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

TL;DR

This paper provides concrete examples of the fibers of the evaluation map in Kontsevich spaces for low degree rational curves on low degree complete intersections, revealing they are Fano or Calabi-Yau varieties with explicit geometric interpretations.

Contribution

It offers explicit, geometric descriptions of moduli fibers for rational curves on complete intersections, bridging abstract moduli theory with concrete projective geometry.

Findings

Fibers are Fano or Calabi-Yau varieties.

Explicit geometric interpretations of moduli fibers.

Examples include surfaces and threefolds.

Abstract

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.