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

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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.