[Paper Review] Kontsevich spaces of rational curves on Fano hypersurfaces

Eric Riedl, David Yang|arXiv (Cornell University)|Sep 12, 2014

Algebraic Geometry and Number Theory27 references3 citations

TL;DR

This paper proves that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}}_{0,0}(X,e)$ of degree $e$ Kontsevich stable maps is an irreducible local complete intersection stack of the expected dimension $e(n-d+1)+n-4$. This resolves nearly all cases of a conjecture by Coskun, Harris, and Starr and confirms that Gromov-Witten invariants for these hypersurfaces are enumerative.

ABSTRACT

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.

Motivation & Objective

• To determine the dimension and geometric structure of the moduli space of rational curves on general Fano hypersurfaces.
• To resolve a conjecture by Coskun, Harris, and Starr on the expected dimension of the space $\overline{\mathcal{M}}_{0,0}(X,e)$ for Fano hypersurfaces.
• To establish that Gromov-Witten invariants for these hypersurfaces are enumerative by proving the moduli space has the expected dimension.
• To analyze the codimension of loci where rational curves fail to have the expected dimension, particularly in the Fano range $n \geq d+2$.
• To extend known results on rational curves to all degrees $e$ and all Fano hypersurfaces in the range $n \geq d+2$.

Proposed method

• Use of the Kontsevich space $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^n,e)$ as a smooth stack of stable maps from rational curves to $\mathbb{P}^n$.
• Interpret $\overline{\mathcal{M}}_{0,0}(X,e)$ as a substack cut out by a section of a rank $ed+1$ vector bundle over $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^n,e)$.
• Apply deformation theory and dimension-theoretic arguments to show that components of $\overline{\mathcal{M}}_{0,0}(X,e)$ have dimension at least $e(n-d+1)+n-4$.
• Use induction and codimension estimates on degenerate loci (e.g., non-injective maps or reducible curves) to prove that the expected dimension is achieved.
• Analyze the fiber dimensions of evaluation maps and use generality assumptions to show that certain loci have high codimension.
• Apply a key inequality involving $n$, $d$, and $e$ to bound the codimension of loci violating the expected dimension, proving irreducibility and local complete intersection structure.

Experimental results

Research questions

• RQ1 Does the space $\overline{\mathcal{M}}_{0,0}(X,e)$ of degree $e$ rational curves on a general Fano hypersurface $X \subset \mathbb{P}^n$ of degree $d$ have the expected dimension $e(n-d+1)+n-4$?
• RQ2 Is $\overline{\mathcal{M}}_{0,0}(X,e)$ irreducible and a local complete intersection stack for general $X$ when $n \geq d+2$?
• RQ3 Are the Gromov-Witten invariants of such hypersurfaces enumerative, i.e., do they count actual rational curves satisfying incidence conditions?
• RQ4 What is the codimension of the locus of non-injective or reducible rational curves in $\overline{\mathcal{M}}_{0,0}(X,e)$, and does it exceed the expected threshold?
• RQ5 Does the conjecture of Coskun, Harris, and Starr hold for all $e$ in the Fano range $n \geq d+2$, excluding the $(n,d)=(3,4)$ case?

Key findings

• For a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}}_{0,0}(X,e)$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.
• The codimension of the locus $S_e$ of non-injective or reducible maps is at least $\binom{n}{2} + d - 2en + \frac{e(e+1)}{2}(n-d+1) - e + 1$ for $e \leq \frac{n+1}{n-d+1}$, which is positive for $n \geq 6$.
• The space of rational curves of degree $e$ through a fixed point $p \in X$ has the expected dimension for all $e$ when $n \geq d+2$, as shown via codimension bounds and induction.
• The result resolves all but one case of the Coskun-Harris-Starr conjecture, specifically excluding the case $(n,d) = (3,4)$.
• The Gromov-Witten invariants of these hypersurfaces are enumerative because the moduli space has the expected dimension and is irreducible.
• The proof relies on showing that the fibers of evaluation maps have high codimension and that degenerate loci do not dominate the moduli space, ensuring the expected dimension is achieved.