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、拟投射簇上的曲线、具有充足法丛的曲线。在代数几何中，用代数群来考虑簇的商是很常见的。最近，Keel和Mori证明了在相当一般的条件下，商作为代数空间存在。特别地，当稳定子族是有限的时，存在一个几何商。该项目的一个目标是推广这些成果，特别是考虑到积极的维度稳定剂。这将适用于模数问题，也适用于粗表示Artin-Stack。光滑射影簇的一个主要发明者是多元，即正则丛(切丛的顶部外幂的对偶)的各种张量幂的整体截面数。一个主要的猜想是，当多样性被投影线的像所覆盖时，多属全部消失。Mori程序的主要结果之一是在三维空间证明了这一点。开(即拟投射)簇也有多元的概念(所谓的对数多元)，Keel和McKernan猜想，当簇被仿射直线的像支配时，这些多元就消失了。基尔和麦科南最近在第二维度证明了这一点，这个项目的目标之一是考虑第三维度的情况。一个有趣的特例是光滑仿射的三重。哈肖恩猜想，如果一个光滑射影簇的光滑子簇有充足的法丛，那么在一个大维族中有一些多重运动。这通常被认为是假的，但对于曲线仍然是开放的。该项目的一个目标是考虑曲线为三折的情况。通过考虑沿曲线方向具有高重数的超曲面，得到了初步的结果，希望这些结果能够得到推广。

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.