A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES
A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES
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DOI：
10.1017/s1474748015000328
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发表时间：
2013-10
影响因子：
0.9
通讯作者：
Benjamin Bakker
中科院分区：
文献类型：
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作者：
Benjamin Bakker
经典地，K3曲面$X$上有效曲线锥中的不可分解类$R$当且仅当$R^{2}=-2$可被光滑有理曲线表示。我们证明了Hassett和Tschinkel猜想的一个高维推广：对于等价于K3曲面上$n$点的Hilbert格式的全纯辛变量$M$变形，当且仅当满足某些相交理论条件时，Mori锥上的H_{2}(M,\mathbb{Z})$中的极值曲线类$R\是lagrange $n$ -平面$\mathbb{P}^{n}\子集M$中的直线。特别地，任何这样的类都满足$(R，R)=-\frac{n+3}{2}$，并且这样的基元类都包含在一个单态轨道中。
Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^{2}=-2$ . We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$ -plane $\mathbb{P}^{n}\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $(R,R)=-\frac{n+3}{2}$ , and the primitive such classes are all contained in a single monodromy orbit.