2 17 [ "A5", "A4", "A4", "A2", "A2" ] [ 1 ] 2 17 [ "A5", "A4", "A4", "A2", "A1", "A1" ] [ 1 ] 2 17 [ "A5", "A3", "A3", "A3", "A2", "A1" ] [ 2 ] 2 17 [ "A4", "A4", "A4", "A4", "A1" ] [ 5 ] 2 17 [ "A4", "A4", "A4", "A3", "A1", "A1" ] [ 1 ] 2 17 [ "A4", "A4", "A4", "A2", "A1", "A1", "A1" ] [ 1 ] 2 17 [ "A4", "A4", "A3", "A2", "A2", "A2" ] [ 1 ] 2 17 [ "A4", "A3", "A3", "A3", "A2", "A2" ] [ 1 ] 2

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A NOTE ON MIRANDA-MORRISON THEORY 3

Acknowledgement. Thanks are due to Professor Klaus Hulek and Professor Michael L¨ onne for their interests in my work.

References

[1] Rick Miranda and David R. Morrison. Embeddings of integral quadratic forms. electronic, 2009, http://web.math.ucsb.edu/ drm/manuscripts/eiqf.pdf .

[2] Ichiro Shimada. Connected components of the moduli of elliptic K3 surfaces, 2016.

arXiv:1610.04706

Department of Mathematics, Graduate School of Science, Hiroshima University, 1- 3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 JAPAN

E-mail address:[email protected]

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