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

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, ${\bar{\mathcal{M}}}_{0,0}({\mathbb{P}}^3,d)$, 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 $d>4$.