Curvas de contato no espa\c{c}o projetivo

TL;DR

This paper studies contact curves in odd-dimensional projective spaces, constructing a parameter space for rational contact curves, deriving a general formula for their enumerative invariants, and computing explicit cases up to degree 4.

Contribution

It introduces a new algebraic stack framework for contact curves, derives a general formula for their enumerative invariants, and computes explicit cases up to degree 4.

Findings

Derived a general formula for the number of contact curves of degree d

Explicitly calculated contact curves counts for degrees 1 to 4

Confirmed known cases and introduced new enumerative invariants for cubics and quartics

Abstract

The odd dimensional projective space $\mathbb{P}^{2n-1}$ admits a contact structure arising from a non integrable distribution of hyperplanes determined by a symplectic form in $\mathbb{C}^{2n}$. Our object of interest is the set of rational curves of degree d which are tangent to that contact distribution in $\mathbb{P}^3$. Such curves are called contact curves or legendrian curves. To explore the geometry of contact curves, we construct the parameter space $\mathcal{L}_d$ using Kontsevich's stable maps, $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^3,d)$, endowed with the structure of algebraic stack. The intersection theory on stacks allows us to define in that space the virtual invariant $N_d$, associated with the number of degree $d$ contact curves incident to $2d+1$ lines. Using graph combinatorics and partitions originated from Bott's localization formula, we determine a general formula for $N_d$. We explicitly calculate it for contact curves up to degree 4 - confirming the known cases of contact lines and conics and introducing the new numbers for cubics and quartics. Finally, we discuss the enumerative significance of these invariants, still conjectural for $d>4$.