论文标题
构建低成符的品种，并应用于一般类型的模量空间
Construction of varieties of low codimension with applications to moduli spaces of varieties of general type
论文作者
论文摘要
在本文中，我们开发了一种新的方式，可以系统地构建任何给定的尺寸$ m $，$ m \ geq 3 $的平滑次视角$ x $的家族，以及任何给定的codimension in $ \ mathbb p^n $，嵌入了完整的亚caronical线性线性系列，特别是在Hartshorne的范围内。我们通过显示无处不在的无折叠方案的存在来实现这一目标，称为绳索，嵌入在$ \ mathbb p^n $中，并通过使它们平滑。在$ 3 \ leq m <n/2 $的范围内，我们构建了平滑的亚变量，该次数由完整的子环线性系列嵌入，这些系列不是完整的交叉点。我们还超越了关于在投影空间中构建简单规范表面的综合问题，并在所有维度上构造简单的规范品种。这些简单的规范品种中许多无限的规范图是有限的，但不是嵌入。最后，我们展示了一般类型品种的模量空间的存在（在所有维度$ m $，$ m $，$ m \ geq 3 $）中，它们是相对于规范映射及其错误的行为的模量曲线的模量曲线的类似物。在许多情况下，这些组件的一般元素是规范嵌入的，它们的编纂在Hartshorne的猜想范围内。
In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions $m$, $m \geq 3$) that are analogues of the moduli space of curves of genus $g > 2$ with respect to the behavior of the canonical map and its deformations. In many cases, the general elements of these components are canonically embedded and their codimension is in the range of Hartshorne's conjecture.
