Rational curves on universal hypersurfaces.

摘要

In this thesis, results are presented toward determining the number and dimension of irreducible components of the space of rational curves on a universal hypersurface. First, the question is reduced to classifying the strata of rational curves in projective space that are defined by the isomorphism class of the restriction of certain vector bundles. Partial results are given for describing these strata. Among these is a conjecture that the general stratum should have balanced isomorphism class. The conjecture is proved in a large infinite family of cases; in the course of the proof, a notion of a balanced vector bundle is introduced for reducible curves. Finally, these results are translated to give partial results about the original question.;A concrete picture of a few rational curves on universal hypersurfaces is given in the supplementary file animations.zip.