不规则代数簇的二典范映射

对代数簇分类以及研究它们的几何性质是代数几何的重要任务。不规则代数簇是特殊而且重要的一类代数簇。本项目拟研究具有极大Albanese 维数的一般型代数簇的二典范映射。对于一般型本原不规则代数簇，最近Barja 等人证明：如果满足q(X)>dim(X)并且它的二典范映射不是双有理的，那么它双有理等价于一个简单的阿贝尔簇的Theta 除子。本项目将研究q(X)=dim（X）的本原簇的二典范映射，此问题在曲面情形已经完全清楚。本项目还将研究非本原不规则代数簇的二典范映射，类比曲面的结果，我们希望能在具有非双有二典范映射的代数簇上得到一些特殊的纤维化。

Claissifying algebraic varieties and studying their geometry are two of the main tasks in algebraic geometry. We intend to study the bicanonical maps of the varieties of maximal Albanese dimension and of general type. Assuming in addition that X is primitive, Barja etc. recently proved that: when q(X)>dim(X) and its bicanonical map is non-birational, then it is birational to a Theta divisor of a simple Abelian variety. We intend to study the case when q(X)=dim(X), and when X is a surface, it is completely clear now. What''s more, we will also try to study the bicanonical maps of non-primitive varieties, and hope to get some special fibrations on those with non-birational bicanonical maps.