Irreducible Contact Curves via Graph Stratification

TL;DR

This paper develops a graph-based stratification of the moduli space of contact stable maps to projective spaces, enabling the enumeration of irreducible rational contact curves with specific conditions, including explicit calculations for certain cases.

Contribution

It introduces a novel stratification approach for contact stable maps, facilitating the enumeration of contact curves with Schubert conditions and providing explicit invariants.

Findings

Determined the number of irreducible rational contact curves in projective spaces.

Provided explicit invariants for contact curves in P^3 and P^5.

Reproved the formula for plane contact curves in P^3.

Abstract

We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with any Schubert condition. We give explicitely some of these invariants for $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$. We give another proof of the formula for the number of plane contact curves in $\mathbb{P}^{3}$ meeting the appropriate number of lines.