Mathematical Sciences: Groupoid Quotients, Rational Curves on Open Varieties, and Curves with Ample Normal Bundle

本课题涉及代数几何中的问题，可分为三类：群群商、拟射影变异上的曲线、有充足法线束的曲线。在代数几何中，用代数群来考虑一个变量的商是很常见的。最近，Keel和Mori证明了商在相当一般的条件下作为代数空间存在。特别地，只要稳定族是有限的，就存在一个几何商。该项目的一个目标是扩展这些结果，特别是考虑到正维度稳定剂。这将适用于模问题，也适用于粗略表示Artin-stacks。光滑投影变异体的一个主要不变项是复数，即正则束的各种张量幂（切束的顶外幂的对偶）的整体截面的个数。一个主要的猜想是，如果品种被投影线的图像覆盖，多属就会全部消失。Mori的程序的主要结果之一是在三维空间中证明了这一点。对于开放（即准投影）品种（即所谓的log plurigenera）也有一个多属的概念，Keel和McKernan推测，如果品种由仿射线的图像主导，这些多属就会消失。Keel和McKernan最近在二维中证明了这一点，这个项目的目标之一是考虑三维的情况。一个有趣的特殊情况是光滑仿射三倍。Harshorne推测，如果光滑射影簇的光滑子簇有充足的法线束，那么在一个大维族中就会有多次移动。这在一般情况下是错误的，但对于曲线仍然是开放的。该项目的一个目标是考虑三倍曲线的情况。通过考虑曲线上具有高多重性的超曲面，得到了一个初步的结果，并希望能推广这一结果。

Keel This project concerns questions in Algebraic Geometry which may be divided into three categories: Quotients by groupoids, Curves on quasi-projective varieties, curves with ample normal bundles. It is common in algebraic geometry to consider the quotient of a variety by an algebraic group. Recently Keel and Mori proved that a quotient exists as an algebraic space under quite eneral conditions. In particular there is a geometric quotient whenever the family of stabilizers is finite. One goal of the project is to extend these results, in particular to allow for positive dimensional stabilizers. This would have applications to moduli problems, and also to coarsely representing Artin-stacks. One of the principal invairants of smooth projective varieties are the plurigenera, the number of global sections of various tensor powers of the canonical bundle (the dual of the top exterior power of the tangent bundle). A main conjecture is that plurigenera all vanish iff the variety is covered by images of the projective line. One of the principal results of Mori's program is the proof of this in dimension three. There is a notion of plurigenera for open (i.e. quasi-projective) varieties as well (the so called log plurigenera), and it has been conjectured by Keel and McKernan that these vanish iff the variety is dominated by images of the affine line. Keel and McKernan recently proved this in dimension two, and one of the goals of this project is to consider the case of dimension three. An interesting special case is that of a smooth affine three-fold. Harshorne conjectured that if smooth subvariety of a smooth projective variety has ample normal bundle, then some multiple moves in a large dimensional family. This is known to be false in general, but remains open for curves. One goal of the project is to consider the case of a curve in a three-fold. A preliminary result has been obtained by considering hypersurfaces with high multiciplicity along the curve, and it is hoped that the results can be extended.