Rational curves on hypersurfaces of low degree

TL;DR

This paper proves that for general hypersurfaces of low degree in projective space, all genus 0 Kontsevich moduli spaces are irreducible, reduced, and have the expected dimension, advancing understanding of rational curves on such hypersurfaces.

Contribution

It establishes irreducibility and expected dimension properties of genus 0 Kontsevich moduli spaces on low-degree hypersurfaces, a new result in algebraic geometry.

Findings

All genus 0 Kontsevich moduli spaces are irreducible.

These moduli spaces are reduced and local complete intersections.

They have the expected dimension.

Abstract

Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.