arXiv math/00052572000-05-25math.AG

Prym簇与一般型曲面的典范映射

Prym varieties and the canonical map of surfaces of general type

C. Ciliberto, R. Pardini, F. Tovena

备注：: 40 pages, LaTeX 2.09

摘要：

设X是一般类型的光滑复曲面，使得X的正则映射$ϕ$的像是曲面$Σ$，且$ϕ$有度$δ\geq 2$ .设$ε\colon S\to Σ$是$Σ$的去奇异化，并假设S的几何亏格不为零。Beauville证明了在这种情况下S是一般型的，并且$ε$是S的标准映射. Beauville还构建了唯一的无限系列的例子$ϕ:X\to Σ$与上述性质是已知的到现在为止。从他的构造出发，我们定义了一个{ \em好生成对}，即一个对$(h:V\to W, L)$，其中h是曲面的有限态射，L是满足一定假设的W的nef大线丛.我们表明，通过应用类似于Beauville的一个良好的生成对的建设，得到一个无限系列的一般类型的表面，其典型的映射是2到1到一个典型的嵌入表面。这样我们就可以构造出更多的无穷级数的这种曲面。此外，我们表明，良好的生成对有界不变量，并存在基本上只有2个例子与$\dim |L|>1$。关键的事实，我们利用获得这些结果是阿尔巴尼亚品种P的V是一个Prym品种和纤维的Prym地图P有积极的层面。

英文摘要：

Let X be a smooth complex surface of general type such that the image of the canonical map $ϕ$ of X is a surface $Σ$ and that $ϕ$ has degree $δ\geq 2$. Let $ε\colon S\to Σ$ be a desingularization of $Σ$ and assume that the geometric genus of S is not zero. Beauville has proved that in this case S is of general type and $ε$ is the canonical map of S. Beauville has also constructed the only infinite series of examples $ϕ:X\to Σ$ with the above properties that was known up to now. Starting from his construction, we define a {\em good generating pair}, namely a pair $(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big line bundle of W satisfying certain assumptions. We show that by applying a construction analogous to Beauville's to a good generating pair one obtains an infinite series of surfaces of general type whose canonical map is 2-to-1 onto a canonically embedded surface. In this way we are able to construct more infinite series of such surfaces. In addition, we show that good generating pairs have bounded invariants and that there exist essentially only 2 examples with $\dim |L|>1$. The key fact that we exploit for obtaining these results is that the Albanese variety P of V is a Prym variety and that the fibre of the Prym map over P has positive dimension.

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。

摘要：

英文摘要：

相关论文