| In this thesis we study the geometry of the space of rational curves on various projective varieties. These varieties include projective spaces, and smooth hypersurfaces contained within them. The parameter space we will use is the Kontsevich moduli space M 0,n(X, beta). In Chapter 2, we first study the space of conics on hypersurfaces without appealing to the bend and break Lemma. We then undertake a thorough study of rational degree e curves on Fermat hypersurfaces. In the bend and break range, we are able to use a detailed understanding of the space of lines on these hypersurfaces to prove that the moduli spaces are irreducible and have the expected dimension. In Chapter 3, we give an upper bound on the largest dimension of a complete family of linearly non-degenerate rational curves contained in projective space. This bound is an improvement over what the Bend and Break Lemma would imply. In Chapter 4, we study the property of strong rational simple connectedness as it relates to smooth cubic hypersurfaces. Using a naturally defined foliation on the moduli space of pointed lines together with a careful understanding of the variety of lines and planes on such a hypersurface, we are able to conclude that cubic hypersurfaces are strongly rationally simply connected in the best possible range. |
