arXiv math/06062892006-06-19math.AGmath-phmath.MP

关于K3曲面与其自身的对应IV

On correspondences of a K3 surface with itself. IV

C. G. Madonna, Viacheslav V. Nikulin

备注：: 14 pages; Var2: Exposition polished
发表信息：: Some results obtained here are part of:: C.G.Madonna and V.V.Nikulin, Explicit correspondences of a K3 surface with itself, Izvestiya: Mathematics 72 (2008), no. 3

摘要：

设$X$是一个K3曲面，其偏振度为$H^2=2rs$，$r,s\ge 1$，各向同性Mukai矢量$v=(r,H,s)$是本原的。具有各向同性Mukai向量$(r,H,s)$的$X$上的层的模空间也是一个K3曲面$Y$。在\cite{Nik2}中，第二作者用$X$的Picard格$N(X)$给出了$Y$与$X$同构的充要条件（在数学AG/0206158、数学AG/0304415和数学AG/0307355中也考虑了一些重要的特殊情况）。在这里，我们表明，这些条件意味着$Y$和$X$之间的同构的存在，这是一个组合的一些普遍的几何同构之间的模的层在$X$，和几何Tyurin的同构之间的模的层在$X$和$X$本身。因此，对于一般的K3曲面$X$，如果它含有$ρ(X)=\text{rk\}N(X)\le 2$和$Y\cong X$，则存在一个同构$Y\cong X$，它是几何泛同构和Tyurin同构的合成.这将我们最近的结果math.AG/0605362和math.AG/0606239推广到一般情况。

英文摘要：

Let $X$ be a K3 surface with a polarization $H$ of the degree $H^2=2rs$, $r,s\ge 1$, and the isotropic Mukai vector $v=(r,H,s)$ is primitive. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface, $Y$. In \cite{Nik2} the second author gave necessary and sufficient conditions in terms of Picard lattice $N(X)$ of $X$ when $Y$ is isomorphic to $X$ (some important particular cases were also considered in math.AG/0206158, math.AG/0304415 and math.AG/0307355). Here we show that these conditions imply existence of an isomorphism between $Y$ and $X$ which is a composition of some universal geometric isomorphisms between moduli of sheaves over $X$, and geometric Tyurin's isomorphsim between moduli of sheaves over $X$ and $X$ itself. It follows that for a general K3 surface $X$ with $ρ(X)=\text{rk\}N(X)\le 2$ and $Y\cong X$, there exists an isomorphism $Y\cong X$ which is a composition of the geometric universal and the Tyurin's isomorphisms. This generalizes our recent results math.AG/0605362 and math.AG/0606239 to a general case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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