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

龙骨 本项目涉及代数几何中的问题，可分为三类：群胚的商，拟投射簇上的曲线，具有丰富法丛的曲线。 在代数几何中，通常考虑一个代数群与一个簇的商。 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。 特别是有一个几何商时，家庭的稳定剂是有限的。 该项目的一个目标是扩展这些结果，特别是允许积极的尺寸稳定器。 这将有应用程序的模数问题，也粗略地表示艺术堆栈。 光滑投射簇的主要不变量之一是plurigenes，即正则丛（切丛的顶外幂的对偶）的各种张量幂的整体截面的数目。 一个主要的猜想是，plurigenes都消失，当且仅当该品种是由图像的投影线。 其中一个主要的结果森的计划是证明这一点的三维。 有一个概念的plurigenes开放（即准投射）品种以及（所谓的日志plurigenes），它已被基尔和麦克南，这些消失，当该品种是占主导地位的形象的仿射线。 Keel和McKernan最近在二维中证明了这一点，这个项目的目标之一就是考虑三维的情况。 一个有趣的特例是光滑仿射三重。 Harshorne证明了如果光滑射影簇的光滑子簇有充足的法丛，则在一个大维族中有多重移动。 这通常是错误的，但对于曲线仍然是开放的。 该项目的一个目标是考虑三重曲线的情况。 通过考虑沿曲线沿着具有高重数的超曲面，得到了一个初步的结果，并希望该结果能得到推广。

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.