Some examples on moduli spaces of low degree rational curves on low degree complete intersections

This paper is to give some concrete examples of the general fibers of the evaluation map of some Kontsevich mapping spaces parametrize low degree rational curves on low degree complete intersection varieties. We prove these examples are Fano or Calabi-Yau vareities of low dimension, e.g, surface and threefold. In usual, the “moduli” fibers are constructed in an abstract way, but here we give a down-to-earth interpretation to some of these fibers from the perspective of projective geometry.