NIKULIN INVOLUTIONS ON K 3 SURFACES
NIKULIN INVOLUTIONS ON K 3 SURFACES
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K 3 表面上的 NIKULIN 对合
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发表时间：
2006
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通讯作者：
A. Sarti
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作者：
A. Sarti
研究了K3曲面上Nikulin（即辛）对合在上同调上诱导的映射.我们parametrize的11维不可约组件的模空间的代数K3表面与Nikulin对合，我们给出的例子一般K3表面的各种组件。最后，我们对MorrisonNikulin对合做了一些评论，这些是在Néron Severi群中交换E8（−1）的两个副本的Nikulin对合。在他的论文[Ni 1] Nikulin开始研究有限群的自同构对K3表面，特别是那些离开全纯两种形式不变，这些被称为辛。他证明了当群G是循环的且作用是辛的时，则G ≠ = Z/nZ，1 ≤ n ≤ 8。文[GS]研究了三阶、五阶和七阶K3曲面的辛自同构。这里我们考虑G = Z/2 Z的情况，由辛对合ι生成。这样的对合被称为Nikulin对合（参见。[Mo，定义5.1]）。在K3曲面X上的尼库林对合有8个不动点，因此商= X/i有8个节点，通过将它们爆破，可以得到K3曲面Y。在文件[莫]莫里森研究这种对合代数K3表面皮卡德数ρ ≥ 17，特别是对那些表面的Néron塞维里群包含两个副本的E8（−1）。这些K3曲面总是允许交换E8（−1）的两个副本的Nikulin对合。我们称这种对合为Morrison-Nikulin对合。莫里森的论文激励我们研究尼库林对合一般。在研究商映射诱导的上同调映射之后，在第二节中我们证明了具有Nikulin对合的代数K3曲面有ρ ≥ 9，并且Néron Severi群包含一个与E8（−2）同构的本原子格。此外，如果ρ = 9（最小可能），则以下两个命题是本文的中心结果：命题2.2。设X是一个具有Nikulin对合的K3曲面，并假设X的Néron Severi群NS（X）的秩为9。设L是E8（−2）<$$> NS（X）的生成元，且L = 2d > 0，设Λ2d：= ZL <$E8（−2）（<$NS（X））。然后我们可以假设L是充足的，并且：（1）在L ≠ 2 mod 4的情况下，我们有Λ2d = NS（X）;第二作者由DFG Research Grant SA 1380/1-1支持。2000年数学学科分类：14 J28、14 J10。
We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on MorrisonNikulin involutions, these are Nikulin involutions which interchange two copies of E8(−1) in the Néron Severi group. In his paper [Ni1] Nikulin started the study of finite groups of automorphisms on K3 surfaces, in particular those leaving the holomorphic two form invariant, these are called symplectic. He proves that when the group G is cyclic and acts symplectically, then G ∼= Z/nZ, 1 ≤ n ≤ 8. Symplectic automorphisms of K3 surfaces of orders three, five and seven are investigated in the paper [GS]. Here we consider the case of G ∼= Z/2Z, generated by a symplectic involution ι. Such involutions are called Nikulin involutions (cf.[Mo, Definition 5.1]). A Nikulin involution on the K3 surface X has eight fixed points, hence the quotient Ȳ = X/ι has eight nodes, by blowing them up one obtains a K3 surface Y . In the paper [Mo] Morrison studies such involutions on algebraic K3 surfaces with Picard number ρ ≥ 17 and in particular on those surfaces whose Néron Severi group contains two copies of E8(−1). These K3 surfaces always admit a Nikulin involution which interchanges the two copies of E8(−1). We call such involutions Morrison-Nikulin involutions. The paper of Morrison motivated us to investigate Nikulin involutions in general. After a study of the maps on the cohomology induced by the quotient map, in the second section we show that an algebraic K3 surface with a Nikulin involution has ρ ≥ 9 and that the Néron Severi group contains a primitive sublattice isomorphic with E8(−2). Moreover if ρ = 9 (the minimal possible) then the following two propositions are the central results in the paper: Proposition 2.2. Let X be a K3 surface with a Nikulin involution ι and assume that the Néron Severi group NS(X) of X has rank nine. Let L be a generator of E8(−2) ⊥ ⊂ NS(X) with L = 2d > 0 and let Λ2d := ZL⊕ E8(−2) (⊂ NS(X)). Then we may assume that L is ample and: (1) in case L ≡ 2 mod 4 we have Λ2d = NS(X); The second author is supported by DFG Research Grant SA 1380/1-1. 2000 Mathematics Subject Classification: 14J28, 14J10.