[Paper Review] Rational curves on hypersurfaces of low degree

Joe Harris, Mike Roth+1 more|ArXiv.org|Mar 8, 2002

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes that for a general complex hypersurface $X \subset \mathbb{P}^n$ of degree $d < \frac{n+1}{2}$ and $n > 2$, the scheme $R_e(X)$ parametrizing smooth rational curves of degree $e$ is an integral, local complete intersection scheme of expected dimension $(n+1-d)e + (n-4)$. The proof combines deformation theory, the Kontsevich moduli space of stable maps, and a combinatorial analysis of boundary components via a generalized bend-and-break lemma.

ABSTRACT

Let n &gt; 2 and let d &lt; (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.

Motivation & Objective

• To determine the geometric structure of the scheme $R_e(X)$ parametrizing smooth rational curves of degree $e$ on a general hypersurface $X \subset \mathbb{P}^n$ of low degree.
• To establish that $R_e(X)$ is an integral, local complete intersection scheme of the expected dimension $(n+1-d)e + (n-4)$ for $d < \frac{n+1}{2}$.
• To extend techniques from stable map moduli spaces and deformation theory to analyze the irreducibility and dimension of $R_e(X)$.

Proposed method

• Embed $R_e(X)$ as an open subscheme in the Kontsevich moduli space $\overline{\mathcal{M}}_{0,0}(X,e)$, which provides a modular compactification.
• Use the Behrend-Manin decomposition to stratify $\overline{\mathcal{M}}_{0,0}(X,e)$ into locally closed subsets indexed by stable $A$-graphs.
• Prove that the evaluation morphism $\overline{\mathcal{M}}_{0,1}(X,e) \to X$ is flat and generically unobstructed, ensuring expected dimension and smoothness.
• Apply a version of Mori’s bend-and-break lemma to show every irreducible component of $\overline{\mathcal{M}}_{0,0}(X,e)$ arises from a basic component parametrizing maps with domain curves mapping to lines in $X$.
• Use a combinatorial equivalence relation on basic components to prove all such components yield the same irreducible component in $\overline{\mathcal{M}}_{0,0}(X,e)$, hence irreducibility.
• Leverage flatness results for the incidence correspondence of pointed lines in $X$ to show that basic components are integral and unibranch with expected dimension.

Experimental results

Research questions

• RQ1 Under what conditions is the scheme $R_e(X)$ parametrizing smooth rational curves on a hypersurface $X \subset \mathbb{P}^n$ of degree $d$ an integral, local complete intersection scheme of expected dimension?
• RQ2 How does the structure of the Kontsevich moduli space $\overline{\mathcal{M}}_{0,0}(X,e)$ reflect the geometry of rational curves on low-degree hypersurfaces?
• RQ3 Can the irreducibility of $\overline{\mathcal{M}}_{0,0}(X,e)$ be established via a combinatorial analysis of stable $A$-graphs and their contractions?
• RQ4 What conditions ensure that the evaluation morphism $\overline{\mathcal{M}}_{0,1}(X,e) \to X$ is flat and generically unobstructed for low-degree hypersurfaces?

Key findings

• For $n > 2$ and $d < \frac{n+1}{2}$, the scheme $R_e(X)$ is an integral, local complete intersection scheme of dimension $(n+1-d)e + (n-4)$.
• The Kontsevich moduli space $\overline{\mathcal{M}}_{0,0}(X,e)$ is irreducible and of the expected dimension for such hypersurfaces.
• The evaluation morphism $\overline{\mathcal{M}}_{0,1}(X,e) \to X$ is flat and generically unobstructed, ensuring the expected dimension and smoothness of the moduli space.
• The incidence correspondence $F_{0,1}(X)$ of pointed lines in $X$ is flat over $X$ of relative dimension $n-d-1$, which implies that basic components of the moduli space are integral and unibranch.
• All irreducible components of $\overline{\mathcal{M}}_{0,0}(X,e)$ arise from a single equivalence class of basic components under a combinatorial equivalence relation, proving irreducibility.
• The moduli space $\overline{\mathcal{M}}_{0,0}(X,e)$ is a reduced, local complete intersection stack of the expected dimension, and $R_e(X)$ is its unique dense stratum.