稳定曲面的有效界

The canonical models of surfaces of general type are called canonical surfaces and various kinds of boundedness are an important part of their classification theory. The notion of stable surfaces generalizes that of canonical surfaces and includes certain degenerations thereof. They are thus used to compactify the moduli spaces of canonical surfaces. Since more types of singularities are allowed, several important boundedness results for canonical surfaces do not generalize readily to stable surfaces. For a better understanding of stable surfaces and their classification we propose to study stable surfaces from the point of view of effective boundedness, concerning their geography, pluricanonical maps and concrete applications in compactifying moduli spaces.

一般型曲面的典范模型称为典范曲面，各种有效界是其分类理论中的重要组成部分。稳定曲面是典范曲面的推广并包括其退化，从而被用来紧化典范曲面的模空间。由于所允许的奇点类型更多了，关于典范曲面的几个重要的有效界结果无法直接推广到稳定曲面上。为了更好地理解稳定曲面及其分类，我们拟从有效界的角度研究稳定曲面的地理问题、多典范映射及在模空间紧化中的具体应用。

本项目研究了稳定曲面的各种有效界以及相关问题，并取得以下成果：(1) 完全解决了带边稳定曲面体积集的聚点问题；(2) 构造例子说明一个猜测的关于稳定曲面的诺特型不等式不成立；(3) 找到具有正几何亏格的带既约边界除子的稳定曲面的最小体积；(4) 构造了具有创纪录小体积 1/48983 的稳定曲面；(5) 证明具有半对数典范奇点的法诺簇是单连通的；(6) 验证广义极化对数典范曲面的不消失猜想；(7) 给出不规则性一般型曲面上数值平凡自同构群的最优界。