Abelian Varieties: Proceedings of the International by Herbert Lange, Wolfgang Barth, Klaus Hulek PDF

By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Publication by way of Barth, Wolf, Hulek, Klaus

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Extra resources for Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993

Example text

Singular points or branch points arise as fixed points of elements of finite order in Γ ι ) η . 2 with non-trivial isotropy group Iso Z. 5 every element of Iso Ζ is conjugate to Jo, I3, 5,±1,T±1 with respect to Sp(4,Z). In particular every element of Iso Ζ contains 1 as an eigenvalue of multiplicity 2. It is shown in [G01, Proof of Lemma 2] that with respect to GL (4, C) the isotropy group is conjugate to a group of matrices °j) with unitary ( 2 x 2 ) - matrix U. 4 to see that the isotropy group is cyclic.

2 ) ) with coset representatives ]12 , (Q (resp. ll2 , (j ). The list C generates the symmetric group S3 isomorphic to the factor group SL(2, Ζ)/Γ (2). Proof. g. the congruence / 1 0 W 1 + 2e [ l 1 ) { c 2b \ ( 1 l+ 2 d ) ^ { l + c 0\ 1 J m o d 2 yields the claim for Γ (2) C Γι (2). 6. Suppose g is a non-trivial element of finite order in Γχ (2), then g is up to sign and conjugacy with respect to Γ (2) exactly the matrix Y = X'2YX~2 = (-1 "2) . Proof. The congruence trace (g) = 0 mod 2 implies that g has to be conjugate to ± y in SL(2,Z).

In particular Μ · Ho = Ho . Proof This is a consequence of [Br 2 , II. 12]. • For every isotropy group IsojF of a fixed variety Τ the normal subgroup Iso2 Τ ·,— < Μ G I s o Τ I M2 — fl4 > is generated by quasi-reflections. To determine the type of a singularity at a fixed point lying on Τ one has to divide out this normal subgroup, since by a well-known theorem of Chevalley the quotient of the action by a quasi-reflection is smooth (compare with [Br 2 , II. 2]). Corollary 4 . 1 0 . i) Every fixed variety intersecting the Humbert surface Ho in Αι,2 is already a subvariety of Ho • Especially Ho has an empty intersection with the union of three surfaces Hi L)H3 UH4 in AI,2 .

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Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 by Herbert Lange, Wolfgang Barth, Klaus Hulek


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